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🗄️

Notes

NameCreatedTags
Unit 6: Market Failure and the Role of GovernmentIn Class
Unit 5: Factor Markets
Unit 4: Imperfect Competition
Unit 3: Production, Cost, and the Perfect Competition ModelIn Class
Unit 2: Supply and DemandIn Class
Unit 1: Basic Economic ConceptsIn Class
\ No newline at end of file diff --git a/notes/AP Microeconomics/Notes 6f2083c54c4f4d288e489d5fea9b8c76/Unit 1 Basic Economic Concepts 7c9752a346774dfe9420f6b61b92c4a2.html b/notes/AP Microeconomics/Notes 6f2083c54c4f4d288e489d5fea9b8c76/Unit 1 Basic Economic Concepts 7c9752a346774dfe9420f6b61b92c4a2.html new file mode 100644 index 0000000..92edacf --- /dev/null +++ b/notes/AP Microeconomics/Notes 6f2083c54c4f4d288e489d5fea9b8c76/Unit 1 Basic Economic Concepts 7c9752a346774dfe9420f6b61b92c4a2.html @@ -0,0 +1,671 @@ +Unit 1: Basic Economic Concepts

Unit 1: Basic Economic Concepts

Created
TagsIn Class

The basic principle of economics is that we have unlimited wants & needs, but a limited supply of resources. This concept is known as scarcity.

What is Economics?

  • Economics is the science of scarcity.
  • Scarcity means that we have unlimited wants but limited resources.
  • Since we are unable to have everything we desire, we must make choices on how we will use our resources.
  • Economics is the study of these choices.
  • In Microeconomics, we will primarily study the choices of individuals and firms.
    • Study of small economic markets such as individuals, firms, and industries (ex: supply and demand in specific markets, production costs, labor markets, etc.)

How is Economics used?

  • Economists use the scientific method to make generalizations and abstractions to develop theories. This is called theoretical economics.
  • These theories are then applied to fix problems or meet economic goals. This is called policy economics.

Positive vs. Normative

  • Positive Statements: Basted on facts. Avoids value judgements (what is).
  • Normative Statements: Includes value judgements (what ought to be).

5 Key Economic Assumptions

  1. Society's wants are unlimited, but ALL resources are limited (scarcity).
  1. Due to scarcity, choices must be made. Every choice has a cost (a trade-off).
  1. Everyone's goal is to maximize their satisfaction. Everyone acts in their own "self-interest."
  1. Everyone makes decisions by comparing the marginal costs and marginal benefits of every choice.
  1. Real-life situations can be modeled through simple graphs.

Marginal Analysis

  • The term marginal means additional, or one more.
  • "Thinking on the margin," or marginal analysis involves making decisions based on the additional benefit vs. the additional cost.
  • The marginal analysis approach to decision making is more commonly used than the "all or nothing" approach.
  • You will continue to do something until the marginal cost outweighs the marginal benefit.

Trade-offs and Opportunity Cost

  • Trade-offs are all the alternatives that we give up whenever we choose one course of action over others.
  • The most desirable alternative given up as a result of a decision is known as the opportunity cost.

The Four Factors of Production

  • Land
  • Labor
  • Capital
  • Entrepreneurship

These four factors of production are all SCARCE resources.

Economic Systems

An economic system is an organized method to provide for the wants & needs of a society.

Every economic system must answer three basic economic questions:

  1. What to produce?
  1. How to produce?
  1. For whom to produce?

The Production Possibilities Curve (PPC)

  • An economic model for explaining the concept of opportunity cost
  • A production possibilities curve is a model that shows alternative ways that an economy can use its scarce resources.
  • This model graphically demonstrates scarcity, trade-offs, opportunity costs, and efficiency.

4 Key Assumptions

  • Only two goods can be produced
  • Full employment of resources
  • Fixed Resources
  • Fixed Technology

Any point which lies on the graph is the most efficient, while any point inside the curve does not make the most efficient use of all available resources.

Law of Increasing Opportunity Cost

  • As you produce more of any good, the opportunity cost will increase
  • Resources are NOT easily adaptable to producing both goods
  • Result is a bowed out (Concave) PPC

Per Unit Opportunity Cost

How much each marginal unit costs=Opportunity costUnits gained\textrm{How much each marginal unit costs} = \frac{\textrm{Opportunity cost}}{\textrm{Units gained}}

Two Types of Efficiency

Productive Efficiency

  • Products are being produced in the least costly way
  • This is any point ON the PPC

Allocative Efficiency

  • The products being produced are the ones most desired by society
  • The optimal point on the PPC depends on the desires of society

Shifting the PPC

  1. Change in resource quantity or quality
  1. Technology improves
  1. More education or training

Comparative Advantage

While the absolute advantage merely compares the efficiency of production, comparative advantage compares the opportunity costs of production.

Output Method

OOO - Output: Other goes Over

Input Method

IOU - Input: Other goes Under

Marginal Utility & Marginal Cost

The formula to find utility maximization is:

MUxPx=MUyPy\frac{MU_x}{P_x} = \frac{MU_y}{P_y}

Factor & Product Markets

Factor Market

  • Factors of production ⇒ money
  • Households sell
  • Firms buy

Product Market

  • Money ⇒ Products / Goods and Services
  • Firms sell
  • Households buy
The circular flow diagram of the Factor and Product markets, representing the circular flow of exchange.
💡
This diagram represents the exchange of goods and services in a pure market economy.

Government Involvement

The government in a mixed-market economy acts as an intermediary between many aspects of the Factor and Product markets.

Financial Institutions & Government

No, just no.

+

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Unit 2: Supply and Demand

Created
TagsIn Class

2.1 - Key Terms

Quantity Demanded

Quantity demanded is the amount of a good demanded by buyers at a given price level.

Buyers who are:

  1. Willing to buy
  1. Able to buy

Combine to make the Effective Demand.

Quantity demanded is the effective demand at a given price level. Quantity demanded has an inverse relationship with price.

💡
Price moves Quantity Demanded. Price is the independent variable in this equation, and Quantity Demanded is the dependent one. Consumers are reactive to changes in price.

Paradox of Demand

Under the paradox of demand, some goods do not have an inverse relationship between price and quantity demanded, but a parallel relationship. Goods are bought at higher prices in order to impress others, for things like luxury goods, therefore demand increases as price increases.

There are two causes for the paradox of demand, either conspicious consumption (buying goods to impress others), or Giffin goods (inferior goods which have no close substitutes).

Demand Schedule

A table showing how much of a good or service consumers will want to buy at different prices.

Graphing Price vs. Quantity Demanded

Price goes on the y-axis, quantity demanded goes on the x-axis.

This is known as a demand curve.

When there is an increase in demand, the demand curve shifts right (outward), with D0D_0 being the original curve and D1D_1 being the new curve. The opposite is also true, when there is a decrease in demand, the demand curve shifts left (inward).

Therefore, when total demand increases, the quantity demanded will increase at ALL POSSIBLE price levels. This means that something other than price has caused consumer demand to increase. The same is, of course, possible for a decrease in demand.

Causes of Change in Demand

Something other than price changes:

  • I: Income - Consumer Income Changes
    • With an increase in Consumer Income: Curve shifts right for Normal Goods (Fettuccini Alfredo) & Left for Inferior Goods (Ramen)
    • With a decrease in Consumer Income: Curve shifts left for Normal Goods & right for Inferior Goods
  • N: Number - Change in the Number of Consumers
    • Increase in number of buyers: Curve shifts right
    • Decrease in number of buyers: Curve shifts left
  • S: Substitute - Change in the Price of Substitute Goods
    • Increase in the price of substitutes: Curve shifts right
    • Decrease in the price of substitutes: Curve shifts left
  • E: Expectations - Change in expectations of future price
    • Expect a rise in price: Curve shifts right
      • Gonna buy up before price goes up
    • Expect a decrease in price: Curve shifts left
      • Not gonna buy until price decreases
  • C: Complement - Change in the price of complement goods
    • Two goods that go together will move in opposite directions as the price of the other changes.
  • T: Taste - Change in consumer taste and preferences
    • Increase in Taste or Preference for the item: Curve moves Right
    • Decrease in Taste or Preference for the item: Curve moves Left

Income Effect

The income effect is the change in the consumption of goods by consumers based on their income. When the price of a good falls, consumers experience an increase in purchasing power from a given income level. When the price of a good increases, consumers experience a decrease in purchasing power.

Normal goods have a positive income effect, that is, more of the good is consumed as the income of it's consumers increases. Inferior goods have a negative income effect, as consumers' income increases, less of the good is purchased.

Substitution Effect

The substitution effect happens when consumers replace cheaper items with more expensive ones when their financial conditions change. When the price of a good falls, consumers will substitute toward that good and away from other goods.

2.2 - Supply

Law of Supply

  • The quantity supplied of any good is the amount that sellers are willing and able to sell.
  • The Law of Supply states that the quantity supplied of a good rises when the price of the good rises, other things being equal.

The Supply Schedule

  • A supply schedule is a table that shows the relationship between the price of a good and the quantity supplied. These tables have two columns, the price of the good and the quantity supplied.

Market Supply vs. Individual Supply

  • The quantity supplied in the market is the sum of the quantities supplied by all the sellers at each price.
  • QsQ^s represents the market quantity supplied

Supply Curve Shifters

  • The supply curve shows how price affects quantity supplied, other things being equal.
  • These other things are the non-price determinants of quantity supplied. \

Input Prices

  • Examples of input prices: wages, prices of raw materials
  • A fall in input prices makes production more profitable at each output price, so firms supply a larger quantity at each price, and the SS curve shifts to the right.
  • An increase in input prices makes production less profitable at each output price, so firms supply a lower quantity at each price, and the SS curve shifts to the left.

Technology

  • Technology determines how much inputs are required to produce a unit of output.
  • A cost-saving technological improvement has the same effect as a fall in input prices, it shifts the SS curve to the right.

Number of Sellers

  • An increase in the number of sellers increases the quantity supplied at each price.
  • This also shifts the SS curve to the right.

Expectations

  • An expectation of a higher price for a good will cause sellers to reduce their quantity supplied and hold on to their inventory until the price increases, temporarily shifting the SS curve to the left.
  • This only applies if the goods are not perishable

Summary: Variables that Influence Sellers

  • Change in price causes a movement along the SS curve

Fancy Acronym

  • R - Resource Input/Cost
  • O - Other goods
  • T - Technology
  • T - Taxes
  • E -

2.3 - Price Elasticity of Demand

Elasticity measures how much one variable responds to changes in another variable.

In economics, elasticity is a numerical measure of the responsiveness or QdQ_d or QSQ_S to one of its determinants.

Price elasticity of demand measures how much QdQ_d responds to a change in PP.

Price elasticity of demand=Precentage change in QdPercentage change in P\textrm{Price elasticity of demand} = \frac{\textrm{Precentage change in } Q_d}{\textrm{Percentage change in } P}

Loosely speaking, it measures the price sensitivity to buyer's demand.

Along a DD curve, PP and QQ move in opposite directions, which would make price elasticity negative.

We will drop the minus sign and write all demand elasticities as positive numbers.

Calculating Percentage Changes

Standard method of computing the percentage change:

end value−start valuestart value×100%\frac{\textrm{end value} - \textrm{start value}}{\textrm{start value}} \times 100\%
🙄
AP Graders are really picky and want to see all of your work so make sure you actually show the formulas and the work with numbers plugged in.

This method sucks though so we will use the midpoint method for elasticity instead.

end value−start valuemidpoint×100%\frac{\textrm{end value} - \textrm{start value}}{\textrm{midpoint}} \times 100\%

Where the midpoint is the number halfway between the start and end values, the average of those values.

end value−start value(end value+start value)/2×100%\frac{\textrm{end value} - \textrm{start value}}{(\textrm{end value} + \textrm{start value})/2} \times 100\%

Determinants of Price Elasticity

  • The extent to which close substitutes are available
  • Whether the good is a necessity or a luxury
  • How broadly or narrowly the good is defined
  • The time horizon -

Inelastic Demand

When the price elasticity of demand is <1, it is considered inelastic demand. That is, the customer's sensitivity to price is relatively low.

When DD is inelastic, a price increase causes revenue to grow.

Unit Elastic Demand

When the price elasticity of demand =1, this is considered unit elastic demand. This means that consumer's sensitivity to price is intermediate. A change in price with unit elastic demand has no effect on revenue for businesses.

Elastic Demand

When the price elasticity of demand >1, this is considered elastic demand. Price sensitivity is relatively high. The change in QDQ_D is greater than the change in PP.

When DD is elastic, a price increase causes revenue to fall.

Perfectly Elastic Demand

When the DD curve is completely horizontal, the price elasticity of demand =∞\infin and consumer's price sensitivity is extreme.

2.4 - Price Elasticity of Supply

Perfectly Inelastic Supply

Price elasticity of supply=% change in Q% change in P=0%10%=0\textrm{Price elasticity of supply} = \frac{\textrm{\% change in Q}}{\textrm{\% change in P}} = \frac{0\%}{10\%} = 0
  • SS curve is vertical
  • Seller's price sensitivity is zero

Inelastic Supply

Price elasticity of supply=% change in Q% change in P=<10%10%<1\textrm{Price elasticity of supply} = \frac{\textrm{\% change in Q}}{\textrm{\% change in P}} = \frac{<10\%}{10\%} < 1
  • SS curve is relatively steep
  • Seller's price sensitivity is relatively low

Unit Elastic Supply

Price elasticity of supply=% change in Q% change in P=10%10%=1\textrm{Price elasticity of supply} = \frac{\textrm{\% change in Q}}{\textrm{\% change in P}} = \frac{10\%}{10\%} = 1
  • SS curve has an intermediate slope
  • Seller's price sensitivity is intermediate

Elastic Supply

Price elasticity of supply=% change in Q% change in P=>10%10%>1\textrm{Price elasticity of supply} = \frac{\textrm{\% change in Q}}{\textrm{\% change in P}} = \frac{>10\%}{10\%} > 1
  • SS curve has a relatively flat slope
  • Seller's price sensitivity is relatively high

Perfectly Elastic Supply

Price elasticity of supply=% change in Q% change in P=any %0%=∞\textrm{Price elasticity of supply} = \frac{\textrm{\% change in Q}}{\textrm{\% change in P}} = \frac{\textrm{any} \space \%}{0\%} = \infin
  • SS curve is horizontal
  • Seller's price sensitivity is infinite

Determinants of Supply Elasticity

  • The more easily sellers can change the quantity they produce, the greater the price elasticity of supply.
    • Example: Supply of beachfront property is harder to vary and thus less elastic than supply of new cars.
  • For many goods, price elasticity of supply is greater in the long run than in the short run, because firms can build new factories, or new firms may be able to enter the market.

2.5 - Other Elasticities

Income Elasticity of Demand

Measures the response of QdQ_d to a change in consumer income.

Income elasticity of demand=Percent change in QdPercent change in income\textrm{Income elasticity of demand} = \frac{\textrm{Percent change in } Q_d}{\textrm{Percent change in income}}
  • For normal goods, the income elasticity of demand is > 0 (positive), while for inferior goods, the income elasticity of demand is < 0 (negative).

Cross-price Elasticity of Demand

Measures the response of demand for one good to changes in the price of another good.

Cross-price elasticity of demand=Percent change in Qd for good 1Percent change in price for good 2\textrm{Cross-price elasticity of demand} = \frac{\textrm{Percent change in } Q_d \textrm{ for good 1}}{\textrm{Percent change in price for good 2}}
  • For substitutes, cross-price elasticity > 0 (e.g. an increase in price of beef causes an increase in demand for chicken)
  • For complements, cross-price elasticity < 0 (e.g. an increase in the price of computers causes a decrease in demand for software)

2.6 - Market Equilibrium and Consumer and Producer Surplus

Surplus (excess supply)

When quantity supplied is greater than quantity demanded, surplus is the excess quantity supplied that is not demanded. Any price above equilibrium creates a surplus. Price will eventually decrease until it stabilizes at equilibrium.

Shortage (excess demand)

When quantity demanded is greater than quantity supplied, the difference between quantity demanded and quantity supplied is the shortage. Any price below equilibrium will create a shortage. Price will eventually increase to reach equilibrium again.

Willingness to Purchase (WTP)

Willingness to purchase is the measure of the highest price at which a buyer is willing to purchase a good. At any QQ, the height of the DD curve is the WTP of the marginal buyer.

Consumer Surplus (CS)

Consumer surplus is the amount a buyer is willing to pay minus the amount the buyer actually pays.

CS=WTP−PCS = WTP - P

On a demand curve, the consumer surplus is the area above the given price but below the demand curve, from 00 to QQ.

Cost and the Supply Curve

Cost is the value of everything a seller must give up to produce a good (i.e. opportunity cost). This includes the cost of all resources used to produce the goods, including the value of the seller's time.

A seller will produce and sell the good/service only if the price exceeds the cost, that is, they can make a profit.

At each QQ, the height of the SS curve is the cost of the marginal seller, the seller who would leave the market if the price were any lower.

Producer Surplus

Producer surplus equals the price of a good minus the cost of it's production.

PS=P−CPS = P - C

On a supply curve, the producer surplus is the area above the supply curve and below the price of the good.

2.7 - Market Disequilibrium and Changes in Equilibrium

Effects of a change in Supply or Demand

Total Surplus & Efficiency

The total surplus is the sum of consumer and producer surplus.

TS=CS+PS\textrm{TS} = \textrm{CS} + \textrm{PS}

Another way of putting this is that the total surplus is the value to buyers minus the cost to sellers.

Total surplus=(value to buyers)−(cost to sellers)\textrm{Total surplus} = (\textrm{value to buyers}) - (\textrm{cost to sellers})

An allocation of resources is efficient if it maximizes the total surplus. Efficiency means:

  • The goods are consumed by the buyers who value them most highly.
  • The goods are produced by the producers with the lowest costs.
  • Raising or lowering the quantity of a good would not increase the total surplus.

2.8 - The Effects of Government Intervention in Markets

When the equilibrium price of a good is deemed too high by a government, they will put in place a price ceiling below the natural equilibrium price (binding). This is a maximum price of a good in a market. Instituting a price ceiling will create a shortage, as now QSQ_S is less than QDQ_D.

Effects of a Tax

The size of a tax can be represented by $T\$T. The government's revenue from tax is represented by:

$T⋅PT\$ T \cdot P_T

With a tax, both CSCS and PSPS are reduced, and the area between them is the tax revenue. The area to the right of the tax revenue becomes dead weight loss.

Taxes can be considered a form of surplus, and therefore are incorporated into total surplus.

Deadweight Loss

The goods between QEQ_E and QTQ_T are not sold, and are known as the deadweight loss.

The size of the deadweight loss is determined by the price elasticities of supply and demand.

When supply is inelastic, it is harder for firms to leave the market when the tax reduces PSP_S, and therefore the DWL is low. The more elastic the supply, the easier it is for firms to leave the market, and thus there is a higher change in QQ and a higher DWL.

When the demand is inelastic, it's harder for consumers to leave the market when the tax raises PBP_B. So, the tax only reduces QQ a little, and the DWL is small. The more elastic the demand, the easier for buyers to leave the market, and thus there is a higher change in QQ and a higher DWL.

Depicted on a Graph

Market without a tax
Effect of a tax on the market

2.9 - International Trade and Public Policy

  • PWP_W= the world price of a good, the price that prevails in world markets.
  • PDP_D = domestic price without trade.
  • If PD<PWP_D \lt P_W
    • country has a comparative advantage in the good
    • under free trade, country exports the good
  • If PD>PWP_D \gt P_W
    • country does not have a comparative advantage in the good
    • under free trade, the country will import the good

The Small Economy Assumption

  • A small economy is a price taker in world markets: Its actions have no effect on PWP_W.
  • Not always true, but simplifies the analysis

The Welfare Effects of Trade

When a good is imported or exported, trade creates winners and losers.

Other Benefits of International Trade

  • Consumers enjoy an increased variety of goods
  • Producers sell to a larger market and may achieve lower costs
  • International competition may increase market power of local producers

Tariff: An Example of a Trade Restriction

  • A tariff is a tax on imports.
  • Example: Cotton shirts
    • PW=$20P_W = \$20
    • Tariff: T=$10/shirtT = \$10/\textrm{shirt}
    • Consumers must pay $30 for an imported shirt.
    • Therefore, domestic producers can charge $30 per shirt.
  • In general, the price facing domestic buyers & sellers equals PW+TP_W + T.

Effect of a Tariff on Supply and Demand

Arguments for Restricting Trade

  • Trade destroys jobs in industries that compete with imports.
    • Look at the data. Do rising imports cause rising unemployment? No.
    • Imports do not destroy jobs, only change which jobs are available.
  • An industry vital to national security should be protected from foreign competition, to prevent dependence on imports.
    • Producers may exaggerate their own importance to national security to get protections
  • A new industry argues for temporary protection until it's mature and can compete with foreign firms
    • This is dumb — Sincerely, Economists
  • Producers argue their competitors in another country have an unfair advantage
    • Ok perfect that sucks we're just gonna import and you suck go die — Economists
  • Nation could use restricting trade to bargain with other nations
    • This is a bad idea, as it places the nation between a rock and a hard place if the other nation declines to act.

Trade Agreements

  • A country can liberalize trade with
    • unilateral reductions in trade restrictions
    • multilateral agreements with other nations
  • Examples of trade agreements:
    • North American Free Trade Agreement (NAFTA), 1993
    • General Agreement on Tariffs and Trade (GATT), ongoing

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Unit 3: Production, Cost, and the Perfect Competition Model

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3.0 - Overview & Market Structures

Perfect Competition

  • Many firms in competition with each other
  • Goods being sold are identical, with no product differentiation
  • Individual firms have no control over prices, they are called "price takers"
  • Low barrier to entry (solely startup cost)
  • Well informed buyers and sellers

Monopolistic Competition

  • Many firms in competition with each other
  • There is some amount of differentiation between products
  • Individual firms have no control over prices, they are called "price takers"
  • Low barrier to entry (solely startup cost)
  • Monopolistic competitors use non-price competition (Advertising, giveaways, promotions, etc.)

Oligopoly

  • Few firms control the market
  • Firms have a strong control over price, and frequently collude to set prices
  • Firms can also engage in price wars and predatory pricing
  • Medium barrier to entry
  • Price Leadership: Unofficial collusion where firms follow a single firm when that firm changes their price

Monopoly

  • One firm has absolute control over the market
  • No variety of goods
  • Firm is a price maker, they are able to set the price to whatever they want
  • High barrier to entry

3.1 - The Production Function

Vocabulary

  • Inputs—Any resources (and, labor, capital, or entrepreneurship) that is used by firms to produce outputs. Ex: a pizza parlor needs inputs such as tomatoes, yeast, and flour.
  • Costs—Money that is spent to purchase inputs. Ex: in order to get workers (labor) you have to pay wages (a type of cost).
  • Outputs—The finished goods and services that firms produce in order to make revenue. Ex: a toy company's outputs are different types of toys.
  • Revenue—This the all money received by a firm (Price x Quantity). Ex: If Mattel sells their Barbie doll for $10 and they sell 100 of the dolls, then their revenue is $1000.

Total, Average, and Marginal Product

Total product (TP)—the total quantity of output produced by a firm in the market.

Average product (AP)—The quantity of total output produced per unit of input used in the production process (Total Product / Units of Inputs).

Marginal product (MP)—The quantity of total output produced by each additional unit of input used in the production process. This is calculated by dividing the change in total product by the change in quantity of input.

The Production Function

The Theory of Production is the idea that input is related to output.

A Production Function shows the relationship between the quantity of inputs used to produce a good and the quantity of output of that good.

The slope of the production function represents the marginal product of labor, the change in output that results from employing an added unit of labor.

MPL=ΔQΔL\textrm {MP}_L = \frac{\Delta Q}{\Delta L}

In continuous terms, the MPL\textrm {MP}_L  is the first derivative of the production function 🤯.

MPL\textrm{MP}_L diminishes as LL increases due to the law of diminishing marginal returns.

Law of Diminishing Marginal Returns

Increasing Marginal Returns—Each additional variable input is more productive than the last. MP and AP increase, as each variable input specializes in its task and utilizes +fixed inputs. TP increases at an increasing rate.

Decreasing Marginal Returns—Each additional variable input is less productive than the last. MP and AP decrease as specialization decreases, and there are not enough fixed inputs. TP increases, but at a slower rate.

Negative Marginal Returns—Each additional variable input gets in the way of production. AP decreases and MP becomes negative, as specialization is impossible with too many +variable inputs. TP decreases.

The Law of Diminishing Marginal Returns dictates that, as variable resources are added to fixed resources, the additional output produced from each new input will eventually fall. Basically, at some point, each additional worker used in the production process becomes less productive.

Stages of Marginal Returns

  • Stage 1: Increasing Marginal Returns. Both TP and MP are increasing.
  • Stage 2: Diminishing Marginal Returns. TP is still increasing, but MP is decreasing.
  • Stage 3: Negative Marginal Returns. Both TP and MP are decreasing.

Graphs of the Production Function

Table calculating the marginal returns of adding additional workers (inputs).
Total, average, and marginal products depicted on a graph.

Relating Total, Average, and Marginal Products

Relationship between Marginal Product and Total Product

The law of variable proportions explains the relationship between Total Product and Marginal Product. It states that when only one variable factor input is allowed to increase and all other inputs are kept constant, the following can be observed:

  • When the Marginal Product (MP) increases, the Total Product is also increasing at an increasing rate. This gives the Total product curve a convex shape in the beginning as variable factor inputs increase. This continues to the point where the MP curve reaches its maximum.
  • When the MP declines but remains positive, the Total Product is increasing but at a decreasing rate. This gives the Total product curve a concave shape after the point of inflection. This continues until the Total product curve reaches its maximum.
  • When the MP is declining and negative, the Total Product declines.
  • When the MP becomes zero, Total Product reaches its maximum.

Relationship between Average Product and Marginal Product

There exists an interesting relationship between Average Product and Marginal Product:

  • When Average Product is rising, Marginal Product lies above Average Product.
  • When Average Product is declining, Marginal Product lies below Average Product.
  • At the maximum of Average Product, Marginal and Average Product equal each other.

3.2 - Short-Run Production Costs

In the short-run, at least 1 resource is fixed, or cannot change. Fixed resources can include plant capacity, or the number of factories in operation.

Fixed, Variable, and Total Costs

Fixed Cost (FC)—The costs of fixed resources used during the production process. These +costs do not change with the amount of output produced. Ex: rent, salaries, insurance.

Variable Cost (VC)—The costs of variable resources used during the production process. These +costs do change with the amount of output produced. The more output produced, the higher the variable costs are and vice versa. Ex: electricity, hourly wages, shipping costs.

Total Cost (TC)—The sum of variable costs and fixed costs.

Marginal Cost (MC)—The additional cost of producing each additional unit of output. (MC=ΔTCΔQ\textrm{MC} = \frac{\Delta TC}{\Delta Q})

  • Profit is maximized when MC=MR\textrm{MC} = \textrm{MR}, so you will continue to produce more output until these two quantities are equal.

Relating Fixed, Variable, and Total Costs

Fixed Costs+Variable Costs=Total Costs\textrm{Fixed Costs} + \textrm{Variable Costs} = \textrm{Total Costs}

Average Costs

Average Fixed Cost (AFC)—Fixed cost divided by the quantity of output

Average Variable Cost (AVC)—Variable cost divided by the quantity of output.

Average Total Cost (ATC)—Total cost divided by the quantity of output.

Average Fixed Costs+Average Variable Costs=Average Total Costs\textrm{Average Fixed Costs} + \textrm{Average Variable Costs} = \textrm{Average Total Costs}

Economic Profit vs. Accounting Profit

Implicit cost—Opportunity costs associated with decisions in the production of goods and services. Ex: If an individual gives up an annual salary to open a business, then that salary is seen as an implicit cost.

Accounting costs—The explicit or "out of pocket" payments paid by firms to use resources during the production process.

Economic costs—The sum of both the implicit costs (opportunity costs) and explicit costs +of production. These costs include both the "out of pocket" payments paid by firms and the opportunity costs of using resources during the production process.

Accounting profits—The profits earned by the firm when the revenue earned by the firm is +greater than the explicit (accounting) costs of production (Total Revenue - Accounting Costs).

Economic profits—The profits earned by the firm when the revenue earned by the firm is +greater than the sum of the explicit (accounting) costs and the implicit (opportunity cost) costs of production (Total Revenue - Economic Costs).

Accounting profits ignore the implicit costs, and therefore will always appear higher than economic profit.

Graphical Depictions of Fixed, Variable, and Total Cost

Example table of a firm's FC, VC, and TC, and how those are related to the firm's output.

What FC, VC, and TC look like on a graph. Notice the fixed costs just move the graph of VC upwards to reach TC.
Graph of marginal cost (MC), average fixed cost (AFC), average variable cost (AVC), and average total cost (ATC).

MC always crosses both AVC and ATC at their lowest point. If fixed costs increase, then both the AFC and ATC would shift up and vice versa. If variable costs increase, then both AVC and ATC will shift upward and vice versa. The MC curve only shifts when variable costs change. It will shift upward for an increase in variable costs and downward for a decrease in variable costs.

Graphs shifting upward with a change in variable cost.

3.3 - Long-Run Production Costs

In the long-run, all resources are flexible, so firms can change both their plant capacity and output level. This allows firms to analyze and compare the average total cost of production at each plant capacity in the short-run, and find the optimal plant capacity that allows them to product output at the lowest possible ATC.

This the LRATC curve is merely the combination of several SRATC curves for different facilities.

The U-shape of the long-run ATC (LRATC) curve is a result of economies of scale and diseconomies of scale that are experienced by the firm.

  • Economies of scale refers to the reduction in total cost-per-unit as a firm increases its production. In this phase, the firm can reduce its total cost-per-unit by boosting its plant capacity and output.
  • Diseconomies of scale refers to the rise in total cost-per-unit as the firm increases its production. In this phase, the firm would be better off reducing its plant capacity and +output in order to lower their per-unit costs.
  • In between these two phases is what we refer to as constant returns to scale. When the firm increases production, costs stay the same. The ATC is at its lowest here.

The light blue area represents economies of scale because as output is increasing, costs are decreasing. The light yellow area represents a constant return to scales because as output continues to increase, costs remain constant. The light green area represents diseconomies of scale because as output increases, costs rise.

3.4 - Types of Profit

In economics, there are a variety of different ways to represent profits. These different methods of calculating profits vary based on what type of costs are being considered in each situation. Profit, in general, is the difference between total revenue and total costs (provided total revenue is greater than total costs).

  • Accounting profit represents a firm's total revenue minus the firm's explicit costs.
  • Economic profit represents a firm's accounting profits minus the firm's explicit and implicit costs. While accounting profit factors in only the explicit costs, economic profit includes implicit costs, and therefore factors in the opportunity costs lost by not pursuing other opportunities.
  • Normal profit occurs when a firm's economic profit is zero. So for example, if a firm's total revenue is $100,000 and the total of that firm's explicit and implicit costs is $100,000, then the firm's economic profit is zero, and it is experiencing normal profit. When a firm is experiencing normal profit, its accounting profit is still positive. Normal profit is also referred to as "breaking even."

If revenue is less than costs, a firm may experience economic losses. For example, if the total revenue is $80 and the total cost is $97, we have an economic loss of $17.

When a firm is experiencing an economic profit, they can increase their production. If a firm is earning an economic loss, they will most likely respond by decreasing their output.

3.5 - Profit Maximization

MR=MC\textrm{MR} = \textrm{MC}

At the point where MR\textrm{MR} equals MC\textrm{MC}, the costs of producing the last unit of output equals the revenue gained when selling it. At this point, a firm's profit is maximized.

For a firm in perfect competition, MR=P\textrm{MR} = \textrm P.

Revenue and Costs for an Example Firm

Using this example firm's revenue and costs, we can see that the profit-maximizing quantity is 3 units, where both the marginal revenue and marginal cost equal $7.

MC and the Firm's Supply Decision

If the price of a good increases from P1P_1 to P2P_2, then the profit maximizing quantity rises to Q2Q_2. The MC\textrm{MC} curve determines the firm's QQ at any price. (Provided the firm is in perfect competition, as the market sets the price).

3.6 - Firms' Short-Run Decisions to Produce and Long-Run Decisions to Enter or Exit a Market

Shut-Down Rule

A shutdown is a short-run decision not to produce anything because of market conditions. The shutdown rule states that a firm should continue to produce and operate so long as price is equal to or above the firm's AVC.

  • Cost of shutting down: revenue loss = TR\textrm{TR}
  • Benefit of shutting down: cost savings = VC\textrm{VC}
    • Firm still has to pay FC\textrm{FC} (rent, electric, other fixed expenses)
  • If TR<VC\textrm{TR} < \textrm{VC}, the cost of shutting down is less than the benefit, and a business should shut down.
  • Dividing both sides by QQ nets the "decision rule"
    TRQ<VCQ\frac{\textrm{TR}}Q \lt \frac{\textrm{VC}}{Q}
    ⁍⁍

Because on the graph above, the profit maximizing quantity (where MR=MC\textrm{MR} = \textrm{MC}) is below the AVC\textrm{AVC} curve, the firm should choose to shutdown and not produce. If the profit maximizing quantity is anywhere above the AVC\textrm{AVC} curve, the firm should produce.

The Irrelevance of Sunk Costs

A sunk cost is a cost that has already been committed and cannot be recovered.

Sunk costs are irrelevant to the shutdown decision, as you have to pay them regardless of your choice. FC\textrm{FC} is a sunk cost, as the firm must pay these costs regardless of if they choose to shut down or not. Therefore, FC\textrm{FC} should not be factored in to the decision to shutdown.

Long-Run Decisions to Enter or Exit the Market

Long-Run Decision to Exit

  • Cost of exiting the market: revenue loss = TR\textrm{TR}
  • Benefit of exiting the market: cost savings = TC\textrm{TC}
    • There are no FC\textrm{FC} in the long run
  • Therefore, a firm will exit the market if TR<TC\textrm{TR} \lt \textrm{TC}
  • Divide both sides by QQ to write the firm's decision rule:
    Exit if:P<ATC\textrm{Exit if:} \quad P < ATC

Long-Run Decision to Enter

  • In the long run, a new firm will enter the market if it is profitable to do so: if TR>TC\textrm{TR} > \textrm{TC}.
  • Divide both sides by QQ to express the firm's entry decision as:
    Enter if:P>ATC\textrm{Enter if:} \quad P > ATC

3.7 - Perfect Competition

Characteristics of Perfect Competition

  • Many, small firms in the industry
  • Firms are "Price Takers"
  • Low barriers to entry
  • Firms break even in the long-run
  • Products sold are identical
  • No non-price competition
  • Firms are perfectly efficient in the long-run

Assumptions in Market Supply

  1. All exisiting firms and potential entrands have identical costs.
  1. Each firm's costs do not change as other firms enter or exit the market.
  1. The number of firms in the market is:
    1. Fixed in the short run
    1. Variable in the long run

Side-by-Side Graphs of Perfect Competiton

Since firms are price-takers in perfect completion, a firm's MC and MR graph is directly related to the market supply and demand graph. The part of the MC curve that is above the MR curve is the same as the firm's supply curve.

In a perfectly competitive market in the short-run, a graph can display three possible scenarios. They can show a short-run profit, short-run loss, or short-run shutdown.

Short-Run Profit

When a firm is making a short-run profit, its profit per unit = P−ATC\textrm P - \textrm{ATC}. Therefore, the firm's total profit is equal to (P−ATC)×Q(\textrm P - \textrm{ATC}) \times Q.

Short-Run Loss

A short-run loss is shown when the ATC is located above the price line at the profit-maximizing point, and the AVC curve is located below the price line at the profit-maximizing point.

When a firm is making short-run losses, its loss per unit equals ATC−P\textrm{ATC} - \textrm P. Therefore, the firm's total loss is equal to (ATC−P)×Q(\textrm{ATC} - \textrm P) \times Q.

Short-Run Shutdown

A business will shutdown when both ATC and AVC are located above the price line at the profit-maximizing point, as shown above.

Long-Run Perfectly Competitive Graph

When a perfectly competitive market is in long-run equilibrium, we show this on the side by side graphs by having ATC tangent to the price line at the profit-maximizing quantity (MR = MC). When a perfectly competitive market is in long-run equilibrium, it is both allocatively efficient and productively efficient. On a graph, allocative efficiency is P(MR) = MC, and productive efficiency is P = minimum ATC.

Shift from Short-Run to Long-Run Equilibrium in a Perfectly Competitive Market

When a short-run perfectly competitive market is earning either a profit or loss, firms will want to either enter or exit the market, thus market shifting the market from short-run to long-run.

When a firm is earning a profit, this provides an incentive for firms not already in the industry to enter because of the possible profit. This will cause the supply curve to shift right due to the increase in supply, which will lower the market price and in turn lower the price for each firm.

This can be illustrated on a graph by shifting the price line down to become tangent with ATC. The profit-maximizing quantity for the firm will decrease. The market quantity will increase as there are now more firms in the industry, but each individual firm is supplying less of that overall quantity.

The below graphs show how a perfectly competitive market goes from a short-run profit to long-run equilibrium.

Since firms joining the market reduces the profit made by each firm, eventually some firms will be incentivized to leave the market, repeating in a cycle until the remaining firms earn zero profit:

The Zero-Profit Condition

  • Long-run equilibrium: The process of entry or exit is complete, and remaining firms earn zero economic profit.
  • Zero economic profit occurs when P=ATC\textrm P = \textrm{ATC}.
  • Since firms produce when P=MR=MC\textrm P = \textrm{MR} = \textrm{MC}, the zero-profit condition is where P=MC=ATC\textrm P = \textrm{MC} = \textrm{ATC}.
  • Recall that MC intersects ATC at the minimum ATC.
  • Hence, in the long run, P=minimum ATC\textrm P = \textrm{minimum ATC}.
❓
Why do firms say in business with zero profits? +Recall that economic profit is revenue minus all costs, including implicit costs like the opportunity cost of the owner's time and money. In zero-profit equilibrium, firms earn enough revenue to cover these costs. Thus, accounting profit is still positive, and firms still have a strong incentive to produce.

Shift from Long-Run to Short-Run back to Long-Run

A market may be in long-run equilibrium, and then experience a change in demand in the market. This shift of demand moves the market into short-run, and then it has to readjust back to long-run.

Let's show how this occurs when there is a scenario that increases demand. +We'll use the market for apples.

Step 1

A market is in long-run equilibrium, when a change in market demand causes the price of the good to increase. We show this on the firm graph by shifting the price line up and identifying the new profit-maximizing point for the firm (MR=MC\textrm{MR} = \textrm{MC}). This causes the firm to go from long-run +equilibrium to short-run profit since the price line is above the ATC curve.

Step 2

Now that the perfectly competitive market is earning a short-run profit, individual firms are incentivized to enter the market. This makes the supply curve shift right, causing the equilibrium price to decrease. This will cause the price line to drop on the firm graph and cause the +profit-maximizing quantity to return to the original one in the firm.

This can happen anytime a perfectly competitive market starts in long-run equilibrium and gets moved to short-run. The market always has to readjust to long-run equilibrium.

Changes Due to the Assumptions of the Perfect Competition Model

  1. Firms have different costs
    • A PP rises, firms with lower costs enter the market before those with higher costs.
    • Further increases in PP make it worthwhile for higher-cost firms to enter the market, which increases market quantity supplied.
    • Hence, the LR\textrm{LR} market supply curve slopes upward.
    • At any PP,
      • For the marginal firm, P=minimum ATCP = \textrm{minimum ATC} and profit=0\textrm{profit} = 0.
      • For lower-cost firms, profit>0\textrm{profit} > 0.
  1. Costs Rise as Firms Enter the Market
    • In some industries, the supply of a key input is limited.
    • The entry of new firms increases demand for this input, causing its price to rise.
    • This increases all firms' costs.
    • Hence, an increase in PP is required to increase the market quantity supplied, so the supply curve is upward-sloping.

Conclusion: The Efficiency of a Competitive Market

  • Profit Maximization: MR=MC\textrm{MR} = \textrm{MC}
  • Perfect Competition: P=MRP = \textrm{MR}
  • So, in the competitive equilibrium, P=MCP = \textrm{MC}
  • Recall, MC\textrm{MC} is cost of producing the marginal unit. PP is the value to buyers of the marginal unit.
  • For a firm in a perfectly competitive market, price = marginal revenue = average revenue.
  • IF P>AVCP > \textrm{AVC}, a firm maximizes profit by producing the quantity where MR=MC\textrm{MR} = \textrm{MC}

+

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Unit 4: Imperfect Competition

Created
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4.1 - Introduction to Imperfectly Competitive Markets

Market Structures

Perfect Competition

  • Many firms in competition with each other
  • Goods being sold are identical, with no product differentiation
  • Individual firms have no control over prices, they are called "price takers"
  • Low barrier to entry (solely startup cost)
  • Well informed buyers and sellers

Monopolistic Competition

  • Many firms in competition with each other
  • There is some amount of differentiation between products
  • Individual firms have no control over prices, they are called "price takers"
  • Low barrier to entry (solely startup cost)
  • Monopolistic competitors use non-price competition (Advertising, giveaways, promotions, etc.)

Oligopoly

  • Few firms control the market
  • Firms have a strong control over price, and frequently collude to set prices
  • Firms can also engage in price wars and predatory pricing
  • Medium barrier to entry
  • Price Leadership: Unofficial collusion where firms follow a single firm when that firm changes their price

Monopoly

  • One firm has absolute control over the market
  • No variety of goods
  • Firm is a price maker, they are able to set the price to whatever they want
  • High barrier to entry

Price Control or Price “Makers”

All imperfectly competitive firms exert some control over price. This ability comes from either the type of product they sell OR the amount of competition.

+

Price-making firms face a downward-sloping demand curve, greater than the marginal revenue curve.

Why Imperfectly Competitive Firms are Inefficient

Defining Efficiency

  • Allocative Efficiency (P=MC\textrm{P} = \textrm{MC}): The price of a good should be equal to the value of the resources (land, labor, capital) used to produce it. Also known as the Socially Efficient quantity of output.
  • Productive Efficiency (P=min ATC\textrm{P} = \textrm{min ATC}): The least costly production techniques are used.

In perfect competition, a firm’s P=MC=min ATC\textrm {P} = \textrm{MC} = \textrm{min ATC}. Therefore, perfectly competitive firms are also 100% efficient.

The same is not true for imperfectly competitive firms, as they are able to make a profit in the long run.

4.2 - Monopolies

A monopoly is a firm that is the sole seller of a product without close substitutes.

Why Monopolies Arise

The main cause of monopolies is barriers to entry—other firms cannot enter the market.

Three sources of barriers to entry:

  1. A single firm owns a key resource. Eg: DeBeers owns most of the world’s diamond mines.
  1. The government gives a single firm the exclusive right to produce the good.
  1. Natural monopoly: a single firm can produce the entire market QQ at a lower cost than could several firms.

For a monopoly, the ATC\textrm{ATC} curve slopes downwards as they have huge FC\textrm{FC} and small MC\textrm{MC}.

Monopoly vs. Perfect Competition: Demand Curves

In a competitive firm, the market supply curve is perfectly elastic.

A monopolist is the only seller, so it faces the market demand curve. To sell a larger QQ, the monopoly must reduce its PP.

In a monopoly, P=AR\textrm P = \textrm {AR} and MR<P\textrm{MR} < \textrm{P}.

Understanding the Monopolist’s MR

  • Increasing QQ has two effects on revenue:
    • Output effect: higher output raises revenue
    • Price effect: lower price reduces revenue
  • To sell a larger QQ, the monopolist must reduce the price on all the units it sells.
  • Hence, MR<P\textrm{MR} < \textrm P.
  • MR\textrm {MR} could even be negative if the price effect exceeds the output effect.

Profit Maximization

  • Like a competitive firm, a monopolist maximizes profit by producing the quantity where MR=MC\textrm{MR} = \textrm{MC}.
  • Once the monopoly has its price maximizing quantity of output, it will identify the price consumers are willing to pay from the demand curve at that quantity QQ.
  • Because this PP must be greater than the MR=MC\textrm{MR} = \textrm{MC} point, monopolies are guaranteed to make long run profits.

A Monopoly Does Not Have an SS Curve

A competitive firm:

  • takes PP as given
  • has a supply curve that shows how its QQ depends on PP

A monopoly firm:

  • is a “price-maker”, not a “price-taker”
  • QQ does not depend on PP
  • QQ and PP are jointly determined by MR\textrm{MR}, MC\textrm{MC}, and the demand curve.

Because of this, there is no supply curve for monopolies.

The Welfare Cost of Monopoly

The value to buyers of an additional unit, PP, exceeds the cost of the resources needed to produce that unit, MC\textrm{MC}. This creates a deadweight loss, which could be recouped were the market in perfect competition. This takes away from the consumer surplus and the total surplus, and brings some additional producer surplus.

In perfect competition, P=MCP = \textrm{MC}, but in a monopoly, P>MCP > \textrm{MC}. The deadweight loss created by the monopoly is the difference between PP and MC\textrm{MC} which would not be present in perfect competition.

Public Policy towards Monopolies

  • Increasing competition with antitrust laws
    • Ban some anticompetitive practices, allow govt to break up monopolies.
    • e.g., Sherman Antitrust Act (1890), Clayton Act (1914)
  • Regulation
    • Government agencies set the monopolist’s price
    • For natural monopolies, MC<ATC\textrm{MC} < \textrm{ATC} at all QQ, so marginal cost pricing would result in losses.
      • For these firms, regulators might subsidize the monopolist or set P=ATCP = \textrm{ATC} for zero economic profit.
  • Public ownership
    • Example: US Postal Service
    • Problem: Public ownership is usually less efficient since no profit motive to minimize costs
      • ^ bullshit
  • Doing nothing
    • The foregoing policies all have drawbacks, so the best policy may be no policy.
      • ^ also bullshit

Conclusion

  • In the real world, pure monopoly is rare.
  • Yet, many firms have market power, due to:
    • selling a unique variety of product
    • having a large market share and few significant competitiors
  • In many such cases, most of the results from this chapter apply, including:
    • markup of price over marginal cost
    • deadweight loss

4.3 - Price Discrimination

  • Discrimination: treating people differently based on some characteristic, e.g. race or gender.
  • Price discrimination: selling the same good at different prices to different buyers.
  • The characteristic used in price determination is willingness to pay (WTP):
    • A firm can increase profit by increasing the price for buyers with a higher WTP.

Perfect Price Discrimination vs. Single Price Monopoly

In a single price monopoly, a firm charges a fixed price for all buyers. This price is greater than the MC for the good, and therefore results in a deadweight loss.

+

With perfect price discrimination, the monopolist produces the competitive quantity, but instead of charging a fixed price, charges every individual buyer their WTP.

In this instance, the monopolist captures all CS as profit. The area which was a deadweight loss for a single-price monopoly instead becomes profit for the monopolist.

Price Discrimination in the Real World

  • Perfect price discrimination is not realistic in the real world:
    • No firm knows every buyer’s WTP
    • Buyers do not reveal it to sellers
  • So, firms divide customers into groups based on some observable trait that is likely related to WTP, such as age.

4.4 - Monopolistic Competition

Monopolistic competition: many firms sell similar, but not identical, products.

Characteristics

  • Many sellers
  • Product differentiation
  • Free entry and exit

Examples

  • apartments
  • books
  • bottled water
  • clothing
  • fast food
  • night clubs

Comparing Perfect and Monopolistic Competiton

  • Number of sellers: both perfect and monopolistic competition have many sellers
  • Free entry and exit: both
  • Long-run economic profits: zero for both, as a result of the freedom to enter and exit the market freely
  • Firms sell: identical products in perfect competition, differentiated products in monopolistic competition
  • Market power: none for perfect competition, some for monopolistic competition as they have the ability to create product differentiation. Location can actually be an important form of “product differentiation” for firms in monopolistic competition.
  • DD curve facing firm: horizontal in perfect competition, downward sloping in monopolistic competition. This is because firms in monopolistic competition have some market power.

Comparing Monopoly and Monopolistic Competition

  • Number of sellers: one in a monopoly, many in monopolistic competition
  • Free entry/exit: no for monopoly, yes for monopolistic competition
  • Long-run economic profits: positive for a monopoly, zero for monopolistic competition
  • Firm has market power?: yes for both, though monopolies obviously have more power
  • DD curve facing firm: both are downward sloping, but monopolies actually face the market demand curve, while firms in monopolistic competition all have unique demand curves.
  • Close substitutes: none for monopoly, many for monopolistic competition

Monopolistic Competition in the Short Run

Short Run Profit
Short Run Loss

At each QQ, MR<P\textrm{MR} < P. To maximize profit, the firm produces QQ where MR=MC\textrm{MR} = \textrm{MC}, and then uses the DD curve to set PP. The area of profit or loss is the difference between PP and ATC\textrm{ATC}, times the quantity QQ produced at the profit maximizing quantity of output.

Going from Short Run to Long Run

In the short run, firms in monopolistic competition behave very similarly to monopolies.

In the long run, however, entry and exit in monopolistic competition drive economic profit to zero, similarly to perfect competition.

  • When monopolistic firms earn a profit in the short run, it compels firms to enter which provides more close substitutes and less market share for the existing firms. This leads to the demand and MR\textrm{MR} curves shifting left together so that the demand curve becomes tangent with ATC\textrm{ATC}.
  • When monopolistic firms earn a loss in the short run, it compels firms to exit which provides less close substitutes and more market share for the existing firms. This leads to the demand and MR\textrm{MR} curves shifting right together so that the demand curve becomes tangent with ATC\textrm{ATC}.

Monopolistic Competition in the Long Run

In the long run, the economies of scale portion of the ATC\textrm{ATC} curve will become tangent to the demand curve at the profit-maximizing price level. This results in a zero-profit equilibrium for firms in monopolistic competition.

Notice that the firm charges a markup of price over marginal cost and does not produce at minimum ATC\textrm{ATC}.

Why Monopolistic Competition is Less Efficient than Perfect Competition

  1. Excess capacity
    • The monopolistic competitor operates on the downward-sloping part of its ATC\textrm{ATC} curve, therefore producing less than the cost-minimizing output.
    • Under perfect competition, firms produce the quantity that minimizes ATC\textrm{ATC}.
  1. Markup over marginal cost
    • Under monopolistic competition, P>MCP > \textrm{MC}.
    • Under perfect competition, P=MCP = \textrm{MC}.

Monopolistic Competition and Welfare

  • Monopolistically competitive markets do not have all the desirable properties of perfectly competitive markets.
  • Because P>MCP > \textrm{MC}, the market quantity is below the socially efficient quantity.
  • Yet, it is not easy for policymakers to fix this problem: firms earn zero profits, so cannot require them to reduce prices.
  • Number of firms in the market may not be optimal, due to external effects from the entry of new firms:
    • The product-variety externality: surplus consumers get from the introduction of new products.
    • The business-stealing externality: losses incurred by existing firms when new firms enter market.
  • The inefficiencies of monopolistic competition are subtle and hard to measure. No easy way for policymakers to improve the market outcome.

Advertising

  • In monopolistically competitive industries, product differentiation and markup pricing lead naturally to the use of advertising.
  • In general, the more differentiated the products, the more advertising firms buy.
  • Economists disagree the social value of advertising.

Critiques of Advertising

  • Society is wasting the resources it devotes to advertising.
  • Firms advertise to manipulate people’s tastes.
  • Advertising impedes competition—it creates the perception that products are more differentiated than they really are, allowing higher markups.

Defense of Advertising

  • It provides useful information to buyers.
  • Informed buyers can more easily find and exploit price differences.
  • Thus, advertising promotes competition and reduces market power.

Advertising as a Signal of Quality

A firm’s willingness to spend huge amounts on advertising may signal the quality of its product to consumers, regardless of the content of ads.

  • Ads may convince buyers to try a product once, but the product must be of high quality for people to become repeat buyers.
  • The most expensive ads are not worthwhile unless they lead to repeat buyers.
  • When consumers see expensive ads, they think the product must be good if the company is willing to spend so much on advertising.

Brand Names

In many markets, brand name products coexist with generic ones. Firms with brand names usually spend more on advertising and charge higher prices for the products. They are able to charge more by creating a brand image.

Critique of Brand Names

  • Brand names cause consumers to perceive differences that do not really exist.
  • Consumers’ willingness to pay more for brand names is irrational, fostered by advertising.
  • Eliminating govt protection of trademarks would reduce influence of brand names, result in lower prices.

Defense of Brand Names

  • Brand names provide information about quality to consumers.
  • Companies with brand names have incentive to maintain quality, to protect the reputation of their brand names.

4.5 - Oligopoly and Game Theory

Measuring Market Concentration

  • Concentration ratio: the percentage of the market’s total output supplied by its four largest firms.
  • The higher the concentration ratio, the less competition.
  • This chapter focuses on oligopoly, a market structure with high concentration ratios.

Oligopoly

  • Oligopoly: a market structure in which only a few sellers offer similar or identical products.
  • Strategic behavior in oligopoly: +A firm’s decisions about PP or QQ can affect other firms and cause them to react. The firm will consider these reactions when making decisions.
  • Game theory: the study of how people behave in strategic situations.
  • Collusion: an agreement among firms in a market about quantities to produce or prices to charge
  • Cartel: a group of firms acting in unison

Collusion vs. Self-Interest

  • In a duopoly, both firms would be better off if both stick to the cartel agreement.
  • But each firm has an incentive to renege on the agreement.
  • Lesson: It is difficult for oligopoly firms to form cartels and honor their agreements.

The Equilibrium for an Oligopoly

  • Nash equilibrium: a situation in which economic participants interacting with one another each choose their best strategy given the strategies that all the others have chosen.

A Comparison of Market Outcomes

When firms in an oligopoly individually choose production to maximize profit,

  • Oligopoly QQ is greater than monopoly QQ but smaller than competitive QQ.
  • Oligopoly PP is greater than competitive PP but less than monopoly PP.

The Output and Price Effects

  • Increasing output has two effects on a firm’s profits:
    • Output effect: +If P>MCP > \textrm{MC}, increasing output raises profits.
    • Price effect: +Raising output increases market quantity, which reduces price and reduces profit on all units sold.
  • If output effect > price effect, then the firm increases production.
  • If price effect > output effect, then the firm reduces production.

The Size of the Oligopoly

As the number of firms in the market increases,

  • the price effect becomes smaller
  • the oligopoly looks more and more like a competitive market
  • PP approaches MC\textrm{MC}
  • the market quantity approaches the socially efficient quantity

Game Theory

  • Game theory helps us understand oligopoly and other situations where “players” interact and behave strategically.
  • Dominant strategy: a strategy that is best for a player in a game regardless of the strategies chosen by the other players.
  • Prisoners’ dilemma: a “game” between two captured criminals that illustrates why cooperation is difficult even when it is mutually beneficial.

Oligopolies as a Prisoners’ Dilemma

  • When oligopolies form a cartel in hopes of reaching the monopoly outcome, they become players in a prisoners’ dilemma.
  • Our earlier example:
    • T-Mobile and Verizon are duopolists in Smalltown.
    • The cartel outcome maximizes profits: Each firm agrees to serve Q = 30 customers.
  • Both firms will end up choosing their dominant strategy, causing a loss for both.

Other Examples of the Prisoners’ Dilemma

Ad Wars

Two firms spend millions on TV ads to steal business from each other. Each firm’s ad cancels out the effects of the other, and both firms’ profits fall by the cost of the ads.

Organization of Petroleum Exporting Countries (OPEC)

Member countries try to act like a cartel, agree to limit oil production to boost prices & profits. But agreements sometimes break down when individual countries renege.

Arms Race between Military Superpowers

Each country would be better off if both disarm, but each has a dominant strategy of arming.

Common Resources

All would be better off if everyone conserved common resources, but each person’s dominant strategy is overusing the resources.

Prisoners’ Dilemma and Society’s Welfare

  • The noncooperative oligopoly equilibrium:
    • Is bad for oligopoly firms, preventing them from achieving monopoly profits
    • Is good for society:
      • QQ is closer to the socially efficient output
      • PP is closer to MC\textrm{MC}
  • In other prisoners’ dilemmas, the inability to cooperate may reduce social welfare.
    • e.g. arms race, overuse of common resources

Why People Sometimes Cooperate

  • When the game is repeated many times, cooperation may be possible.
  • These strategies may lead to cooperation:
    • If your rival reneges in one round, you renege in all subsequent rounds.
    • “Tit-for-tat”: Whatever your rival does in one round (whether renege or cooperate), you do in the following round.

Public Policy Toward Oligopolies

  • In oligopolies, production is too low and prices are too high, relative to the social optimum.
  • Role for policymakers: +Promote competition, prevent cooperation to move the oligopoly outcome closer to the efficient outcome.

Controversies over Antitrust Policy

  • Most people agree that price-fixing agreements among competitors should be illegal.
  • Some economists are concerned that policymakers go too far when using antitrust laws to stifle business practices that are not necessarily harmful, and may have legitimate objectives.
  • We consider three such practices:
    • Resale Price Maintenance (“Fair Trade”)
      • Occurs when a manufacturer imposes lower limits on the prices retailers can charge.
      • Is often opposed because it appears to reduce competition at the retail level.
      • Yet, any market power the manufacturer has is at the wholesale level; manufacturers do not gain from restricting competition at the retail level.
      • The practice has a legitimate objective: preventing discount retailers from free-riding on the services provided by full-service retailers.
    • Predatory Pricing
      • Occurs when a firm cuts prices to prevent entry or drive a competitor out of the market, so that it can charge monopoly prices later.
      • Illegal under antitrust laws, but hard for the courts to determine when a price cut is predatory and when it is competitive & beneficial to consumers.
      • Many economists doubt that predatory pricing is a rational strategy:
        • It involves selling at a loss, which is extremely costly for the firm.
        • It can backfire.
    • Tying
      • Occurs when a manufacturer bundles two products together and sells them for one price (e.g., Microsoft including a browser with its operating system)
      • Critics argue that tying gives firms more market power by connecting weak products to strong ones.
      • Others counter that tying cannot change market power: Buyers are not willing to pay more for two goods together than for the goods separately.
      • Firms may use tying for price discrimination, which is not illegal, and which sometimes increases economic efficiency.

+

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Unit 5: Factor Markets

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5.1 - Introduction to Factor Markets

Key Vocabulary

Marginal Product

The additional product that is produced by each additional input. We calculated this number by dividing the change in the total product by the change in the number of inputs.

MP=ΔYΔX\textrm{MP} = \frac{\Delta Y}{\Delta X}

Total Revenue

This is calculated by price times the total product.

TR=Q⋅P\textrm{TR} = Q \cdot P

Marginal Revenue Product (MRP)

The change in total revenue resulting from employing an additional unit of a resource (e.g. labor).

MRP=ΔTRΔQ\textrm{MRP} = \frac{\Delta \textrm{TR}}{\Delta Q}

Marginal Resource Cost (MRC)

The additional cost incurred by hiring or employing one more unit of the resource.

MRC=Wage\textrm{MRC} = \textrm{Wage}

Hiring Labor and Other Resources

The rule for hiring labor and any other resources is that firms will continue to hire workers and resources as long as marginal revenue product (MRP) > marginal resource cost (MRC) and until marginal revenue product (MRP) = marginal resource cost (MRC). A firm will never hire when marginal resource cost (MRC) > marginal revenue product (MRP).

This is just like profit maximization, where MC=MR\textrm{MC} = \textrm{MR}. In the Factor Market, we say that MRC=MRP\textrm{MRC} = \textrm{MRP}.

5.2 - Changes in Factor Demand and Factor Supply

Changes in Factor Demand

The demand curve for a resource can shift, based on chances in the following determinants:

  • Demand for the product
  • Productivity of the resource
  • Changes in the Prices of Other Resources
    • Substitute Resources
    • Complementary Resources

Demand for the Product

Change in the demand for the good or service can cause a shift in demand in the factor market. For example, if there is an increase in the demand for pizza, then there will be greater demand for all the resources that are involved in the production of pizza, including cheese, sauce, dough, and workers. Resource demand can also change when the price of a product changes. For example, if the price of pizza decreases, then the worker who is trained to make a pizza generates a smaller MRP (because MRP = MP x price), so the demand for these workers will decrease.

Productivity of the Resource

A change in productivity can also cause a shift in demand in the factor market. Let's take a situation where a new technique is developed that cuts production time in half. Since labor productivity has increased, each worker can make a greater quantity of the goods than they used to. This leads to each worker generating a greater marginal revenue product which increases their value to the firm or business. As a result, this increases the demand for labor.

Price of Related Inputs

A change in the price of related inputs can also shift the demand in the factor market. In this determinant, we are referring to substitute resources and complementary resources that are used in the production of goods and services. If the price of one resource becomes more expensive, the firm will increase their demand for the substitute resource. For example, if the price of copper piping increases, home builders will more willing to demand plastic piping. In looking at complementary resources, we can look at the production of soft drinks. Both aluminum and sugar are used in the production of soft drinks. If the price of aluminum increases, then we would see the demand for sugar decrease since both products are used to produce soft drinks.

Graphing Change in Factor Demand

Factor Supply

Just like with the demand for resources, there are several things that can change the supply of resources. When we are looking at the supply of resources, we generally focus on workers.

Number of Qualified Workers

The number of qualified workers that are available in a particular industry is one determinant of supply in the factor market. This can be influenced by immigration, education, training, and abilities. Here are some examples of this determinant:

  • If a country enacts stricter immigration laws than that will shift the labor supply to the left because of the smaller pool of workers.
  • The number of graduates with engineering degrees soar. This would cause the labor supply to shift right.

Government Regulations and Licensing

The second determinant of the supply of workers is government regulations and +licensing. Here are some examples of this determinant:

  • If the government establishes a certification process that makes it harder to be an electrician than we would see the supply of electricians decrease shifting the labor supply curve to the left.

Change in Societal Values

The third determinant is personal values regarding leisure time and societal roles. Here are some examples of this determinant.

  • The increase in the labor force, especially women, during WWII because people saw it as a patriotic duty to help produce the goods that would help in the war efforts. This would increase the labor supply and shift the curve right.
  • Low-skill workers decide that working at minimum wage isn't worth their time. This would shift the labor supply curve to the left due to the decrease in the amount of workers.

Graphing Change in Factor Supply

5.3 - Profit-Maximizing Behavior in Perfectly Competitive Factor Markets

There are two types of factor markets. The first type is what is known as a perfectly competitive factor/resource market. There is significant use of labor as the type of factor (resource) in describing this type of factor market in AP Micro.

Characteristics of Perfectly Competitive Labor (Factor) Markets

Perfectly competitive labor (factor markets) are very similar the perfectly competitive market structure EXCEPT that we are dealing with resources instead of goods and services.

The characteristics of this type of factor (resource) market include:

  • Many, small firms hiring workers
  • Workers are "wage takers"
  • Skill level of workers is identical (i.e. workers are perfect substitutes)
  • Firms can hire as many workers as they need or want at the wage set in the market
  • Firms will hire workers as long as MRP>MRC\textrm{MRP} > \textrm{MRC}, or until MRP=MRC\textrm{MRP} = \textrm{MRC}. +MRC=wage\textrm{MRC} = \textrm{wage} in this type of factor market.

Perfectly Competitive Labor Market Graphs

In the perfectly competitive labor market, there is a downward-sloping demand curve because of the law of diminishing marginal returns. This means that each additional worker generates less revenue (MRP), and, therefore, is worth less to the firm. The supply curve for the labor market graph is upward-sloping because of the incentive to earn higher wages and greater income. If there are higher wages, it gives workers the incentive to give up leisure time and offer more of their time as workers. The same can be said for lower wages, which will deter workers from wanting to work more.

+

💡
Notice that SLS_L and DLD_L are used to describe the supply and demand of labor. It is important to use the subscript LL when you are drawing graphs.

Firm Graph in a Perfectly Competitive Labor Market

The perfect labor market firm graph looks a little different than it did in the product market. The demand for labor, otherwise known as MRP\textrm{MRP}, is downward sloping. The supply of labor, otherwise known as MRC, is perfectly elastic. This shows that workers are wage takers and that firms hire all workers at the same wage level set by the market. The quantity of labor that each individual firm will hire is where MRC=MRP\textrm{MRC} = \textrm{MRP}.

Relating the Market and Firm Graphs

When there is a change in the market graph for either labor demand or labor supply, we have to show the corresponding changes in the firm graph. For example, if the supply of labor increases, that means the equilibrium wage will decrease. This will move the MRC\textrm{MRC} curve in the firm graph down and increase the number of workers each firm will hire. The graph below illustrates this change.

Cost Minimizing Combination of Resources

During the production process, firms must be careful to choose a combination of resources that will minimize their costs. This is sometimes referred to as the Least-Cost Rule. In order for a firm to be using the combination of resources that will reduce its costs, they have to satisfy the following formula.

+

MPxPx=MPyPy\frac{\textrm{MP}_x}{P_x} = \frac{\textrm{MP}_y}{P_y}

Where MP\textrm{MP} is marginal product and PP is price. The xx and yy represent different resources.

Profit-Maximizing Combination of Resources

Another method that firms can look at when determining the combination of resources that they can use is what is known as the profit-maximizing rule for combining resources. In order to adhere to this rule, the firm must satisfy the following formula:

MRPxMRCx=MRPyMRCy=1\frac{\textrm{MRP}_x}{\textrm{MRC}_x} = \frac{\textrm{MRP}_y}{\textrm{MRC}_y} = 1

This means that the firm is hiring where MRP=MRC\textrm{MRP} = \textrm{MRC} for each resource. If they are not currently at this particular point for each resource, they can either increase or decrease the number of resources they use to satisfy this formula and the rule.

5.4 - Monopsony Markets

A monopsony market is a type of imperfectly competitive factor (resource) market where only a single firm buys resources.

Characteristics of Monopsonies

  • One, large firm hires all workers in a single labor market and is large enough to control the labor market.
  • The market is imperfectly competitive.
  • The firm is a wage maker.
  • Firms must increase wages in order to hire additional wages.
  • MRC > wage per worker (This is because when you hire an additional worker, you must pay them a higher wage than the previous worker. However, you cannot wage discriminate, so you not only have the additional cost of that worker, but also the cost of bringing all the earlier workers up to the current wage rate).
  • The firm will hire the quantity of labor where MRP=MRC\textrm{MRP} = \textrm{MRC}.
  • The firm will pay workers a wage that they are willing and able to work for below their MRP\textrm{MRP}.

Differences between Perfect Competition and Monopsony

Graphing a Monopsony Market

In a monopsony, MRC\textrm{MRC} is greater than SS, and thus appears to the left graphically. We determine the number of workers by finding where MRP=MRC\textrm{MRP} = \textrm{MRC} and then going down to the horizontal axis. We determine wage by finding MRP=MRC\textrm{MRP} = \textrm{MRC} and then going down to the supply curve and over to the vertical axis.

The competitive solution would result in a higher wage and greater employment than the monopsonistic labor market. Monopsonists maximize profits by hiring a smaller number of workers and thereby paying a less-than-competitive wage rate.

5.5 - Unions and Collective Bargaining

Unions attempt to monopolize the sale of labor, so the competitive model breaks down in this case. Union membership is only a small and declining portion of the American labor force, however.

Unionization is much less prevalent in America than it is in most other industrialized countries.

Unions as Labor Monopolies

Unions cam monopolize the supply of labor, but they are not all powerful.

Unions try to increase the demand for their labor.

  • Featherbedding: forcing management to employ more workers than they really need
  • Institute a campaign to increase worker productivity
  • Raise the demand for the company’s product
    • Flex political muscle (for example, by obtaining legislation to reduce foreign competition)
    • Appeal to the public to buy union products

Three Union Models

Demand Enhancement Model

Increase worker wages by increasing product / derived demand, increasing worker productivity, and by increasing the price of substitute labor.

Exclusive or Craft Model

Use restrictive membership policies, restrictions of the labor supply of the economy, and occupational licensing to reduce the supply of qualified laborers.

Inclusive or Industrial Model

Focuses on organizing virtually all workers and thereby controlling the supply curve for labor.

Wage Increases and Unemployment

  • Union members receive about 15-20% higher wages on average
  • Negative impact on the level of employment

Bilateral Monopoly Model in the Labor Market

Economic theory cannot determine the actual outcome of a bilateral monopoly in the labor market. The actual wage will be determined through bargaining between the labor union and the monopolist.

Minimum Wage Controversy

A minimum wage acts as a government-enforced price floor in the factor market. This has the potential to create a labor surplus, where supply for labor is greater than the demand for labor.

Additional Factor Market Concepts

The Determination of Rent

Some resources (like land) are fixed. Supply→Demand\textrm{Supply} \rarr \textrm{Demand} is the only active determinant of land rent.

Economic rent: an “extra” payment for a factor of production (such as land) that does not change the amount of the factor that is supplied.

If demand decreases, rent decreases.

Entrepreneurship and Profit

  • When economists calculate profits, they consider both explicit and implicit costs.
  • Essentially, profits are what remains from revenue after all other factors have been paid.

Risk and Profit

There are two types of risks in business, insurable risks and uninsurable risks. While most risks are insurable, risks like changes in economy, structural changes, government policy, and rival producers are examples of uninsurable risks.

Clarification of Resources

Money is NOT a resource. You cannot directly produce any goods or services with it. Money can, however, be used to fund the acquisition of productive resources.

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Unit 6: Market Failure and the Role of Government

Created
TagsIn Class

6.1 - Socially Efficient and Inefficient Market Outcomes

Definitions

  • Market Failures +A situation in which the free-market system fails to satisfy society’s wants. (When the invisible hand doesn’t work)
  • Social Efficiency +This is the optimal distribution of resources in society, taking into account all external costs and benefits as well as internal costs and benefits. Occurs where Marginal Social Benefit (MSB) = Marginal Social Costs (MSC).
  • Marginal Social Benefit (MSB) +The additional benefit received by all members of society due to the consumption of an additional unit of a good or service.
    • Social benefit=private benefit+external benefit\textrm{Social benefit} = \textrm{private benefit} + \textrm{external benefit}
  • Marginal Social Cost (MSC) +The additional cost incurred by all members of society due to the consumption of an additional unit of a good or service
    • Social cost=private cost+external cost\textrm{Social cost} = \textrm{private cost} + \textrm{external cost}
  • Pareto Improvement +When at least one individual becomes better off without anyone becoming worse off.
  • Pareto Efficiency +The idea that there is a point where it is impossible to make anyone better off without making someone worse off. Occurs when an economy is operating on a simple production possibility frontier, it is not possible to increase output of goods without reducing output of services.

Social Efficiency and the Market Equilibrium

The market equilibrium quantity is equal to the socially optimal quantity only when all social benefits and costs are internalized by individuals in the market.

Total Economic Surplus is maximized at that quantity.

Socially Efficient Point

The Socially Efficient Point occurs when you are producing at the quantity and price level where MSB=MSC\textrm{MSB} = \textrm{MSC}, where all economic surplus is maximized.

If we were to produce either at a quantity above or below the equilibrium, we would be producing at an inefficient point where we are either underproducing or overproducing the good or service. The government sometimes has to take action or make policies to correct these inefficiencies.

Factors that affect Socially Efficient Outcomes

  • Externalities
  • Market Power (Market Structure)
  • Public Goods
  • Inequality

6.2 - Externalities

One of the Ten Principles from Chapter 1: Markets are usually a good way to organize economic activity.

In the absence of market failures, the competitive market outcome is efficient and maximizes total surplus. However, there is the possibility for market failures:

One type of market failure is an externality: the uncompensated impact of one person’s actions on the wellbeing of a bystander.

Externalities can be negative or positive, depending on whether impact on bystander is adverse or beneficial.

Self-interested buyers and sellers neglect the external costs or benefit of their actions, so the market outcome is not efficient. In the presence of externalities, government and public policy can often improve outcomes.

Examples of Negative Externalities

  • Air pollution from a factory
  • The neighbor’s barking dog
  • Late-night stereo blasting from the dorm room next to yours
  • Noise pollution form construction projects
  • Health risk to others from second-hand smoke
  • Talking on cell phone while driving makes the roads less safe for others

Graph of a Negative Externality

When a firm chooses to produce a good or service, it does not take into effect the external costs to society. It just looks at its own production costs. This can create a deadweight loss in the graph, where the social cost exceeds the social benefit. The quantity produced in the free market creates a negative social benefit.

Positive Externalities

In the presence of a positive externality, the social value of a good includes:

  • private value — the direct value to buyers
  • external benefit — the value of the positive impact on bystanders

The socially optimal QQ maximizes welfare:

  • At any lower QQ, the social value of additional units exceeds their cost.
  • At any higher QQ, the cost of the last unit exceeds its social value.

You can see in the graph above that at the free market quantity (QFM), marginal social benefit (MSB) is greater than marginal private benefit (MPB). This is marked by point C on the graph above. This essentially means that society is experiencing more benefits than the firm at that quantity. Society would like the firm to produce at point B, which is where marginal social benefit (MSB) equals marginal social cost (MSC). By producing at the free market quantity, a deadweight loss is created. This is marked by the triangle ABC on the graph.

Consumers only consider the marginal private benefit (MPB) of consumption, so consumers will consume until MPB = MSC, and that is where the good will be supplied. To account for the spillover benefits, the government provides a per-unit subsidy to consumers for each unit of the good. When this is done, it makes the good less expensive to buyers which increases the demand for the good. Firms will increase the production of the good to the socially optimal quantity to meet the +increased demand.

Internalizing the Externality

Internalizing the Externality: altering incentives so that people take account of the external effects of their actions.

When market participants must pay social costs, the market equilibrium will change to meet the social optimum.

(Imposing the tax on buyers would achieve the same outcome: market QQ would equal optimal QQ).

Effects of Externalities: Summary

If negative externality:

  • market quantity larger than socially desirable

If positive externality:

  • market quantity smaller than socially desirable

To remedy the problem:

  • tax goods with negative externalities
  • subsidize goods with positive externalities

Public Policies Toward Externalities

  • Command and Control Policies regulate behavior directly. Examples:
    • limits on quantity of pollution emitted
    • requirements that firms adopt a particular technology to reduce emissions
  • Market-based policies provide incentives so that private decision-makers will choose to solve the problem on their own. Examples:
    • corrective taxes and subsidies
    • tradable pollution permits

Corrective Taxes and Subsidies

  • Corrective Tax: a tax designed to induce private decision-makers to take account of the social costs that arise from a negative externality
  • Also called Pigouvian taxes after Arthur Pigou (1877-1959).
  • The ideal corrective tax = external cost
  • For activities with positive externalities, ideal corrective subsidy = external benefit
  • Other taxes and subsidies distort incentives and move economy away from the social optimum.
  • Corrective taxes & subsidies
    • align private incentives with society’s interests
  • Different firms have different costs of pollution ab abatement.
  • Efficient outcome: Firms with the lowest abatement costs reduce pollution the most.
  • A pollution tax is efficient:
    • Firms with low abatement costs will reduce pollution to reduce their tax burden.
    • Firms with high abatement costs have greater willingness to pay tax.

Corrective Taxes vs. Regulations

Corrective taxes are better for the environment:

  • The corrective tax gives firms incentive to continue reducing pollution as long as the costs of doing so is less than the tax.
  • If a cleaner technology becomes available, the tax gives firms incentive to adopt it.
  • In contrast, firms have no incentive for further reduction beyond the level specified in a regulation.

Private Solutions to Externalities

  • The Coase Theorem +If private parties can costlessly bargain over the allocation of resources, they can solve the externalities problem on their own.

Why Private Solutions Do Not Always Work

  • Transaction Costs +The costs parties incur in the process of agreeing to and following through on a bargain. These costs may make it impossible to reach a mutually beneficial agreement.
  • Stubbornness +Even if a beneficial agreement is possible, each party may hold out for a better deal.
  • Coordination problems +

6.3 - Public and Private Goods

Public and Common Goods

We consume many goods without paying, such as parks, national defense, clean air and water. When goods have no prices, the market forces that normally allocate resources are absent. Therefore, the private market may fail to provide the socially efficient quantity of these goods.

Important Characteristics of Goods

  • A good is excludable if a person can be prevented from using it.
    • Excludable: fish tacos, wireless Internet access
    • Not excludable: FM radio signals, national defense
  • A good is rival in consumption if one person’s use of it diminishes another person’s use.
    • Rival: fish tacos
    • Not rival: An MP3 file of Kanye West’s donda

Different Kinds of Goods

  • Private goods: goods which are both excludable and rival in consumption
    • Example: food
  • Public goods: goods which are not excludable, and not rival in consumption
    • Example: national defense
  • Common resources: goods which are rival in consumption, but not excludable
    • Example: fish in the ocean
  • Club goods: goods which are excludable but not rival in consumption
    • Example: cable TV

Public Goods

  • Public goods are difficult for private markets to provide because of the free-rider problem.
  • Free rider: a person who receives the benefit of a good but avoids paying for it.
    • If a good is not excludable, people have incentive to be free riders, because firms cannot prevent non-payers from consuming the good.
  • Result: The good is not produced, even if buyers collectively value the good higher than the cost of providing it.
  • If the benefit of a public goods exceeds the cost of providing it, the government should provide the good and pay for it with a tax on people who benefit.
  • Problem: Measuring the benefit is usually difficult
  • Cost-benefit analysis: a study that compares the costs and benefits of providing a public good.
  • Cost-benefit analyses are imprecise, so the efficient provision of public goods is more difficult than that of private goods.
  • Examples of important public goods:
    • National defense
    • Knowledge created through basic research
    • Fighting poverty

Common Resources

  • Like public goods, common resources are not excludable
    • Cannot prevent free-riders from using them
    • Little incentive for private firms to produce
    • Role for government: seeing that they are provided
  • Additional problem with common resources: rival in consumption
    • Each person’s use reduces others’ ability to use
    • Role for government: ensuring they are not overused

The Tragedy of the Commons

  • A parable that illustrates why common resources get more use than is socially desirable.
  • Setting: a medieval town where sheep graze on common land
  • As the population of the town grows, so does the number of sheep in the town
  • The amount of land is fixed, the grass begins to disappear from overgrazing
  • The private incentives (using the land for free) outweigh the social incentives (using it carefully)
  • Result: People can no longer raise sheep
  • The tragedy is due to an externality: +Allowing one’s flock to graze on the common land reduces its quality for other families.
  • People neglect this external cost, resulting in overuse of the land

Policy Options to Prevent Overconsumption

  • Regulate the resource
  • Impose a corrective tax to internalize the externality
    • Example: hunting and fishing licenses, entrance fees for congested national parks
  • Auction off permits allowing use of the resource
    • Example: Spectrum auctions by the U.S. FCC
  • If the resource is land, convert to a private good by dividing and selling parcels to individuals

Conclusion

  • Public goods tend to be under-provided, while common resources tend to be over-consumed.
  • These problems arise because property rights are not well-established:
    • Nobody owns the air, so no one can charge polluters. Result: too much pollution
    • Nobody can charge people who benefit from national defense. Result: too little defense
  • The government can potentially solve these problems with appropriate policies.

6.4 - The Effects of Government Intervention in Different Market Structures

Perfect Competition

In a perfectly competitive market, a tax will shift the supply curve to the left, resulting in a new higher price, a lower quantity demanded, and an associated deadweight loss:

A subsidy in a perfectly competitive market will shift the supply curve to the right, resulting in a new lower price, a higher quantity demanded, and an associated deadweight loss:

Natural Monopoly

A natural monopoly is a market where the most efficient number of firms in the industry is only one. This is often due to high start-up costs. An example of a natural monopoly would be an electric company; it is more efficient for 1 firm to provide power to an entire city rather than having multiple firms with overlapping power grids.

The graph above shows a natural monopoly. Point A is where a monopoly would produce when they are unregulated by the government. Point B represents the fair-return point, where the monopoly would earn a normal economic profit or break even. Point C represents the perfectly competitive or socially optimal point on a monopoly graph. There is no deadweight loss at Point C.

The government can set a price ceiling that can cause a natural monopoly to produce the socially optimal output. The monopoly will need a lump-sum subsidy to produce here.

Per-unit taxes and subsidies affect variable costs, thus affecting the MC\textrm{MC}, ATC\textrm{ATC}, and AVC\textrm{AVC} curves, while lump-sum taxes and subsidies only affect fixed costs, thus affecting only the ATC\textrm{ATC} and AFC\textrm{AFC} curves.

A lump-sum tax will cause an upward shift in the ATC\textrm{ATC} and AFC\textrm{AFC} curves, while a lump-sum subsidy will cause a downward shift in the ATC\textrm{ATC} and AFC\textrm{AFC} curves.

A per-unit tax will cause an upward shift in the ATC\textrm{ATC} and AVC\textrm{AVC} curves, as well as a leftward shift in the MC\textrm{MC} curve. A per-unit subsidy will cause a downward shift in the ATC\textrm{ATC} and AVC\textrm{AVC} curves, as well as a rightward shift in the MC\textrm{MC} curve.

Per-Unit Subsidy - Monopoly
Per-Unit Tax - Monopoly

6.5 - Inequality

There are two types of economic inequality: (1) income inequality, and (2) wealth inequality. Income inequality looks at how annual earnings are distributed and wealth inequality looks at how assets are distributed.

There are several sources of these two types of inequality:

  1. Tax Structure
  1. Human Capital (training and skills)
  1. Social Capital
  1. Inheritance
  1. Effects of Discrimination
  1. Access to Financial Markets
  1. Mobility
  1. Bargaining Power within Economic and Social Units

Income Inequality

Governments use simulation to measure the income inequality level. A graph known as the Lorenz Curve can be used to graphically represent income inequality.

The Lorenz curve shows the actual income distribution in a society. The larger the gap between perfect equality and the Lorenz curve, the greater the amount of income inequality that exists.

The Gini Coefficient is a numerical measurement of income inequality. It is a statistical measurement of income equality where perfect equality is 0 and perfect inequality is 1. The government can help with income inequality by either increasing the amount it taxes wealthier citizens or by increasing transfer payments to the poor. Transfer payments are government payments to individuals or businesses designed to meet a specific objective rather than pay for goods or resources (Ex: welfare).

Types of Taxes

There are several types of taxes that can contribute to or help with income inequality:

  • Progressive taxes take a larger percentage of income from high-income groups than from +low-income groups.
    • The most common example of a progressive tax is the Federal Income Tax System in the United States. This system divides people into tax brackets based on their income level, and taxes people increasing amounts as income increases. To demonstrate, someone making between $0 and $14,100 this year is taxed 10% of that income, while someone making between $14,101 and $53,700 this year is taxed 12% of that income.
  • Regressive taxes take a larger percentage from low-income groups than from high-income +groups.
    • The most common example of a regressive tax is sales tax in the United States. Let's say the sales tax is 5% in a state, and Sally and Bob both spend $200 in sales taxes. Sally only makes $10,000 a year, so this $200 takes a larger percentage of her income than Bob, who makes $100,000 a year. Although this sales tax is the same for everyone, someone in a low-income group will feel the burden of this tax more than someone in a high-income group.
  • Proportional taxes takes the same percentage of income from all income groups.
    • For example, if there was a 20% flat income tax on all income groups, each income group is equally affected by the tax.

+

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Notes

NameCreatedTags
Unit 30: Induction and InductanceTextbook
Unit 29: Magnetic Fields Due to CurrentsTextbook
Unit 28: Magnetic FieldsTextbook
Unit 27: CircuitsTextbook
Unit 26: Current and ResistanceTextbook
Unit 25: CapacitanceTextbook
Unit 24: Electric PotentialTextbook
Unit 23: Gauss’ LawTextbook
Unit 22: Electric FieldsTextbook
Unit 21: Coulomb’s LawTextbook
Unit 15: Oscillations
Unit 13: GravitationTextbook
Unit 11: Rolling, Torque, and Angular MomentumTextbook
Unit 10: Rotation
Unit 9: Center of Mass and Linear MomentumTextbook
Unit 8: Potential Energy and Conservation of EnergyTextbook
Unit 7: Kinetic Energy and WorkTextbook
Unit 6: Force and Motion - IITextbook
Unit 5: Force and Motion - I
Unit 4: Motion in Two and Three DimensionsTextbook
Unit 3: VectorsTextbook
Video 2: Now with CalculusVideo
Video 1: MotionVideo
\ No newline at end of file diff --git a/notes/AP Physics C/Notes 7f9dac375011471f8386ef7d43d664b0/Unit 10 Rotation c40c3eba937b45578ff8c3875b58cfe8.html b/notes/AP Physics C/Notes 7f9dac375011471f8386ef7d43d664b0/Unit 10 Rotation c40c3eba937b45578ff8c3875b58cfe8.html new file mode 100644 index 0000000..c3c38c1 --- /dev/null +++ b/notes/AP Physics C/Notes 7f9dac375011471f8386ef7d43d664b0/Unit 10 Rotation c40c3eba937b45578ff8c3875b58cfe8.html @@ -0,0 +1,747 @@ +Unit 10: Rotation

Unit 10: Rotation

10.1 - Rotational Variables

To describe the rotation of a rigid body about a fixed axis, called the rotation axis, we assume a reference line is fixed in the body, perpendicular to that axis and rotating with the body. We measure the angular position θ\theta of this line relative to a fixed direction. When θ\theta is measured in radians,

θ=sr\theta = \frac{s}{r}

where ss is the arc length of a circular path of radius rr and angle θ\theta.

Radians are related to angular measurements by:

1 rev=360°=2π rad1 \textrm{ rev} = 360\degree = 2 \pi \textrm{ rad}

Angular Displacement

A body that rotates about a rotation axis, changing its angular position from θ1\theta_1 to θ2\theta_2 , undergoes an angular displacement:

Δθ=θ2−θ1\Delta \theta = \theta_2 - \theta_1

where Δθ\Delta \theta is positive for counterclockwise rotation and negative for clockwise rotation.

Angular Velocity

If a body rotates through an angular displacement Δθ\Delta \theta in a time interval Δt\Delta t, its average angular velocity ωavg\omega_{avg} is:

ωavg=ΔθΔt\omega_{avg} = \frac{\Delta \theta}{\Delta t}

The instantaneous rate of change (angular velocity) can thus be represented as:

ω=dθdt\omega = \frac{d\theta}{dt}

Bothωavg\omega_{avg} and ω\omega  are vectors, with directions given by a right-hand rule. They are positive for counterclockwise rotation and negative for clockwise rotation. The magnitude of the body’s angular velocity is the angular speed.

Angular Acceleration

If the angular velocity of a body changes from ω1\omega_1 to ω2\omega_2 in a time interval Δt\Delta t, then the average angular acceleration αavg\alpha_{avg} of the body is

αavg=ΔωΔt\alpha_{avg} = \frac{\Delta \omega}{\Delta t}

The instantaneous rate of change (angular acceleration) can thus be represented as:

α=dωdt\alpha = \frac{d\omega}{dt}

Both αavg\alpha_{avg} and α\alpha are vectors.

10.2 - Constant Angular Acceleration

There are a bunch of kinematic equations for rotational motion, which can all be found here :)

ω=ω0+αt\omega = \omega_0 + \alpha t
θ−θ0=ω0t+12αt2\theta - \theta_0 = \omega_0t + \frac{1}{2}\alpha t^2
ω2=ω02+2α(θ−θ0)\omega^2 = \omega_0^2 + 2\alpha (\theta - \theta_0)
θ−θ0=12(ω0+ω)t\theta - \theta_0 = \frac{1}{2}(\omega_0 + \omega)t
θ−θ0=ωt−12αt2\theta - \theta_0 = \omega t - \frac{1}{2}\alpha t^2

10.3 - Relating the Linear and Angular Variables

Arc Length

A point in a rigid rotating body, at a perpendicular distance rr from the rotation axis, moves in a circle with radius rr. If the body rotates through an angle θ\theta, the point moves along an arc with length ss given by

s=θrs = \theta r

where θ\theta is in radians.

Linear Velocity

The linear velocity v⃗\vec v of the point is tangent to the circle; the point's linear speed vv is given by

v=ωrv = \omega r

where ω\omega is the angular speed (in radians per second) of the body, and thus also the point.

Linear Acceleration

The linear acceleration a⃗\vec a of the point has both tangential and radial components. The tangential component is

at=αra_t = \alpha r

where α\alpha is the magnitude of the angular acceleration (in radians per second-squared) of the body. The radial component of a⃗\vec a is

ar=v2r=ω2ra_r = \frac{v^2}{r} = \omega^2r

Period

If the point moves in uniform circular motion, the period TT of the motion for the point and the body is

T=2πrv=2πωT = \frac{2\pi r}{v} = \frac{2\pi}{\omega}

10.4 - Kinetic Energy of Rotation

Kinetic Energy of Rotation

The kinetic energy KK of a rigid body rotating around a fixed axis is given by

K=12Iω2K = \frac{1}{2}I\omega^2

in which II is the rotational inertia of the body, defined as

I=∑miri2I = \sum m_ir_i^2

for a system of discrete particles.

Rotational Inertia

We call the quantity II the rotational inertia (or moment of inertia) I of the body with respect to the axis of rotation. It is a constant for a particular rigid body and a particular rotation axis.

⛔
The axis of rotation must be specified if the value of II is to be meaningful.

10.5 - Calculating the Rotational Inertia

The rotational inertia of a body II can be defined as

I=∑miri2I = \sum m_ir_i^2

for a system of discrete particles, or as

I=∫r2 dmI = \int r^2 \space dm

for a body with a continuously distributed mass. The rr and rir_i in these expressions represent the perpendicular distance from the axis of rotation to each mass element in the body, and the integration is carried out over the entire body so as to include every mass element.

Parallel Axis Theorem

The parallel-axis theorem relates the rotational inertia II of a body about any axis to that of the same body about a parallel axis through the center of mass:

I=Icom+Mh2I = I_{com} + Mh^2

Here, hh is the perpendicular distance between the two axes, and IcomI_{com} is the rotational inertia about the axis through the com. We can describe hh as being the distance the actual rotation axis has been shifted from the rotation axis through the com.

Common Rotational Inertias

10.6 - Torque

Torque is a turning or twisting action on a body about a rotation axis due to a force F⃗\vec F. If F⃗\vec F is exerted at a point given by the position vector r⃗\vec r relative to the axis, then the magnitude of the torque is

τ=rFt=r⊥F=rFsin⁡ϕ\tau = rF_t = r_\perp F = rF \sin \phi

where FtF_t is the component of F⃗\vec F perpendicular to r⃗\vec r and ϕ\phi is the angle between r⃗\vec r and F⃗\vec F. The quantity r⊥r_\perp is the perpendicular distance between the rotation axis and an extended line running through the F⃗\vec F vector. This line is called the line of action of F⃗\vec F, and r⊥r_\perp is called the moment arm of F⃗\vec F. Similarly, rr is the moment arm of FtF_t.

The SI unit of torque is the newton-meter (N*m). A torque τ\tau is positive if it tends to rotate a body at rest counterclockwise and negative if it tends to rotate the body clockwise.

10.7 - Newton's Second Law for Rotation

The rotational equivalent of Newton's second law is:

τnet=Iα\tau_{net} = I\alpha

where τnet\tau_{net} is the net torque acting on a particle or rigid body, II is the rotational inertia of the particle or body about the rotation axis, and α\alpha is the resulting angular acceleration about that axis.

10.8 - Work and Rotational Kinetic Energy

The equations used for calculating work and power in rotational motion correspond to equations used for translational motion and are:

W=∫θiθfτ dθW = \int_{\theta_i}^{\theta_f} \tau \space d\theta
P=dWdt=τωP = \frac{dW}{dt} = \tau\omega

Constant Torque

When you have a constant τ\tau, the integral reduces to:

W=τ(θf−θi)W = \tau(\theta_f - \theta _i)

Work—Kinetic Energy Theorem for Rotating Bodies

The form of the work-kinetic energy theorem used for rotating bodies is:

ΔK=Kf−Ki=12Iωf2−12Iωi2=W\Delta K = K_f - K_i = \frac{1}{2}I\omega_f^2 - \frac{1}{2}I\omega_i^2 = W
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Unit 11: Rolling, Torque, and Angular Momentum

11.1 - Rolling as Translation and Rotation Combined

For a wheel of radius RR rolling smoothly,

vcom=ωRv_{com} = \omega R

where vcomv_{com} is the linear speed of the wheel's center of mass (moving forward), and ω\omega is the angular speed of the wheel about its center.

Rolling as Pure Rotation

There is another way to look at the rolling motion of a wheel, as pure rotation about an axis which is always at the base of the wheel (where it touches the surface it is rolling on). Rolling motion can be treated as pure rotation passing about that axis, as shown by point PP in the diagram to the right.

11.2 - Forces and Kinetic Energy of Rolling

A smoothly rolling wheel has kinetic energy defined as:

K=12Icomω2+12Mvcom2K = \frac{1}{2} I_{com} \omega^2 + \frac{1}{2} M v_{com}^2

where IcomI_{com} is the rotational inertia of the wheel about its center of mass and MM is the mass of the wheel.

The first term in this equation represents the kinetic energy of the actual rotation of the wheel, while the second term is the linear velocity of the wheel rolling forward.

Linear Acceleration

If the wheel is being accelerated but is still rolling smoothly, the acceleration of the center of mass a⃗com\vec a_{com} is related to the angular acceleration α\alpha about the center with:

acom=αRa_{com} = \alpha R

Rolling Down a Ramp

If a wheel is rolling smoothly down a ramp of angle θ\theta, its acceleration along an xx axis extending up the ramp is:

acom,x=−gsin⁡θ1+IcomMR2a_{com, x} = -\frac{g \sin \theta}{1 + \frac{I_{com}}{MR^2}}

11.3 - The Yo-Yo

A yo-yo, which travels vertically up or down a string, can be treated as a wheel rolling along an inclined plane at angle θ=90°\theta = 90\degree.

11.4 - Torque Revisited

In three dimensions, torque τ⃗\vec \tau is a vector quantity defined relative to a fixed point (usually an origin); it is:

τ⃗=r⃗×F⃗\vec \tau = \vec r \times \vec F

where F⃗\vec F is a force applied to a particle and r⃗\vec r is a position vector locating the particle relative to the fixed point.

11.5 - Angular Momentum

The angular momentum l⃗\vec l of a particle with linear momentum p⃗\vec p, mass mm, and linear velocity v⃗\vec v is a vector quantity defined relative to a fixed point (usually an origin) as:

l⃗=r⃗×p⃗=m(r⃗×v⃗)\vec l = \vec r \times \vec p = m(\vec r \times \vec v)

The magnitude of l⃗\vec l is given by:

l=rmvsin⁡ϕl = rmv \sin \phi

The direction of l⃗\vec l is given by the right-hand rule: Position your right hand so that the fingers are in the direction of r⃗\vec r .Then rotate them around the palm to be in the direction of p⃗\vec p. Your outstretched thumb gives the direction of l⃗\vec l.

11.6 - Newton's Law in Angular Form

Newton's Second Law can be written in angular form as:

τ⃗net=dl⃗dt\vec \tau_{net} = \frac{d \vec l}{dt}

where τ⃗net\vec \tau_{net} is the net torque acting on the particle and l⃗\vec l is the angular momentum of the particle.

11.7 - Angular Momentum of a Rigid Body

The angular momentum, L⃗\vec L of a system of particles is the vector sum of the angular momenta of the individual particles:

L⃗=l⃗1+l⃗2+l⃗3+⋯+l⃗n=∑i=1nl⃗i\vec L = \vec l_1 + \vec l_2 + \vec l_3 + \dots + \vec l_n = \sum_{i = 1}^n \vec l_i

Effect of Torque on a System of Particles

The time rate of change of this angular momentum is equal to the net external torque on the system (the vector sum of the torques due to interactions of the particles of the system with particles external to the system):

τ⃗net=dL⃗dt\vec \tau_{net} = \frac{d\vec L}{dt}

Angular Momentum of a Rigid Body Rotating About a Fixed Axis

For a rigid body rotating about a fixed axis, the component of its angular momentum parallel to the rotation axis is:

L=IωL = I\omega

11.8 - Conservation of Angular Momentum

The angular momentum L⃗\vec L of a system remains constant if the net external torque acting on the system is zero. Therefore,

L⃗i=L⃗f\vec L_i = \vec L_f

in an isolated system. This is the law of conservation of angular momentum.

11.9 - Precession of a Gyroscope

A spinning gyroscope can precess about a vertical axis through its support at the rate:

Ω=MgrIω\Omega = \frac{Mgr}{I\omega}

Where MM is the gyroscope's mass, rr is the moment arm, II is the rotational inertia, and ω\omega is the spin rate.

Explanation

In this scenario, the torque causing the downward rotation (causing the gyroscope to fall) changes the angular momentum L⃗\vec L of the gyroscope, and is caused by the downward force of gravity Mg⃗M\vec g.

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Unit 13: Gravitation

13.1 - Newton's Law of Gravitation

Any particle in the universe attracts any other particle with a gravitational force whose magnitude is:

F=Gm1m2r2F = G\frac{m_1m_2}{r^2}

where m1m_1 and m2m_2 are the masses of the particles, rr is their separation, and GG is the gravitational constant. With standard SI units (N, m, and kg), the gravitational constant is 6.67×10−11N⋅m2/kg26.67\times10^{-11} N \cdot m^2 / kg^2.

Shell Theorem

Although Newton’s law of gravitation applies strictly to particles, we can also apply it to real objects as long as the sizes of the objects are small relative to the distance between them. The Moon and Earth are far enough apart so that, to a good approximation, we can treat them both as particles—but what about an apple and Earth? From the point of view of the apple, the broad and level Earth, stretching out to the horizon beneath the apple, certainly does not look like a particle.

Newton solved the apple–Earth problem with the shell theorem:

⭐
A uniform spherical shell of matter attracts a particle that is outside the shell as if all the shell’s mass were concentrated at its center.

13.2 - Gravitation and the Principle of Superposition

Gravitational forces obey the principle of superposition; that is, if nn particles interact, the net force F⃗1,net\vec F_{1,net} on a particle labeled particle 1 is the sum of the forces on it from all the other particles taken one at a time:

F⃗1,net=∑i=2nF⃗1i\vec F_{1, net} = \sum_{i=2}^{n} \vec F_{1i}

in which the sum is a vector sum of the forces on particle 1 from particles 2, 3, . . . , nn.

💡
This is a very fancy way of saying that you can add all the forces to get a net force.

The gravitational force on a particle from an extended body is found by first dividing the body into units of differential mass dmdm, each of which produces a differential force dF⃗d \vec F on the particle, and then integrating over all those units to find the sum of those forces:

F⃗1=∫dF⃗\vec F_1 = \int d \vec F

13.3 - Gravitation Near Earth's Surface

The gravitational acceleration aga_g of a particle (of mass mm) is due solely to the gravitational force acting on it. When the particle is at distance rr from the center of a uniform, spherical body of mass MM, the magnitude FF of the gravitational force on the particle can be found using Newton's Law of Gravitation. Thus, by Newton’s second law,

Fg=magF_g = ma_g

and thus,

ag=GMr2a_g = \frac{GM}{r^2}

Because Earth’s mass is not distributed uniformly, because the planet is not perfectly spherical, and because it rotates, the actual free-fall acceleration g⃗\vec g of a particle near Earth differs slightly from the gravitational acceleration a⃗g\vec a_g, and the particle’s weight (equal to mgmg) differs (very slightly) from the magnitude of the gravitational force on it.

13.4 - Gravitation Inside Earth

A uniform shell of matter exerts no net gravitational force on +a particle located inside it. Therefore, an object at the center of the earth would experience no net gravitational force, as all parts of the earth would be pulling on it equally.

The gravitational force on a particle inside a uniform solid sphere, at a distance rr from the center, is due only to mass MinsM_{ins} in an “inside sphere” with that radius rr:

Mins=43πr3ρ=MR3r3M_{ins} = \frac{4}{3} \pi r^3 \rho = \frac{M}{R^3}r^3

where ρ\rho is the solid sphere's density, RR is its radius, and MM is its mass. We can assign this inside mass to be that of a particle at the center of the solid sphere and then apply Newton’s law of gravitation for particles. We find that the magnitude of the force acting on mass mm is:

F=GmMR3rF = \frac{GmM}{R^3}r

13.5 - Gravitational Potential Energy

The gravitational potential energy U(r)U(r) of a system of two particles, with masses MM and mm and separated by a distance rr, is the negative of the work that would be done by the gravitational force of either particle acting on the other if the separation between the particles were changed from infinite (very large) to rr. This energy is the gravitational potential energy:

U=−GMmrU = -\frac{GMm}{r}

Systems of Several Particles

If a system contains more than two particles, its total gravitational potential energy UU is the sum of the terms representing the potential energies of all the pairs. Here's an example for three particles, with masses m1m_1, m2m_2, and m3m_3:

U=−(Gm1m2r12+Gm1m3r13+Gm2m3r23)U = -(\frac{Gm_1m_2}{r_{12}}+\frac{Gm_1m_3}{r_{13}}+\frac{Gm_2m_3}{r_{23}})

Escaping the Gravitational Pull

An object will escape the gravitational pull of an astronomical body of mass MM and radius RR (that is, it will reach an infinite distance) if the object’s speed near the body’s surface is at least equal to the escape speed, given by:

v=2GMRv = \sqrt{\frac{2GM}{R}}

Common Escape Speeds

13.6 - Planets and Satellites: Kepler's Laws

The motion of satellites, both natural and artificial, is governed by Kepler's laws:

  1. The Law of Orbits: All planets move in elliptical orbits with the Sun at one focus.
  1. The Law of Areas: A line joining any planet to the Sun sweeps out equal areas in equal time intervals. (This statement is equivalent to conservation of angular momentum.)
  1. The Law of Periods: The square of the period TT of any planet is proportional to the cube of the semimajor axis aa of its orbit. For circular orbits with radius rr:
    T2=(4π2GM)r3T^2 = (\frac{4\pi^2}{GM})r^3

    where MM is the mass of the attracting body—the Sun in the case of the solar system. For elliptical planetary orbits, the semimajor axis aa is substituted for rr.

13.7 - Satellites: Orbits and Energy

When a planet or satellite with mass m moves in a circular +orbit with radius rr, its potential energy UU and kinetic energy KK +are given by

U=−GMmr and K=GMm2rU = -\frac{GMm}{r} \textrm{ and } K = \frac{GMm}{2r}

The mechanical energy E=K+UE = K + U is then

E=−Gmm2rE = -\frac{Gmm}{2r}

For an elliptical orbit of semimajor axis aa,

E=−GMm2aE = -\frac{GMm}{2a}

13.8 - Einstein and Gravitation

Einstein pointed out that gravitation and acceleration are equivalent. This principle of equivalence led him to a theory of gravitation (the general theory of relativity) that explains gravitational effects in terms of a curvature of space.

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Unit 15: Oscillations

15.1 - Simple Harmonic Motion

Important Stuff to Know

  • The frequency, ff, of periodic, or oscillatory, motion is the number of oscillations per second. In SI units, it is measured in hertz: 1 Hz=1 s−11 \textrm{ Hz} = 1 \textrm{ s}^{-1}.
  • The period, TT, is the time required for one complete oscillation, or cycle. It is related to the frequency by T=1/fT = 1/f.
  • In simple harmonic motion (SHM), the displacement x(t)x(t) of a particle from its equilibrium position is described by the equation:
    x=xmcos⁡(ωt+ϕ)x = x_m \cos(\omega t + \phi)

    in which xmx_m is the amplitude of the displacement, ωt+ϕ\omega t + \phi is the phase of the motion, and ϕ\phi is the phase constant. The angular frequency ω\omega is related to the period and the frequency of motion by ω=2π/T=2πf\omega = 2 \pi/T = 2\pi f.

  • Differentiating x(t)x(t) leads to equations for the particle's SHM velocity and acceleration as functions of time:
    v(t)=−ωxmsin⁡(ωt+ϕ)v(t) = -\omega x_m \sin(\omega t + \phi)
    a(t)=−ω2xmcos⁡(ωt+ϕ)a(t) = -\omega^2 x_m \cos(\omega t + \phi)
    a(t)=−ω2x(t)a(t) = -\omega^2 x(t)

    In the velocity function, the positive quantity ωxm\omega x_m is the velocity amplitude vmv_m. In the acceleration function, the positive quantity ω2xm\omega^2x_m is the acceleration magnitude ama_m.

  • A particle with mass mm that moves under the influence of a Hooke's law restoring force given by F=−kxF = -kx is a linear simple harmonic oscillator with:
    ω=km(angular frequency)\omega = \sqrt{\frac{k}{m}} \hspace{1.5em} \small{\textrm{(angular frequency)}}
    T=2πmk(period)T = 2\pi\sqrt{\frac{m}{k}} \hspace{1.5em} \small{\textrm{(period)}}

15.2 - Energy in Simple Harmonic Motion

A particle in simple harmonic motion has, at any time, kinetic energy K=12mv2K = \frac{1}{2} m v^2 and potential energy U=12kx2U = \frac{1}{2} kx^2. If no friction is present, the mechanical energy E=K+UE = K + U remains constant even though KK and UUchange.

Total Energy

E=U+K=12kxm2E = U + K = \frac{1}{2} kx^2_m

15.3 - An Angular Simple Harmonic Oscillator

A torsion pendulum consists of an object suspended by a wire. When the wire is twisted and released, the object oscillates in angular simple harmonic motion with a period given by:

T=2πIκT = 2\pi \sqrt{\frac{I}{\kappa}}

where II is the rotational inertia of the spinning object about the axis of rotation and κ\kappa is the torsion constant of the wire.

15.4 - Pendulums, Circular Motion

Simple Pendulum

A simple pendulum consists of a rod of negligible mass that pivots about its upper end, with a particle (the bob) attached at its lower end. If the rod swings through only small angles, its motion is approximately simple harmonic motion with a period given by:

T=2πImgLT = 2\pi \sqrt{\frac{I}{mgL}}

where II is the particle's rotational inertia about the pivot, mm is the particle's mass, and LL is the rod's length.

Physical Pendulum

A physical pendulum has a more complicated distribution of mass. For small angles of swinging, its motion is simple harmonic motion with a period given by:

T=2πImghT = 2\pi \sqrt{\frac{I}{mgh}}

where II is the pendulum's rotational inertia about the pivot, mm is the pendulum's mass, and hh is the distance between the pivot and the pendulum's center of mass.

Relating SHM and UCM

Simple harmonic motion corresponds to the projection of uniform circular motion onto a diameter of the circle.

15.5 - Damped Simple Harmonic Motion

The mechanical energy EE in a real oscillating system decreases during the oscillations because external forces, such as a drag force, inhibit the oscillations and transfer mechanical energy to thermal energy. The real oscillator and its motion are then said to be damped.

Equations

If the dampening force is given by F⃗d=−bv⃗\vec F_d = -b\vec v, where v⃗\vec v is the velocity of the oscillator and bb is a dampening constant, then the displacement of the oscillator is given by:

x(t)=xme−bt/2mcos⁡(ω′t+ϕ)x(t) = x_m e^{-bt/2m} \cos(\omega't + \phi)

where ω′\omega', the angular frequency of the dampened oscillator, is given by:

ω′=km−b24m2\omega' = \sqrt{\frac{k}{m} - \frac{b^2}{4m^2}}

If the dampening constant is small (b≪kmb \ll \sqrt{km}), then ω′≈ω\omega' \approx \omega, where ω\omega is the angular frequency of the undampened oscillator. For a small bb, the mechanical energy EE of the oscillator is given by:

E(t)≈12kxm2e−bt/mE(t) \approx \frac{1}{2} kx^2_m e^{-bt/m}

15.6 - Forced Oscillations and Resonance

  • If an external driving force with angular frequency ωd\omega_d acts on an oscillating system with natural angular frequency ω\omega, the system oscillates with angular frequency ωd\omega_d.
  • The velocity amplitude vmv_m of the system is greatest when ωd=ω\omega_d = \omega, a condition called resonance. The amplitude xmx_m of the system is (approximately) greatest under the same condition.

+

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Unit 21: Coulomb’s Law

21.1 - Coulomb’s Law

The strength of a particle’s electrical interaction with objects around it depends on its electric charge (usually represented as qq), which can be either positive or negative. Particles with the same sign of charge repel each other, and particles with opposite signs of charge attract each other.

An object with equal amounts of the two kinds of charge is electrically neutral, whereas one with an imbalance is electrically charged and has an excess charge.

Conductors are materials in which a significant number of electrons are free to move. The charged particles in nonconductors (insulators) are not free to move.

Electric current (ii) is the rate dq/dtdq/dt at which charge passes a point:

i=dqdti = \frac{dq}{dt}

Coulomb’s law describes the electrostatic force (or electric force) between two charged particles. If the particles have charges q1q_1 and q2q_2, are separated by distance rr, and are at rest (or moving only slowly) relative to each other, then the magnitude of the force acting on each due to the other is given by:

F=14πϵ0∣q1∣∣q2∣r2F = \frac{1}{4\pi\epsilon_0} \frac{|q_1||q_2|}{r^2}

where ϵ0=8.85×10−12 C2/N⋅m2\epsilon_0 = 8.85 \times 10^{-12} \space C^2/N \cdot m^2 is the permittivity constant. The ratio 1/4πϵ01/4\pi\epsilon_0 is often replaced with the electrostatic constant (or Colulomb’s constant) k=8.99×109N⋅m2/C2k = 8.99 \times 10^9 N \cdot m^2/C^2.

Units

The coulomb, the SI unit of charge, is derived from the SI unit ampere for electric current.

1 C=(1 A)(1 s)1 \space \textrm{C} = (1\space\textrm{A})(1\space\textrm{s})

The electrostatic force vector acting on a charged particle due to a second charged particle is either directly toward the second particle (opposite signs of charge) or directly away from it (same sign of charge).

If multiple electrostatic forces act on a particle, the net force is the vector sum (not scalar sum) of the individual forces.

Shell Theorems

  • Shell Theorem 1: A charged particle outside a shell with charge uniformly distributed on its surface is attracted or repelled as if the shell's charge were concentrated as a particle at its center.
  • Shell Theorem 2: A charged particle inside a shell with charge uniformly distributed on its surface has no net force acting on it due to the shell.
  • Charge on a conducting spherical shell spreads uniformly over the (external) surface.

21.2 - Charge is Quantized

Electric charge is quantized, or restricted to certain values. Because each particle (proton and electron) has the same charge:

e=1.602×10−19 Ce = 1.602 \times 10^{-19} \space \textrm{C}

then every possible electric charge is some multiple of this base charge, the elementary charge. Therefore, any positive or negative charge qq that can be detected can be written as:

q=ne,n=±1,±2,±3,...,q = ne, \hspace{2em} n = \pm 1, \pm 2, \pm 3, ...,

21.3 - Charge is Conserved

💡
Key Idea +The net electric charge of any isolated system is always conserved.

Annihilation Process

An annihilation process occurs when an electron e−e^- (charge −e-e) and its antiparticle, the positron e+e^+ (charge ee), undergo , transforming into two gamma rays (high-energy light):

If two charged particles undergo an annihilation process, they have opposite signs of charge.

If two charged particles appear as a result of a pair production process, they have opposite signs of charge.

e−+e+→γ+γe^- + e^+ \rightarrow \gamma + \gamma

+

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Unit 22: Electric Fields

22.1 - The Electric Field

A charged particle sets up an electric field (represented as a vector) in the surrounding space. If a second charged particle is located in that space, an electrostatic force acts on it due to the magnitude and direction of the field at its location.

The electric field, E⃗\vec E, at any point is defined as the electrostatic force, F⃗\vec F, that would be exerted on a positive test charge, q0q_0, placed there:

E⃗=F⃗q0\vec E = \frac{\vec F}{q_0}

Electric field lines are a way to visualize the direction and magnitude of electric fields. The electric field vector at any point is tangent to the field line through that point. The density of field lines in that region is proportional to the magnitude of the electric field there. Thus, closer field lines represent a stronger field.

Electric field lines originate on positive charges and terminate on negative charges. So, a field line extending from a positive charge must end on a negative charge.

22.2 - The Electric Field Due to a Charged Particle

The magnitude of the electric field E⃗\vec E set up by a particle with a charge qq at a distance rr from the particle is:

E=14πϵ0∣q∣r2E = \frac{1}{4\pi\epsilon_0} \frac{|q|}{r^2}

(this should be familiar)

The electric field vectors set up by a positively charged particle all point directly away from the particle. Those set up by a negatively charged particle all point directly toward the particle.

If more than one charged particle sets up an electric field at a point, the net electric field is the vector sum of the individual electric fields—electric fields obey the superposition principle. (This is how we get repulsion between particles, as shown on the electric field above).

22.3 - The Electric Field Due to a Dipole

An electric dipole consists of two particles with charges of equal magnitude qq but opposite signs, separated by a small distance dd.

The electric dipole moment p⃗\vec p has magnitude qdqd and points from the negative charge to the positive charge.

The magnitude of the electric field set up by an electric dipole at a distant point on the dipole axis (which runs through both particles) can be written in terms of either the product qdqd of the magnitude pp of the dipole moment:

E=12πϵ0qdz3=12πϵ0pz3E = \frac{1}{2\pi\epsilon_0} \frac{qd}{z^3} = \frac{1}{2\pi\epsilon_0} \frac{p}{z^3}

where zz is the distance between the pint and the center of the dipole.

Because of the 1/z31/z^3 dependence, the field magnitude of an electric dipole decreases more rapidly with distance than the field magnitude of either of the individual charges forming the dipole, which depends on 1/r21/r^2.

22.4 - The Electric Field Due to a Line of Charge

The equation for the electric field set up by a particle does not apply to an extended object with charge (said to have a continuous charge distribution).

To find the electric field of an extended object at a point, we first consider the electric field set up by a charge element dqdq in the object, where the element is small enough for us to apply the equation for a particle. Then we sum, via integration, components of the electric fields dE⃗d\vec E from all the charge elements.

Because the individual electric fields dE⃗d\vec E have different magnitudes and point in different directions, we first see if symmetry allows us to cancel out any of the components of the fields, to simplify the integration.

By thinking of each little piece as a point charge, dqdq, the field created by that point charge can be defined as:

dE⃗=kdqd3d⃗d\vec E = k\frac{dq}{d^3}\vec d

where d⃗\vec d is the vector from the source (the ring) to the target (point P in the diagram above). The total electric field can then be found through integration:

E⃗=∫dE⃗=∫kdqd3d⃗\vec E = \int d\vec E = \int k\frac{dq}{d^3} \vec d

To make this actually possible to integrate, we need to figure out what dqdq and d⃗\vec d are.

The charge of segment dqdq is equal to its line charge density λ(x)\lambda (x) times its length dLdL, and we can write its length as dxdx because it lies along the xx axis. Therefore

dq=λ(x)dxdq = \lambda(x)dx

For d⃗\vec d, just use the height and distance given (RR and zz in the diagram above) to create a vector pointing upwards from the ring to the point PP.

With all this newfound knowledge, here’s what the integral of the diagram above will look like:

E⃗=∫dE⃗=∫−L/2L/2kλ(x)dxr3(−R i^+z j^)\vec E = \int d\vec E = \int_{-L/2}^{L/2} k\frac{\lambda(x)dx}{r^3} (-R \space \hat i + z \space \hat j)

This is the integral of a vector, and you may be wondering how the hell you’re supposed to do that. The answer is “simple”: the unit vectors i^\hat i, j^\hat j, and k^\hat k are all constants: they always point in their respective directions no matter what xx is. Therefore, we can pull them out of the integral:

E⃗=−kRi^∫−L/2L/2λ(x)dxr3+kzj^∫−L/2L/2λ(x)dxr3\vec E = -kR\hat i \int_{-L/2}^{L/2} \frac{\lambda(x)dx}{r^3} + kz\hat j \int_{-L/2}^{L/2} \frac{\lambda(x)dx}{r^3}
💡
Tip +If the line charge density, λ\lambda, is constant (it usually is in these problems to make your brain hurt less), then you can replace λ(x)\lambda(x) in all the equations above with just the constant λ\lambda, which can then be pulled out of the integrals, leaving just dxdx on top within the integral.
E⃗=−kRλi^∫−L/2L/2dxr3+kzλj^∫−L/2L/2dxr3\vec E = -kR\lambda\hat i \int_{-L/2}^{L/2} \frac{dx}{r^3} + kz\lambda\hat j \int_{-L/2}^{L/2} \frac{dx}{r^3}

22.5 - The Electric Field Due to a Charged Disk

On the central axis through a uniformly charged disk,

E=σ2ϵ0(1−zz2+R2)E = \frac{\sigma}{2\epsilon_0}(1 - \frac{z}{\sqrt{z^2 + R^2}})

gives the electric field magnitude. Here, zz is the distance along the axis from the center of the disk, RR is the radius of the disk, and σ\sigma is the surface charge density.

22.6 - A Point Charge in an Electric Field

If a particle with charge qq is placed in an external electric field E⃗\vec E, an electrostatic force F⃗\vec F acts on the particle:

F⃗=qE⃗\vec F = q\vec E

If charge qq is positive, the force vector is in the same direction as the field vector. If charge qq is negative, the force vector is in the opposite direction (the minus sign in the equation reverses the force vector from the field vector).

22.7 - A Dipole in an Electric Field

The torque on an electric dipole of dipole moment p⃗\vec p when placed in an external electric field E⃗\vec E is given by a cross product:

τ=p⃗×E⃗\tau = \vec p \times \vec E

A potential energy UU is associated with the orientation of the dipole moment in the field, as given by a dot product:

U=−p⃗⋅E⃗U = -\vec p \cdot \vec E

If the dipole orientation changes, the work done by the electric field is:

W=−ΔUW = -\Delta U

If the change is due to an external agent, the work done by the agent is:

Wa=−WW_a = -W

+

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Unit 23: Gauss’ Law

23.1 - Electric Flux

💡
Key Idea +The electric flux Φ\Phi through a surface is the amount of electric field that pierces the surface.

The area vector dA⃗\vec {dA} for an area element (patch element) on a surface is a vector that is perpendicular to the element and has a magnitude equal to the area dAdA of the element.

The electric flux dΦd\Phi through a patch element with area vector dA⃗d \vec A is given by a dot product:

dΦ=E⃗⋅dA⃗d\Phi = \vec E \cdot d\vec A

Total Flux

The total flux through a surface is given by:

Φ=∫E⃗⋅dA⃗\Phi = \int \vec E \cdot d\vec A

where the integration is carried out over the surface.

For a uniform flat surface, this can be written as:

Φ=(Ecos⁡ϕ)A\Phi = (E \cos \phi) A

Net Flux

The net flux through a closed surface (which is used in Gauss’ law) is given by:

Φ=∮E⃗⋅dA⃗\Phi = \oint \vec E \cdot d\vec A

23.2 - Gauss’ Law

Gauss’ law relates the net flux Φ\Phi penetrating a closed surface to the net charge qencq_{enc} enclosed by the surface:

ϵ0Φ=qenc\epsilon_0 \Phi = q_{enc}

Gauss’ law can also be written in terms of the electric field piercing the enclosing Gaussian surface:

ϵ0∮E⃗⋅dA⃗=qenc\epsilon_0 \oint \vec E \cdot d\vec A = q_{enc}

23.3 - A Charged Isolated Conductor

💡
Key Idea +An excess charge on an isolated conductor is located entirely on the outer surface of the conductor.

The internal electric field of a charged, isolated conductor is zero, and the external field (at nearby points) is perpendicular to the surface and has a magnitude that depends on the surface charge density σ\sigma:

E=σϵ0E = \frac{\sigma}{\epsilon_0}

23.4 - Applying Gauss’ Law: Cylindrical Symmetry

The electric field at a point near an infinite line of charge (or charged rod) with uniform linear charge density λ\lambda is perpendicular to the line and has magnitude:

E=λ2πϵ0rE = \frac{\lambda}{2\pi \epsilon_0 r}

where rr is the perpendicular distance from the line to the point.

23.5 - Applying Gauss’ Law: Planar Symmetry

Nonconducting Sheet

The electric field due to an infinite nonconducting sheet with uniform surface charge density σ\sigma is perpendicular to the plane of the sheet and has magnitude:

E=σ2ϵ0E = \frac{\sigma}{2\epsilon_0}

Two Conducting Plates

The external electric field just outside the surface of an isolated charged conductor with surface charge density σ\sigma is perpendicular to the surface and has magnitude:

E=σϵ0E = \frac{\sigma}{\epsilon_0}

Inside the conductor, the electric field is zero.

23.6 - Applying Gauss’ Law: Spherical Symmetry

Spherical Shells

Outside a spherical shell of uniform charge qq, the electric field due to the shell is radial (inward or outward, depending on the sign of the charge) and has magnitude:

E=14πϵ0qr2E = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2}

where rr is the distance to the point of measurement from the center of the shell. The field is the same as though all of the charge is concentrated as a particle at the center of the shell.

At any point inside the shell, the field due to the shell is zero.

Uniform Spheres

Inside a sphere with a uniform volume charge density, the field is radial and has the magnitude:

E=14πϵ0qR3rE = \frac{1}{4\pi\epsilon_0} \frac{q}{R^3} r

where qq is the total charge, RR is the sphere’s radius, and rr is the radial distance from the center of the sphere to the point of measurement.

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Unit 24: Electric Potential

24.1 - Electric Potential

The electric potential VV at a point PP in the electric field of a charged object is:

V=−W∞q0=Uq0V = \frac{-W_\infin}{q_0} = \frac{U}{q_0}

where W∞W_\infin is the work that would be done by the electric force on a positive test charge q0q_0 were it brought from an infinite distance to PP, and UU is the electric potential energy that would then be stored in the test charge-object system.

Electric Potential Energy

If a particle with charge qq is placed at a point where the electric potential of a charged object is VV, the electric potential energy UU of the particle–object system is

U=qVU = qV

If the particle moves through a potential difference ΔV\Delta V, the change in electric potential energy is

ΔU=qΔV=q(Vf−Vi)\Delta U = q \Delta V = q(V_f-V_i)

If, instead, an applied force acts on the particle, doing work WappW_{app}, the change in kinetic energy is

ΔK=−qΔV+Wapp\Delta K = -q\Delta V + W_{app}

In the special case when ΔK=0\Delta K = 0, the work of an applied force involves only the motion of the particle through a potential difference:

Wapp=qΔVW_{app} = q\Delta V
⛔
Warnings to Consider +1. VV is called a potential, but it should not be confused with potential energy. Yes, the two quantities are related, but they are very different and not interchangeable. +2. Electric potential is a scalar, not a vector.

Language

A potential energy is a property of a system (or configuration) of objects, but sometimes we can get away with assigning it to a single object. For example, the gravitational potential energy of a baseball hit to outfield is actually a potential energy of the baseball–Earth system (because it is associated with the force between the baseball and Earth). However, because only the baseball noticeably moves (its motion does not noticeably affect Earth), we might assign the gravitational potential energy to it alone. In a similar way, if a charged particle is placed in an electric field and has no noticeable effect on the field (or the charged object that sets up the field), we usually assign the electric potential energy to the particle alone.

Units

The SI unit for potential that follows from Eq. 24-2 is the joule per coulomb. This combination occurs so often that a special unit, the volt (abbreviated VV), is used to represent it. Thus,

1 volt=1 joule per coulomb.1 \textrm{ volt} = 1 \textrm{ joule per coulomb}.

With two unit conversions, we can now switch the unit for electric field from newtons per coulomb to a more conventional unit:

1 N/C=1 V/m1 \textrm{ N/C} = 1 \textrm{ V/m}

From now on, express values of the electric field in volts per meter rather than in newtons per coulomb.

Electron-volts

In atomic and subatomic physics, energy measures in the SI unit of joules often require awkward powers of ten. A more convenient (but non-SI unit) is the electron-volt (eVeV), which is defined to be equal to the work required to move a single elementary charge ee (such as that of an electron or proton) through a potential difference VV of exactly one volt.

1 eV=e(1 V)=(1.602×10−19 C)(1 J/C)=1.602×10−19 J1 \textrm{ eV} = e(1 \textrm{ V}) = (1.602 \times 10^{-19} \textrm{ C})(1 \textrm{ J/C}) \newline = 1.602 \times 10^{-19} \textrm{ J}

24.2 - Equipotential Surfaces and the Electric Field

An equipotential surface is a surface in which every point on the surface has the same electric potential. The work done on a test charge in moving it from one such surface to another is independent of the locations of the initial and final points on these surfaces and of the path that joins the points. The electric field is always directed perpendicularly to corresponding equipotential surfaces.

Electric Potential Difference

The electric potential difference between two points ii and ff is

Vf−Vi=−∫ifE⃗⋅ds⃗V_f - V_i = -\int_i^f \vec E \cdot d\vec s

where the integral is taken over any path connecting the points. If the integration is difficult along a particular path, we can choose a different path along which the integration might be easier.

If we choose Vi=0V_i = 0, we have, for the potential at a particular point

V=−∫ifE⃗⋅ds⃗V = -\int_i^f \vec E \cdot d \vec s

In a uniform field of magnitude EE, the change in potential from a higher equipotential surface to a lower one, separated by distance Δx\Delta x, is

ΔV=−EΔx\Delta V = -E \Delta x

24.3 - Potential due to a Charged Particle

The electric potential due to a single charged particle at a distance rr from that charged particle is

V=14πϵ0qrV = \frac{1}{4\pi\epsilon_0} \frac{q}{r}

where VV has the same sign as qq.

Collections of Charged Particles

The potential due to a collection of charged particles is

V=∑i=1nVi=14πϵ0∑i=1nqiriV = \sum_{i = 1}^{n} V_i = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^{n} \frac{q_i}{r_i}

Thus, the potential is the algebraic sum of the individual potentials, with no consideration of directions.

24.4 - Potential due to an Electric Dipole

At a distance rr from an electric dipole with dipole moment magnitude p=qdp = qd, the electric potential of the dipole is

V=14πϵ0pcos⁡θr2V = \frac{1}{4\pi\epsilon_0} \frac{p\cos\theta}{r^2}

for r≫dr \gg d, the angle θ\theta lies between the dipole moment vector and a line extending from the dipole moment to the point of measurement.

+

+

Induced Dipole Moment

Many molecules, such as water, have permanent electric dipole moments. In other molecules (called nonpolar molecules) and in every isolated atom, the centers of the positive and negative charges coincide (Fig. a) and thus no dipole moment is set up. However, if we place an atom or a nonpolar molecule in an external electric field, the field distorts the electron orbits and separates the centers of positive and negative charge (Fig. b). Because the electrons are negatively charged, they tend to be shifted in a direction opposite the field. This shift sets up a dipole moment that points in the direction of the field. This dipole moment is said to be induced by the field, and the atom or molecule is then said to be polarized by the field (that is, it has a positive side and a negative side). When the field is removed, the induced dipole moment and the polarization disappear.

24.5 - Potential due to a Continuous Charge Distribution

For a continuous distribution of charge (over an extended object), the potential is found by (1) dividing the distribution into charge elements dqdq that can be treated as particles and then (2) summing the potential due to each element by integrating over the full distribution:

V=14πϵ0∫dqrV = \frac{1}{4\pi\epsilon_0} \int \frac{dq}{r}

In order to carry out the integration, dqdq is replaced with the product of either a linear charge density λ\lambda and a length element (such as dxdx), or a surface charge density σ\sigma and area element (such as dx/dydx/dy).

In some cases where the charge is symmetrically distributed, a two-dimensional integration can be reduced to a one-dimensional integration.

24.6 - Calculating the Field from the Potential

The component E⃗\vec E in any direction is the negative of the rate at which the potential changes with distance in that direction:

ES=−∂V∂sE_S = -\frac{\partial V}{\partial s}

The xx, yy, and zz components of E⃗\vec E may be found from:

Ex=−∂V∂xE_x = -\frac{\partial V}{\partial x}
Ey=−∂V∂yE_y = -\frac{\partial V}{\partial y}
Ez=−∂V∂zE_z = -\frac{\partial V}{\partial z}

When the electric field E⃗\vec E is uniform, this all reduces to:

E=−ΔVΔsE = -\frac{\Delta V}{\Delta s}

where ss is perpendicular to the equipotential surfaces.

The electric field is zero parallel to an equipotential surface, as the field points directly perpendicular to the surface of the equipotential surface.

24.7 - Electric Potential Energy of a System of Charged Particles

The electric potential energy of a system of charged particles is equal to the work needed to assemble the system with the particles initially at rest and infinitely distant from each other. For two particles at separation rr:

U=W=14πϵ0q1q2rU = W = \frac{1}{4\pi \epsilon_0} \frac{q_1q_2}{r}

24.8 - Potential of a Charged Isolated Conductor

💡
Key Idea +An excess charge placed on a conductor will, in the equilibrium state, be located entirely on the outer surface of the conductor.

The entire conductor, including interior points, is at a uniform potential.

If an isolated charged conductor is placed in an external electric field, then at every internal point, the electric field due to the charge cancels the external electric field that otherwise would have been there.

Also, the net electric field at every point on the surface is perpendicular to the surface.

+

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Unit 25: Capacitance

25.1 - Capacitance

A capacitor consists of two isolated conductors (the plates) with charges +q+q and −q-q. Its capacitance CC is defined from:

q=CVq = CV

where VV is the potential difference between the plates.

Therefore,

C=qVC = \frac{q}{V}

When a circuit with a battery, an open switch, and an uncharged capacitor is completed by closing the switch, conduction electrons shift, leaving the capacitor plates with opposite charges.

25.2 - Calculating the Capacitance

We generally determine the capacitance of a particular capacitor configuration by:

  1. Assuming a charge qq to have been placed on the plates
  1. Finding the electric field due to this charge
  1. Evaluating the potential difference VV between the plates
  1. Calculating CC from q=CVq = CV

Some results are the following:

Parallel-Plate Capacitor

A parallel-plate capacitor with flat parallel plates of area AA and spacing dd has capacitance:

C=ϵ0AdC = \frac{\epsilon_0A}{d}

Cylindrical Capacitor

A cylindrical capacitor (two long coaxial cylinders) of length LL and radii aa and bb has capacitance:

C=2πϵ0Lln⁡(ba)C = 2\pi\epsilon_0 \frac{L}{\ln\left(\frac{b}{a}\right)}

Spherical Capacitor

A spherical capacitor with concentric spherical plates of radii aa and bb has capacitance:

C=4πϵ0abb−aC = 4\pi\epsilon_0 \frac{ab}{b-a}

Isolated Sphere

An isolated sphere of radius RR has capacitance:

C=4πϵ0RC = 4\pi\epsilon_0 R

25.3 - Capacitors in Parallel and in Series

Capacitors in Parallel

The equivalent capacitance CeqC_{eq} of combinations of individual capacitors connected in parallel can be found from:

Ceq=∑j=1nCj(n capacitors in parallel)C_{eq} = \sum_{j=1}^n C_j \hspace{2em} (n \textrm{ capacitors in parallel})

Capacitors in parallel share the same voltage, but different currents.

Capacitors in Series

The equivalent capacitance CeqC_{eq} of combinations of individual capacitors connected in series can be found from:

1Ceq=∑j=1n1Cj(n capacitors in series)\frac{1}{C_{eq}} = \sum_{j=1}^n \frac{1}{C_j} \hspace{2em} (n \textrm{ capacitors in series})

Equivalent capacitances can also be used to calculate the capacitances of more complicated series–parallel combinations.

Capacitors in series have different voltages, but share the same current.

Total capacitance in a series is always less than the smallest individual capacitor.

25.4 - Energy Stored in an Electric Field

The electric potential energy UU of a charged capacitor,

U=q22C=12CV2U = \frac{q^2}{2C} = \frac{1}{2} CV^2

is equal to the work required to charge the capacitor. This energy can be associated with the capacitor’s electric field E⃗\vec E.

Every electric field, in a capacitor or from any other source, has an associated stored energy. In vacuum, the energy density uu (potential energy per unit volume) in a field of magnitude EE is:

u=12ϵ0E2u = \frac{1}{2} \epsilon_0 E^2

25.5 - Capacitor with a Dielectric

A dielectric is an insulating material such as mineral oil or plastic, which can change the capacitance of a capacitor.

If the space between the plates of a capacitor is completely filled with a dielectric material, the capacitance CC in vacuum (or, effectively, in air) is multiplied by the material’s dielectric constant κ\kappa, which is a number greater than 1.

In a region that is completely filled by a dielectric, all electrostatic equations containing the permittivity constant ϵ0\epsilon_0 must be modified by replacing ϵ0\epsilon_0 with κϵ0\kappa \epsilon_0.

When a dielectric material is placed in an external electric field, it develops an internal electric field that is oriented opposite the external field, thus reducing the magnitude of the electric field inside the material.

When a dielectric material is placed in a capacitor with a fixed amount of charge on the surface, the net electric field between the plates is decreased.

25.6 - Dielectrics and Gauss’ Law

Inserting a dielectric into a capacitor causes induced charge to appear on the faces of the dielectric, weakening the electric field between the plates.

The induced charge is less than the free charge on the plates.

When a dielectric is present, Gauss’ law may be generalized to:

ϵ0∮κE⃗⋅dA⃗=q\epsilon_0 \oint \kappa \vec E \cdot d\vec A = q

where qq is the free charge. Any induced surface charge is accounted for by including the dielectric constant κ\kappa inside the integral.

+

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Unit 26: Current and Resistance

26.1 - Electric Current

Electrons are always bouncing around at random in a circuit. Sometimes, however, these electrons travel generally in a common direction as a sort of biased movement. Electrons themselves move very slowly, however a conductor is always full of electrons, which move in a way very similar to a liquid in a pipe.

An electric current ii in a conductor is defined by:

i=dqdti = \frac{dq}{dt}

where dqdq is the amount of positive charge that passes in time dtdt.

By convention, the direction of electric current is taken as the direction in which positive charge carriers would move even though (normally) only conduction electrons can move.

Current is measured in coulombs per second, or as they are better known, amperes (AA):

1 ampere=1 A=1 coulomb per second=1 C/s1 \space \textrm{ampere} = 1 \space \textrm{A} = 1 \space \textrm{coulomb per second} = 1 \space \textrm{C/s}

16.2 - Current Density

Current ii (a scalar quantity) is related to the current density J⃗\vec J (a vector quantity), by:

i=∫J⃗⋅dA⃗i = \int \vec J \cdot d\vec A

where dA⃗d\vec A is a vector perpendicular to a surface element of area dAdA and the integral is taken over any surface cutting across the conductor. The current density J⃗\vec J has the dame direction as the velocity of the moving charges if they are positive and the opposite direction if they are negative.

If the current is uniform across the surface and parallel to dA⃗d\vec A, then J⃗\vec J is also uniform and parallel to dA⃗d\vec A, simplifying the formula for JJ:

J=iAJ = \frac{i}{A}

When an electric field E⃗\vec E is established in a conductor, the charge carriers (assumed positive) acquire a drift speed vdv_d in the direction of E⃗\vec E.

The drift velocity is related to the current density by:

J⃗=(ne)v⃗d\vec J = (ne)\vec v_d

where nene is the carrier charge density.

26.3 - Resistance and Resistivity

The resistance RR of a conductor is defined as:

R=ViR = \frac{V}{i}

where VV is the potential difference across the conductor and ii is the current.

The resistivity ρ\rho and conductivity σ\sigma of a material are related by:

ρ=1σ=EJ\rho = \frac{1}{\sigma} = \frac{E}{J}

where EE is the magnitude of the applied electric field and JJ is the magnitude of the current density.

The electric field and current density are related to the resistivity by:

E⃗=ρJ⃗\vec E = \rho \vec J

The resistance RR of a conducting wire of length LL and uniform cross section is:

R=ρLAR = \rho \frac{L}{A}

where AA is the cross sectional area.

The resistivity ρ\rho for most materials changes with temperature. For many materials, including metals, the relationship between ρ\rho and temperature TT is approximated by the equation:

ρ−ρ0=ρ0α(T−T0)\rho - \rho_0 = \rho_0 \alpha(T - T_0)

Here T0T_0 is a reference temperature, ρ0\rho_0 is the resistivity at T0T_0, and α\alpha is the temperature coefficient of resistivity for the material.

26.4 - Ohm’s Law

A given device (conductor, resistor, or any other electrical device) obeys Ohm’s law if its resistance R(=V/i)R (= V/i) is independent of the applied potential difference VV.

A given material obeys Ohm’s law if its resistivity r(=E/J)r (= E/J) is independent of the magnitude and direction of the applied electric field .

The assumption that the conduction electrons in a metal are free to move like the molecules in a gas leads to an expression for the resistivity of a metal:

ρ=me2nτ\rho = \frac{m}{e^2n\tau}

where nn is the number of free electrons per unit volume and τ\tau is the mean time between the collisions of an electron with the atoms of the metal.

Metals obey Ohm’s law because the mean free time τ\tau is approximately independent of the magnitude EE of any electric field applied to the metal.

26.5 - Power, Semiconductors, Superconductors

Power

The power PP, or rate of energy transfer, in an electrical device across which a potential difference VV is maintained is:

P=iVP = iV

If the device is a resistor, the power can also be written as:

P=i2R=V2RP = i^2R = \frac{V^2}{R}

In a resistor, electric potential energy is converted to internal thermal energy via collisions between charge carriers and atoms.

Semiconductors

Semiconductors are materials that have few conduction +electrons but can become conductors when they are doped +with other atoms that contribute charge carriers.

Superconductors

Superconductors are materials that lose all electrical resistance. Most such materials require very low temperatures, but some become superconducting at temperatures as high as room temperature.

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Unit 27: Circuits

27.1 - Single Loop Circuits

To produce a steady flow of charge, you need a “charge pump,” a device that—by doing work on the charge carriers—maintains a potential difference between a pair of terminals. We call such a device an emf device, and the device is said to provide an emf E\mathscr{E}, which means that it does work on charge carriers.

An emf device is sometimes called a seat of emf. The term emf comes from the outdated phrase electromotive force, which was adopted before scientists clearly understood the function of an emf device.

An emf device does work on charges to maintain a potential difference between its output terminals. If dWdW is the work the device does to force positive charge dqdq from the negative to the positive terminal, then the emf (work per unit charge) of the device is:

E=dWdq\mathscr{E} = \frac{dW}{dq}

Ideal vs. Real EMF Devices

An ideal emf device is one that lacks any internal resistance. The potential difference between its terminals is equal to the emf.

A real emf device has internal resistance. The potential difference between its terminals is equal to the emf only if there is no current through the device.

Resistance Rule

The change in potential in traversing a resistance RR in the direction of the current is −iR-iR; in the opposite direction it is +iR+iR.

EMF Rule

The change in potential in traversing an ideal emf device in the direction of the emf arrow is +E+\mathscr{E}; in the opposite direction it is −E-\mathscr{E}.

Loop Rule

Conservation of energy leads to the Loop Rule: The algebraic sum of the changes in potential encountered in a complete traversal of any loop of a circuit must be zero.

Junction Rule

Conservation of charge leads to the Junction Rule: The sum of the currents entering any junction must be equal to the sum of the currents leaving that junction.

Internal Resistance (Real Batteries)

When a real battery of emf E\mathscr{E} and internal resistance rr does work on the charge carriers in an current ii through the battery, the rate PP of energy transfer to the charge carriers is:

P=iVP = iV

where VV is the potential across the terminals of the battery.

The rate PrP_r at which energy is dissipated as thermal energy in the battery is:

Pr=i2rP_r = i^2r

The rate PemfP_{\textrm{emf}} at which the chemical energy in the battery changes is:

Pemf=iEP_{\textrm{emf}} = i\mathscr{E}

When resistances are in series, they have the same current. The equivalent resistance that can replace a series combination of resistances is:

Req=∑j=1nRjR_{eq} = \sum_{j=1}^n R_j

(this is from last unit)

27.2 - Multiloop Circuits

When resistances are in parallel, they have the same potential difference (VV). The equivalent resistance that can replace a parallel combination of resistances is given by:

1Req=∑j=in1Rj\frac{1}{R_{eq}} = \sum_{j=i}^n \frac{1}{R_j}

27.3 - The Ammeter and the Voltmeter

An instrument used to measure currents is called an ammeter. To measure the current in a wire, you usually have to break or cut the wire and insert the ammeter.

A meter used to measure potential differences is called a voltmeter. To find the potential difference between any two points in the circuit, the voltmeter terminals are connected between those points without breaking or cutting the wire.

Often a single meter is packaged so that, by means of a switch, it can be made to serve as either an ammeter or a voltmeter—and usually also as an ohmmeter, designed to measure the resistance of any element connected between its terminals. Such a versatile unit is called a multimeter.

27.4 - RC Circuits

When an emf E\mathscr{E} is applied to a resistance RR and capacitance CC in series, the charge on the capacitor increases according to:

q=CE(1−e−t/RC)q = C\mathscr{E}\left(1 - e^{-t/RC}\right)

in which CE=q0C\mathscr{E} = q_0 is the equilibrium (final) charge and RC=τRC = \tau is the capacitive time constant of the circuit.

💡
A capacitor that is being charged initially acts like ordinary connecting wire relative to the charging current. A long time later, it acts like a broken wire.

During charging, the current is:

i=dqdt=(ER)e−t/RCi = \frac{dq}{dt} = \left(\frac{\mathscr{E}}{R}\right) e^{-t/RC}

When a capacitor discharges through a resistance RR, the charge on the capacitor decays according to:

q=q0e−t/RCq = q_0 e^{-t/RC}

During the discharging, the current is equal to:

i=dqdt=−(q0RC)e−t/RCi = \frac{dq}{dt} = -\left(\frac{q_0}{RC}\right)e^{-t/RC}

+

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Unit 28: Magnetic Fields

28.1 - Magnetic Fields and the Definition of B⃗\vec B

What Produces a Magnetic Field?

Because an electric field is produced by an electric charge, we might reasonably expect that a magnetic field is produced by a magnetic charge. Although individual magnetic charges (called magnetic monopoles) are predicted by certain theories, their existence has not been confirmed. How then are magnetic fields produced? There are two ways.

Moving electronically charged particles, such as a current in a wire, can be used to make an electromagnet. The current produces a magnetic field which can be controlled by controlling the flow of current through the wire.

The other way to produce a magnetic field is by means of elementary particles such as electrons because the particles have an intrinsic magnetic field around them. The magnetic fields of the electrons in certain materials add together to give a net magnetic field around the material. These materials are known as permanent magnets.

The Definition of B⃗\vec B

The magnetic field B⃗\vec B is defined in terms of the magnetic force F⃗B\vec F_B exerted on a moving electrically charged test particle.

In principle, we do this by firing a charged particle through the point at which B⃗\vec B is to be defined, using various directions and speeds for the particle and determining the force that acts on the particle at that point. After many such trials we would find that when the particle’s velocity v⃗\vec v is along a particular axis through the point, force F⃗B\vec F_B is zero. For all other directions of v⃗\vec v, the magnitude of F⃗B\vec F_B is always proportional to vsin⁡ϕv \sin \phi, where ϕ\phi is the angle between the zero-force axis and the direction of v⃗\vec v.

We can then define a magnetic field B⃗\vec B to be a vector quantity that is directed along the zero-force axis. We can next measure the magnitude of F⃗B\vec F_B when v⃗\vec v is directed perpendicular to that axis and then define the magnitude of B⃗\vec B in terms of that force magnitude:

B=Fb∣q∣vB = \frac{F_b}{|q|v}

where qq is the charge of the particle.

Finding the Magnetic Force on a Particle

The force F⃗B\vec F_B on a particle from a magnetic field can be represented as a cross product:

F⃗B=qv⃗×B⃗\vec F_B = q\vec v \times \vec B

that is, the force F⃗B\vec F_B on the particle is equal to the charge qq times the cross product of its velocity v⃗\vec v and the field B⃗\vec B.

Thus, we can write the magnitude of F⃗B\vec F_B as:

FB=∣q∣vBsin⁡ϕF_B = |q|vB \sin \phi

where ϕ\phi is the angle between the directions of the velocity v⃗\vec v and the magnetic field B⃗\vec B.

28.2 - Crossed Fields: Discovery of the Electron

If a charged particle moves through a region containing both an electric field and a magnetic field, it can be affected by both an electric force and a magnetic force.

If the fields are perpendicular to each other, they are said to be crossed fields.

If the forces are in opposite directions, a particular speed will result in no deflection of the particle.

28.3 - Crossed Fields: The Hall Effect

When a uniform magnetic field BB is applied to a conducting strip carrying current ii, with the field perpendicular to the direction of the current, a Hall-effect potential difference VV is set up across the strip.

The electric force F⃗E\vec F_E on the charge carriers is then balanced by the magnetic force F⃗B\vec F_B on them.

The number density n of the charge carriers can then be determined from:

n=BiVlen = \frac{Bi}{Vle}

where ll is the thickness of the strip (parallel to B⃗\vec B).

When a conductor moves through a uniform magnetic field B⃗\vec B at speed vv, the Hall-effect potential difference VV across it is:

V=vBdV = vBd

where dd is the width perpendicular to both velocity v⃗\vec v and field B⃗\vec B.

28.4 - A Circulating Charged Particle

A charged particle with mass mm and charge magnitude ∣q∣|q| moving with velocity v⃗\vec v perpendicular to a uniform magnetic field B⃗\vec B will travel in a circle.

Applying Newton’s second law to the circular motion yields:

∣q∣vB=mv2r|q|vB = \frac{mv^2}{r}

from which we find the radius rr of the circle to be

r=mv∣q∣Br = \frac{mv}{|q|B}

The frequency of revolution ff, the angular frequency ω\omega, and the period of the motion TT are given by:

f=ω2π=1T=∣q∣B2π⋅mf = \frac{\omega}{2\pi} = \frac{1}{T} = \frac{|q|B}{2\pi \cdot m}

If the velocity of the particle has a component parallel to the magnetic field, the particle moves in a helical path around the field vector B⃗\vec B.

28.5 - Cyclotrons and Synchrotrons

A cyclotron is a form of particle accelerator in which particles are made to circulate using an an electrical oscillator to induce magnetic fields that slowly increase the speed of the particles (e.g. protons).

Frequency

The key to the operation of the cyclotron is that the frequency ff at which the proton circulates in the magnetic field (and that does not depend on its speed) must be equal to the fixed frequency foscf_{osc} of the electrical oscillator:

f=fosc(resonance condition)f = f_{osc} \hspace{1.5em} \textrm{(resonance condition)}

which can be used in combination with the above equation for frequency to obtain:

∣q∣B=2πmfosc|q|B = 2\pi m f_{osc}

The Proton Synchrotron

At proton energies above 50 MeV, the conventional cyclotron begins to fail because one of the assumptions of its design—that the frequency of revolution of a charged particle circulating in a magnetic field is independent of the particle’s speed—is true only for speeds that are much less than the speed of light.

The proton synchrotron corrects for the relativity and feasibility problems of a traditional cyclotron. The magnetic field BB and the oscillator frequency foscf_{osc}, instead of having field values as in the conventional cyclotron, are made to vary with time during accelerating cycle.

28.6 - Magnetic Force on a Current-Carrying Wire

A straight wire carrying a current ii in a uniform magnetic field experiences a sideways force:

F⃗B=iL⃗×B⃗\vec F_B = i\vec L \times \vec B

The force acting on a current element i dL⃗i \space d\vec L in a magnetic field is:

dF⃗B=i dL⃗×B⃗d \vec F_B = i \space d\vec L \times \vec B

The direction of the length vector L⃗\vec L or dL⃗d\vec L is that of the current ii.

28.7 - Torque on a Current Loop

Various magnetic forces act on the sections of a current-carrying coil lying in a uniform external magnetic field, but the net force is zero.

The net torque acting on the coil has a magnitude given by:

τ=NiABsin⁡θ\tau = NiAB \sin \theta

where NN is the number of turns in the coil, AA is the area of each turn, ii is the current, BB is the field magnitude, and θ\theta is the angle between the magnetic field B⃗\vec B and the normal vector to the coil n⃗\vec n.

28.8 - The Magnetic Dipole Moment

A coil (of area AA and NN turns, carrying current ii) in a uniform magnetic field will experience a torque given by:

τ⃗=μ⃗×B⃗\vec \tau = \vec \mu \times \vec B

Here μ⃗\vec \mu is the magnetic dipole moment of the coil, with magnitude μ=NiA\mu = NiA and direction given by the right-hand rule.

The orientation energy of a magnetic dipole in a magnetic field is given by:

U(θ)=−μ⃗⋅B⃗U(\theta) = -\vec \mu \cdot \vec B

If an external agent rotates a magnetic dipole from an initial orientation θi\theta_i to some other orientation θf\theta_f and the dipole is stationary both initially and finally, the work WaW_a done on the dipole by the agent is:

Wa=ΔU=Uf−UiW_a = \Delta U = U_f - U_i
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Unit 29: Magnetic Fields Due to Currents

29.1 - Magnetic Field Due to a Current

The magnetic field set up by a current-carrying conductor can be found from the Biot-Savart law. This law asserts that the contribution dB⃗d\vec B to the field produced by a current-length element i ds⃗i \space d\vec s at a point PP located a distance rr from the current element is:

dB⃗=μ04πids⃗×r^r2d\vec B = \frac{\mu_0}{4\pi} \frac{id\vec s \times \hat r}{r^2}

Here, r^\hat r is a unit vector that points from the element towards PP. The quantity μ0\mu_0, called the permeability constant, has the value:

4π×10−7 T⋅m/A≈1.26×10−6 T⋅m/A4\pi \times 10^{-7} \textrm{ T} \cdot \textrm{m}/\textrm{A} \approx 1.26 \times 10^{-6} \textrm{ T} \cdot \textrm{m}/\textrm{A}

Long Straight Wire

For a long straight wire carrying current ii, the Biot-Savart law gives, for the magnitude of the magnetic field at a perpendicular distance RR from the wire:

B=μ0i2πR(long straight wire)B = \frac{\mu_0 i}{2\pi R} \hspace{1em} \textrm{(long straight wire)}

Circular Arc of Wire

The magnitude of the magnetic field at the center of a circular arc, of radius RR and central angle ϕ\phi (in radians), carrying current ii, is:

B=μ0iϕ4πR(at center of circular arc)B = \frac{\mu_0 i \phi}{4\pi R} \hspace{1em} \textrm{(at center of circular arc)}

29.2 - Force Between Two Parallel Currents

Parallel wires carrying currents in the same direction attract each other, whereas parallel wires carrying currents in opposite directions repel each other. The magnitude of the force on a length LL of either wire is:

Fba=ibLBasin⁡90°=μ0Liaib2πdF_{ba} = i_bLB_a \sin{90\degree} = \frac{\mu_0 L i_a i_b}{2\pi d}

where dd is the wire separation, and iai_a and ibi_b are the currents in the wires.

29.3 - Ampere’s Law

Ampere’s Law states that:

∮B⃗⋅ds⃗=μ0ienc\oint \vec B \cdot d\vec s = \mu_0 i_{enc}

The line integral in this equation is evaluated around a closed loop called an Amperian loop. The current ii on the right side is the net current encircled by the loop.

29.4 - Solenoids and Toroids

Solenoids

A solenoid is a long, tightly wound helical coil of wire, in which the length is far greater than the diameter. The magnetic field produced within the solenoid is the vector sum of the fields produced by the individual turns (windings) that make up the solenoid.

Inside a long solenoid carrying current ii, at points not near its ends, the magnitude BB of the magnetic field is:

B=μ0in(ideal soldnoid)B = \mu_0 in \hspace{1em} \textrm{(ideal soldnoid)}

where nn is the number of turns per unit length.

Toroids

A toroid is a hollow solenoid that has been curved until its two ends meet, forming a sort of hollow bracelet. Ampere’s law and the symmetry of the bracelet can be used to calculate the magnetic field B⃗\vec B within the toroid.

At a point inside a toroid, the magnitude BB of the magnetic field is:

B=μ0iN2π1r(toroid)B = \frac{\mu_0 iN}{2\pi} \frac{1}{r} \hspace{1em} \textrm{(toroid)}

where rr is the distance from the center of the toroid to the point.

29.5 - A Current-Carrying Coil as a Magnetic Dipole

The magnetic field produced by a current-carrying coil, which is a magnetic dipole, at a point PP located a distance zz along the coil’s perpendicular central axis is parallel to the axis and is given by:

B⃗(z)=μ02πμ⃗z3\vec B(z) = \frac{\mu_0}{2\pi} \frac{\vec \mu}{z^3}

where μ⃗\vec \mu is the dipole moment of the coil. This equation only applies when zz is much greater than the dimensions of the coil.

+

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Unit 3: Vectors

Scalars and Vectors

Scalars, such as temperature, have magnitude only. They are specified by a number with a unit (10°C) and obey the rules of arithmetic and ordinary algebra. Vectors, such as displacement, have both magnitude and direction (5 m, north) and obey the rules of vector algebra.

Adding Vectors Geometrically

Two vectors a⃗\vec a and b⃗\vec b may be added geometrically by drawing them to a common scale and placing them head to tail. The vector connecting the tail of the first to the head of the second is the vector sum s⃗\vec s. To subtract b⃗\vec b from a⃗\vec a, reverse the direction of b⃗\vec b to get −b⃗-\vec b; then add −b⃗-\vec b to a⃗\vec a. Vector addition is commutative

a⃗+b⃗=b⃗+a⃗\vec a + \vec b = \vec b + \vec a

and obeys the associative law.

(a⃗+b⃗)+c⃗=a⃗+(b⃗+c⃗)(\vec a + \vec b) + \vec c = \vec a + (\vec b + \vec c)

Components of a Vector

The (scalar) components axa_x and aya_y of any two-dimensional vector along the coordinate axes are found by dropping perpendicular lines from the ends of onto the coordinate axes. The components are given by

ax=acos⁡θanday=asin⁡θa_x = a \cos \theta \hskip{1em} \textrm{and} \hskip{1em} a_y = a \sin \theta

where θ\theta is the angle between the positive direction of the xx axis and the direction of a⃗\vec a. The algebraic sign of a component indicates its direction along the associated axis. Given its components, we can find the magnitude and orientation (direction) of the vector by using

a=ax2+ay2andtan⁡θ=ayaxa = \sqrt{a_x^2 + a_y^2} \hskip{1em} \textrm{and} \hskip{1em} \tan \theta = \frac{a_y}{a_x}

Unit Vectors

A unit vector is a vector that has a magnitude of exactly 1 and points in a particular direction. It lacks both dimension and unit. Its sole purpose is to point, that is, to specify a direction. The unit vectors in the positive directions of the x, y, and z axes are labeled i^\hat i, j^\hat j, and k^\hat k, where the hat ^\hat{} is used instead of an overhead arrow as for other vectors.

Essentially, these can be used to give direction by multiplying with numerical magnitudes.

A vector a⃗\vec a can be written in terms of unit vectors as

a⃗=axi^+ayj^+azk^\vec a = a_x\hat i + a_y\hat j + a_z\hat k

in which axi^a_x\hat i, ayj^a_y\hat j, and azk^a_z\hat k are the vector components of a⃗\vec a and axa_x, aya_y, and aza_z are its scalar components.

Adding Vectors by Components

We can add vectors geometrically on a sketch or directly on a vector-capable calculator. A third way is to combine their components axis by axis.

To add vectors in component form, we use the rules

rx=ax+bxr_x = a_x + b_x
ry=ay+byr_y = a_y + b_y
rz=az+bzr_z = a_z + b_z

Here a⃗\vec a and b⃗\vec b are the vectors being added, and r⃗\vec r is the vector sum. After summing each of the separate components of the vector, it can be represented in unit-vector notation or magnitude-angle notation.

Product of a Scalar and a Vector

The product of a scalar ss and a vector v⃗\vec v is a new vector whose magnitude is svsv and whose direction is the same as that of v⃗\vec v if ss is positive, and opposite that of v⃗\vec v if ss is negative. +To divide v⃗\vec v by ss, multiply v⃗\vec v by 1s\frac{1}{s}.

The Scalar Product

The scalar (or dot) product of two vectors a⃗\vec a and b⃗\vec b is written a⃗⋅b⃗\vec a \cdot \vec b and is the scalar quantity given by a⃗⋅b⃗=ab cos ϕ\vec a \cdot \vec b = ab \space cos \space \phi, in which ϕ\phi is the angle between the directions of a⃗\vec a and b⃗\vec b. +A scalar product is the product of the magnitude of one vector and the scalar component of the second vector along the direction of the first vector. In unit-vector notation,

a⃗⋅b⃗=(axi^+ayj^+azk^)⋅(bxi^+byj^+bzk^)\vec a \cdot \vec b = (a_x\hat i + a_y\hat j + a_z\hat k) \cdot (b_x\hat i + b_y\hat j + b_z\hat k)

which may be expanded according to the distributive law. Note that a⃗⋅b⃗=b⃗⋅a⃗\vec a \cdot \vec b = \vec b \cdot \vec a.

The Vector Product

The vector (or cross) product of two vectors a⃗\vec a and b⃗\vec b is written a⃗×b⃗\vec a \times \vec b and is a vector c⃗\vec c whose magnitude cc is given by:

c=absin⁡ϕc = ab \sin \phi

in which ϕ\phi is the smaller of the angles between the directions of a⃗\vec a and b⃗\vec b. The direction of c⃗\vec c is perpendicular to the plane defined by a⃗\vec a and b⃗\vec b and is given by a right-hand rule. Note that a⃗×b⃗=−(b⃗×a⃗)\vec a \times \vec b = -(\vec b \times \vec a). In unit-vector notation,

a⃗×b⃗=(axi^+ayj^+azk^)×(bxi^+byj^+bzk^)\vec a \times \vec b = (a_x\hat i + a_y\hat j + a_z\hat k) \times (b_x\hat i + b_y\hat j + b_z\hat k)

which we may expand with the distributive law.

The right hand rule for cross products.
💡
In nested products, where one product is buried inside another, follow the normal algebraic procedure by starting with the innermost product and working outward.

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Unit 30: Induction and Inductance

30.1 - Faraday’s Law and Lenz’s Law

Faraday’s Law of Induction

Faraday's law of induction describes how an electric current produces a magnetic field and, conversely, how a changing magnetic field generates an electric current in a conductor.

Magnetic Flux

To put Faraday’s law to work, we need a way to calculate the amount of magnetic field that passes through a loop. Suppose a loop enclosing an area AA is placed in a magnetic field B⃗\vec B. Then the magnetic flux through the loop is:

ΦB=∫B⃗⋅dA\Phi_B = \int \vec B \cdot dA
💡
As with electric flux, we want the component of the field that pierces the surface (not skims along it).

Uniform, Parallel Magnetic Field

Suppose that the loop lies in a plane and that the magnetic field is perpendicular to the plane of the loop. This simplifies the above equation to:

ΦB=BA(B⃗⊥area,B⃗ uniform)\Phi_B = BA \hspace{2em} (\vec B \perp \textrm{area}, \vec B \textrm{ uniform})

Magnetic flux is measured in Webers:

1 weber=1 Wb=1 T⋅m21 \space \textrm{weber} = 1 \space \textrm{Wb} = 1 \space \textrm T \cdot \textrm m^2

Faraday’s Law

The magnitude of the emf E\mathscr E induced in a conducting loop is equal to the rate at which the magnetic flux B⃗\vec B through that loop changes with time.

E=−dΦBdt\mathscr E = -\frac{d\Phi_B}{dt}

If we change the magnetic flux through a coil of NN turns, an induced emf appears in every turn and the total emf induced in the coil is the sum of these individual induced emfs. If the coil is tightly wound (closely packed), so that the same magnetic flux ΦB\Phi_B passes through all the turns, the total emf induced in the coil is:

E=−NdΦBdt(coil of N turns)\mathscr E = -N \frac{d\Phi_B}{dt} \hspace{2em} (\textrm{coil of } N \textrm{ turns})

Lenz’s Law

Soon after Faraday proposed his law of induction, Heinrich Friedrich Lenz devised a rule for determining the direction of an induced current in a loop:

⭐
An induced current has a direction such that the magnetic field due to the current opposes the change in the magnetic flux that induces the current.

30.2 - Induction and Energy Transfers

The induction of a current by a changing flux means that energy is being transferred to that current. The energy can then be transferred to other forms, such as thermal energy.

Eddy Currents

Suppose we move a solid conducting plate out of a magnetic field, as shown above. The relative motion of the field and the conductor again induces a current in the conductor. Thus, we again encounter an opposing force and must do work because of the induced current. With the plate, however, the conduction electrons making up the induced current do not follow one path as they do with the loop. Instead, the electrons swirl about within the plate as if they were caught in an eddy (whirlpool) of water. Such a current is called an eddy current, and can be represented as if it followed a single path as shown above.

30.3 - Induced Electric Fields

An emf is induced by a changing magnetic flux even if the loop through which the flux is changing is not a physical conductor but an imaginary line. The changing magnetic field induces an electric field E⃗\vec E at every point of such a loop; the induced emf is related to E⃗\vec E by:

E=∮E⃗⋅ds⃗\mathscr{E} = \oint \vec E \cdot d\vec s

Using the induced electric field, we can write Faraday’s law in its most general form as:

∮E⃗⋅ds⃗=−dΦbdt\oint \vec E \cdot d\vec s = -\frac{d\Phi_b}{dt}

30.4 - Inductors and Inductance

An inductor is a device (coil, toroid, or solenoid) that can be used to produce a known magnetic field in a specified region. If a current ii is established through each of the NN windings of an inductor, a magnetic flux ΦB\Phi_B links those windings. The inductance LL of the inductor is:

L=NΦBiL = \frac{N\Phi_B}{i}

The SI unit of inductance is the henry (H), where:

1 henry=1 H=1 T⋅m2/A1 \space \textrm{henry} = 1 \space \textrm{H} = 1 \space \textrm{T} \cdot \textrm m ^2 / \textrm A

The inductance per unit length near the middle of a long solenoid of cross-sectional area AA and nn turns per unit length is:

Ll=μ0n2A\frac{L}{l} = \mu_0n^2A

30.5 - Self-Induction

If a current ii in a coil changes with time, an emf is induced in the coil. This self-induced emf is:

EL=−Ldidt\mathscr E_L = -L\frac{di}{dt}

The direction of EL\mathscr E_L is found from Lenz’s law: The self-induced emf acts to oppose the charge that produces it.

30.6 - RL Circuits

If a constant emf E\mathscr E is introduced into a single-loop circuit containing a resistance RR and an inductance LL, the current rises to an equilibrium value of E/R\mathscr E / R according to:

i=ER(1−e−t/τL)i = \frac{\mathscr E}{R}(1-e^{-t/\tau_L})

Here τL(=L/R)\tau_L (=L/R) governs the rate of rise of the current and is called the inductive time constant of the circuit.

When the course of constant emf is removed, the current decays from a value i0i_0 according to:

i=i0e−t/τLi = i_0 e^{-t/\tau_L}

30.7 - Energy Stored in a Magnetic Field

If an inductor LL carries a current ii, the inductor’s magnetic field stores an energy given by:

UB=12Li2U_B = \frac{1}{2} Li^2

30.8 - Energy Density of a Magnetic Field

If BB is the magnitude of a magnetic field at any point (in an inductor or anywhere else), the density of stored magnetic energy at that point is:

uB=B22μ0u_B = \frac{B^2}{2\mu_0}

30.9 - Mutual Induction

If coils 1 and 2 are near each other, a changing current in either coil can induce an emf in the other. This mutual induction is described by:

E2=−Mdi1dt\mathscr E_2 = -M \frac{di_1}{dt}
E1=−Mdi2′dt\mathscr E_1 = -M \frac{di_2'}{dt}

where MM (measured in henries) is the mutual inductance.

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Unit 4: Motion in Two and Three Dimensions

Position and Displacement

One general way of locating a particle (or particle-like object) is with a position vector r⃗\vec r, which is a vector that extends from a reference point (usually the origin) to the particle. In the unit-vector notation:

r⃗=xi^+yj^+zk^\vec r = x\hat i + y\hat j + z\hat k

where xi^x\hat i, yj^y\hat j, and zk^z\hat k are the vector components of r⃗\vec r and the coefficients xx, yy, and zz are its scalar components.

As a particle moves, its position vector changes in such a way that the vector always extends to the particle from the reference point (the origin). If the position vector changes—say, from r⃗1\vec r_1 to r⃗2\vec r_2 during a certain time interval—then the particle’s displacement Δr⃗\Delta \vec r during that time interval is

Δr⃗=r⃗2−r⃗1 +\Delta \vec r = \vec r_2 - \vec r_1

This can be rewritten using unit vector notation as follows:

Δr⃗=(x2−x1)i^+(y2−y1)j^+(z2−z1)k^\Delta \vec r = (x_2 - x_1)\hat i + (y_2 - y_1)\hat j + (z_2 - z_1)\hat k

or more simply

Δr⃗=Δxi^+Δyj^+Δzk^\Delta \vec r = \Delta x \hat i + \Delta y \hat j + \Delta z \hat k

Average and Instantaneous Velocity

If a particle moves through a displacement Δr⃗\Delta \vec r in a time interval Δt\Delta t, then its +average velocity v⃗avg\vec v_{avg} is

v⃗avg=Δr⃗Δt\vec v_{avg} = \frac{\Delta \vec r}{\Delta t}

The direction of v⃗avg\vec v_{avg} must be the same as that of the displacement Δr⃗\Delta \vec r. This formula can also be written in components:

v⃗avg=ΔxΔti^+ΔyΔtj^+ΔzΔtk^\vec v_{avg} = \frac{\Delta x}{\Delta t}\hat i + \frac{\Delta y}{\Delta t}\hat j + \frac{\Delta z}{\Delta t}\hat k

When we speak of the velocity of a particle, we usually mean the particle’s instantaneous velocity v⃗\vec v at some instant. This v⃗\vec v is the value that v⃗avg\vec v_{avg} approaches in the limit as we shrink the time interval Δt\Delta t to 0 about that instant. Using the language of calculus, we may write v⃗\vec v as the derivative.

v⃗=dr⃗dt\vec v = \frac{d \vec r}{dt}

Average Acceleration and Instantaneous Acceleration

When a particle’s velocity changes from v⃗1\vec v_1 to v⃗2\vec v_2 in a time interval Δt\Delta t, its average acceleration a⃗vg\vec a_{vg} during Δt\Delta t is:

a⃗avg=Δv⃗Δt\vec a_{avg} = \frac{\Delta \vec v}{\Delta t}

If we shrink Δt\Delta t to zero about some instant, then in the limit a⃗avg\vec a_{avg} approaches the instantaneous acceleration (or acceleration) a⃗\vec a at that instant; that is,

a⃗=dv⃗dt\vec a = \frac{d \vec v}{dt}

If the velocity changes in either magnitude or direction (or both), the particle must have an acceleration.

This equation can also be written in vector form as

a⃗=dv⃗xdti^+dv⃗ydtj^+dv⃗zdtk^\vec a = \frac{d\vec v_x}{dt}\hat i + \frac{d\vec v_y}{dt}\hat j + \frac{d\vec v_z}{dt}\hat k

The scalar components of a⃗\vec a can be found by differentiating the scalar components of v⃗\vec v.

Particle Motion

+

Uniform Circular Motion

+

Relative Motion

The velocity of a particle depends on the reference frame of whoever is observing or measuring the velocity. For our purposes, a reference frame is the physical object to which we attach our coordinate system. In everyday life, that object is the ground. For example, the speed listed on a speeding ticket is always measured relative to the ground. The speed relative to the police officer would be different if the officer were moving while making the speed measurement.

When two frames of reference A and B are moving relative to each other at constant velocity, the velocity of a particle P as measured by an observer in frame A usually differs from that measured from frame B. The two measured velocities are related by:

v⃗PA=v⃗PB+v⃗BA\vec v_{PA} = \vec v_{PB} + \vec v_{BA}

Where v⃗BA\vec v_{BA} is the velocity of BB with respect to AA. Both observers measure the same acceleration for the particle:

a⃗PA=a⃗PB\vec a_{PA} = \vec a_{PB}

Two Dimensions

+

In this scenario, the head-to tail vectors can be arranged to get the following formula:

r⃗PA=r⃗PB+r⃗BA\vec r_{PA} = \vec r_{PB} + \vec r_{BA}

Taking the time derivative gives a similar formula for velocity:

v⃗PA=v⃗PB+v⃗BA\vec v_{PA} = \vec v_{PB} + \vec v_{BA}

And the following equation for acceleration:

a⃗PA=a⃗PB\vec a_{PA} = \vec a_{PB}

As for one-dimensional motion, we have the following rule: Observers on different frames of reference that move at constant velocity relative to each other will measure the same acceleration for a moving particle.

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Unit 5: Force and Motion - I

5.1 - Newton's First and Second Laws

First Law

If no net force acts on a body (F⃗net=0\vec F_{net} = 0), the body’s velocity cannot change; that is, the body cannot accelerate. External forces are required to cause acceleration.

Inertial Reference Frames

Newton’s first law is not true in all reference frames, but we can always find reference frames in which it (as well as the rest of Newtonian mechanics) is true. Such special frames are referred to as inertial reference frames, or simply inertial frames.

Second Law

The net force on a body is equal to the product of the body’s mass and its acceleration.

F⃗net=ma⃗\vec F_{net} = m \vec a

Where F⃗net\vec F_{net} is the sum of all forces acting on the body (the net force).

This does not necessarily mean that there are no forces on the objects, but rather that all of the forces on the body are in equilibrium, meaning that the net force is zero.

5.2 - Some Particular Forces

The Gravitational Force

Fg=mgF_g = mg

The gravitational force is a force directly downward and only in the y direction, which equals the mass of an object times the acceleration due to gravity (9.8m/s).

Weight

The weight W of a body is the magnitude of the net force required to prevent the body from falling freely, as measured by someone on the ground.

The Normal Force

The force of a solid object pushing back on an object pushing into it. Results in a balanced force, when, for example, an object affected by gravity is sitting on a flat surface.

FN−Fg=mayF_N - F_g = ma_y

Friction

Friction is any force which opposes motion. There are two forms of friction, kinetic and static, which we will learn more about later.

Tension

When a cord (or a rope, cable, or other such object) is attached to a body and pulled taut, the cord pulls on the body with a force T⃗\vec T directed away from the body and along the cord. The force is often called a tension force because the cord is said to be in a state of tension (or to be under tension), which means that it is being pulled taut. The tension in the cord is the magnitude T of the force on the body. For example, if the force on the body from the cord has magnitude T=50NT = 50 N, the tension in the cord is 50 N.

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Unit 6: Force and Motion - II

6.1 - Friction

  • When a force tends to slide a body along a surface, a frictional force from the surface acts on the body. The frictional force is parallel to the surface and directed so as to oppose the sliding. It is due to bonding between the body and the surface. If the body does not slide, the frictional force is a static frictional force f⃗s\vec f_s. If there is sliding, the frictional force is a kinetic frictional force f⃗k\vec f_k.
  • If a body does not move, the static frictional force f⃗s\vec f_s and the component of F⃗\vec F parallel to the surface are equal in magnitude, and f⃗s\vec f_s is directed opposite that component. If the component increases, f⃗s\vec f_s also increases.
  • The magnitude of f⃗s\vec f_s has a maximum value, f⃗s,max\vec f_{s,max} given by:
    fs,max=μsFNf_{s,max} = \mu_sF_N

    where μs\mu_s is the coefficient of static friction and FNF_N is the magnitude of the normal force. If the component of F⃗\vec F parallel to the surface exceeds f⃗s,max\vec f_{s,max}, the body slides on the surface.

  • If the body begins to slide on the surface, the magnitude of the frictional force rapidly decreases to a constant value fkf_k given by:
    fk=μkFNf_k = \mu_kF_N

    where μk\mu_k is the coefficient of kinetic friction.

6.2 - The Drag Force and Terminal Speed

Drag Force

When there is relative motion between air (or some other fluid) and a body, the body experiences a drag force D⃗\vec D that opposes the relative motion and points in the direction in which the fluid flows relative to the body. The magnitude of D⃗\vec D is related to the relative speed vv by an experimentally determined drag coefficient CC according to:

D=12CρAv2D = \frac{1}{2} C \rho A v^2

where ρ\rho is the fluid density (mass per unit volume) and AA is the effective cross-sectional area of the body (the area of a cross section taken perpendicular to the relative velocity v⃗\vec v).

Terminal Speed

When a blunt object has fallen far enough through the air, the magnitudes of the drag force D⃗\vec D and the gravitational force F⃗g\vec F_g on the body become equal. The body then falls at a constant terminal speed vtv_t given by:

vt=2FgCρAv_t = \sqrt{\frac{2F_g}{C \rho A}}

6.3 - Uniform Circular Motion

If a particle moves in a circle or a circular arc of radius RR at constant speed vv, the particle is said to be in uniform circular motion. It then has a centripetal acceleration with magnitude given by:

a=v2Ra = \frac{v^2}{R}

This acceleration is due to a net centripetal force on the particle, with magnitude given by:

F=mv2RF = \frac{mv^2}{R}

where mm is the particle's mass. The vector quantities a⃗\vec a and F⃗\vec F are directed toward the center of the curvature of the particle's path.

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Unit 7: Kinetic Energy and Work

7.1 - Kinetic Energy

The kinetic energy KK associated with the motion of a particle of mass mm and speed vv, where vv is well below the speed of light, is:

K=12mv2K = \frac{1}{2}mv^2

The SI unit of kinetic energy is the joule (J), defined as:

1 joule=1J=1kg∗m2/s2\textrm{1 joule} = 1 \textrm{J} = 1 \textrm{kg} * \textrm{m}^2/\textrm{s}^2

7.2 - Work and Kinetic Energy

Work WW is energy transferred to or from an object via a force acting on the object. Energy transferred to the object is positive work, and from the object, negative work.

When two or more forces act on an object, their net work is the sum of the individual works done by the forces, which is also equal to the work that would be done on the object by the net force F⃗net\vec F_{net} of those forces.

For a particle, a change ΔK\Delta K in the kinetic energy equals the net work WW done on the particle:

ΔK=Kf−Ki=W\Delta K = K_f - K_i = W

In which KiK_i is the initial kinetic energy of the particle and KfK_f is the kinetic energy after the work is done. This is known as the work-kinetic energy theorem. This formula can also be rearranged as the following:

Kf=Ki+WK_f = K_i + W

The work done on a particle by a constant force F⃗\vec F during displacement d⃗\vec d is:

W=Fdcos⁡ϕ=F⃗⋅d⃗W = Fd \cos \phi = \vec F \cdot \vec d

in which ϕ\phi is the constant angle between the directions of F⃗\vec F and d⃗\vec d.

Only the component of F⃗\vec F that is along the displacement d⃗\vec d can do work on the object.

7.3 - Work Done by the Gravitational Force

The work WgW_g done by the gravitational force FgF_g on a particle-like object of mass mm as the object moves through a displacement d⃗\vec d is given by:

Wg=mgdcos⁡ϕW_g = mgd \cos \phi

in which ϕ\phi is the angle between F⃗g\vec F_g and d⃗\vec d.

The work WaW_a done by an applied force as a particle-like object is either lifted or lowered is related to the work WgW_g done by the gravitational force and the change ΔK\Delta K in the object's kinetic energy by:

ΔK=Kf−Ki=Wa+Wg\Delta K = K_f - K_i = W_a + W_g

If Kf=KiK_f = K_i, that is, the object's kinetic energy did not change, then the equation reduces to:

Wa=−WgW_a = -W_g

or in other words, the applied force transfers as much energy to the object as the gravitational force transfers from it.

7.4 - Work Done by a Spring Force

The force F⃗s\vec F_s from a spring is:

F⃗s=−kd⃗\vec F_s = -k\vec d

where d⃗\vec d is the displacement of the spring's free end from its position when the spring is in its relaxed state (not compressed or extended), and kk is the spring constant (a measure of the spring's stiffness). If an xx axis lies along the spring, with the origin at the location of the spring's free end when the spring is in its relaxed state, we can write:

Fx=−kxF_x = -kx

A spring force is thus a variable force: It varies with the displacement of the spring's free end. (Yay calculus!)

If an object is attached to the spring's free end, the work WsW_s done on the object by the spring force when the object is moved from an initial position xix_i to a final position xfx_f is:

Ws=12kxi2−12kxf2W_s = \frac{1}{2}kx_i^2 - \frac{1}{2}kx_f^2

If xi=0x_i = 0 and xf=xx_f = x, the equation could be written as:

Ws=−12kx2W_s = -\frac{1}{2}kx^2

Written as an integral makes this easier to work with:

Ws=∫xixf−Fx dxW_s = \int_{x_i}^{x_f} -F_x \space dx
Ws=∫xixf−kx dxW_s = \int_{x_i}^{x_f} -kx \space dx

7.5 - Work Done by a General Variable Force

When the force F⃗\vec F on a particle-like object depends on the position of the object, the work done by F⃗\vec F on the object while the object moves from an initial position rir_i with coordinates (xix_i, yiy_i, ziz_i) to a final position rfr_f with coordinates (xfx_f, yfy_f, zfz_f) must be found by integrating the force.

If we assume that components only depend on their respective coordinates, then the work is:

W=∫xixfFx dx+∫yiyfFy dy+∫zizfFz dzW = \int_{x_i}^{x_f} F_x \space dx + \int_{y_i}^{y_f} F_y \space dy + \int_{z_i}^{z_f} F_z \space dz

If F⃗\vec F only has an xx component, then this reduces to:

W=∫xixfF(x) dxW = \int_{x_i}^{x_f} F(x) \space dx

7.6 - Power

The power due to a force is the rate at which that force does work on an object. If the force does work WW during a time interval Δt\Delta t, the average power due to the force over that time interval is:

Pavg=WΔtP_{avg} = \frac{W}{\Delta t}

Instantaneous power is the instantaneous rate of doing work:

P=dWdtP = \frac{dW}{dt}

In other words, Power is the derivative of Work.

For a force F⃗\vec F at an angle ϕ\phi to the direction of travel of the instantaneous velocity v⃗\vec v, the instantaneous power is:

P=Fvcos⁡ϕ=F⃗⋅v⃗P = Fv \cos \phi = \vec F \cdot \vec v
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Unit 8: Potential Energy and Conservation of Energy

8.1 - Potential Energy

Conservative Forces

A force is a conservative force if the net work it does on a particle moving around any closed path, from an initial point and then back to that point, is zero. Equivalently, a force is conservative if the net work it does on a particle moving between two points does not depend on the path taken by the particle.

The gravitational force and the spring force are examples of conservative forces; while the kinetic frictional force is a non-conservative force.

Potential Energy

Potential energy is energy that is associated with the configuration of a system in which a conservative force acts.

When the conservative force does work WW on a particle within the system, the change ΔU\Delta U in the potential energy of the system is:

ΔU=−W\Delta U = -W

As a practical example of this, when an object is lifted up with work WW, the resulting change in its potential energy ΔU\Delta U is equal to the inverse of WW.

If a particle moves from point xix_i to xfx_f, the change in potential energy can be calculated using an integral:

ΔU=−∫xixfF(x) dx\Delta U = -\int_{x_i}^{x_f} F(x) \space dx

Gravitational Potential Energy

The potential energy associated with a system consisting of Earth and a nearby particle if gravitational potential energy.

If the particle moves from height yiy_i to height yfy_f, the change in gravitational potential energy is:

ΔU=mg Δy\Delta U = mg \space \Delta y

Elastic Potential Energy

Elastic potential energy is the energy associated with the state of compresssion or extension of an elastic object (spring). For a spring that exerts a spring force F=−kxF = -kx when its free end has displacement xx, the elastic potential energy is:

U(x)=12kx2U(x) = \frac{1}{2} k x^2

In this reference configuration, the spring is at its relaxed length where x=0x = 0 and U=0U = 0.

8.2 - Conservation of Mechanical Energy

The mechanical energy EmecE_{mec} of a system is the sum of its kinetic energy KK and potential energy UU:

Emec=K+UE_{mec} = K + U

In an isolated system with no external forces causing energy changes (no friction, only conservative forces), them the mechanical energy EmecE_{mec} of the system cannot change. This is known as the conservation of mechanical energy, and can be written as:

K2+U2=K1+U1K_2 + U_2 = K_1 + U_1

in which the subscripts refer to different moments in time within the system. This principle can also be written as:

ΔEmec=ΔK+ΔU=0\Delta E_{mec} = \Delta K + \Delta U = 0

8.3 - Reading a Potential Energy Curve

If we know the potential energy function U(x)U(x) for a systemin which a one-dimensional force F(x)F(x) acts on a particle, we can find the force as:

F(x)=−dU(x)dxF(x) = - \frac{dU(x)}{dx}

If U(x)U(x) is given on a graph, then at any value of xx, the force F(x)F(x) is the inverse of the slope of the curve at that point, and the kinetic energy of the particle is given by:

K(x)=Emec−U(x)K(x) = E_{mec} - U(x)

where EmecE_{mec} is the total mechanical energy in the system.

A "turning point" is a point xx at which the particle reverses its motion (where K=0K = 0).

A particle is at equilibrium at points where the slope of the U(x)U(x) curve is zero (where F(x)=0F(x) = 0).

8.4 - Work Done on a System by an External Force

Important Ideas

  • Work WW is the energy transferred to or from a system by means of an external force acting on the system.
  • When more than one force acts on a system, their net work is the transferred energy.
  • When friction is not involved, the work done on the system and the change ΔEmec\Delta E_{mec} in the mechanical energy of the system are equal:
    W=ΔEmec=ΔK+ΔUW = \Delta E_{mec} = \Delta K + \Delta U
  • When a kinetic frictional force acts within the system, then the thermal energy EthE_{th} of the system changes. Therefore, the work done on the system is then:
    W=ΔEmec+ΔEthW = \Delta E_{mec} + \Delta E_{th}

    The change ΔEth\Delta E_{th} is related to the magnitude FkF_k of the frictional force and the magnitude dd of the displacement caused by the external force:

    ΔEth=Fkd\Delta E_{th} = F_k d
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Unit 9: Center of Mass and Linear Momentum

9.1 - Center of Mass

Important idea: the center of mass of a system of nn particles, with a total mass MM is the point with coordinates:

xcom=1M∑i=1nmixix_{com} = \frac{1}{M} \sum_{i=1}^n m_i x_i
ycom=1M∑i=1nmiyiy_{com} = \frac{1}{M} \sum_{i=1}^n m_i y_i
zcom=1M∑i=1nmiziz_{com} = \frac{1}{M} \sum_{i=1}^n m_i z_i

Which can also be represented in vector form:

r⃗com=1M∑i=1nmir⃗i\vec r_{com} = \frac{1}{M} \sum_{i=1}^n m_i \vec r_i

For a solid mass, such as a ball, which contains a massive number of particles, the object can be treated as a continuous distribution of matter. The "particles" then become differential mass elements dmdm, and the coordinates for the center of mass can be represented as:

xcom=1M∫x dmx_{com} = \frac{1}{M} \int x \space dm
ycom=1M∫y dmy_{com} = \frac{1}{M} \int y \space dm
zcom=1M∫z dmz_{com} = \frac{1}{M} \int z \space dm

where MM is now the mass of the object.

In order to evaluate these integrals with common densities, we use:

ρ=dmdV=MV\rho = \frac{dm}{dV} = \frac{M}{V}

and substitute into the above equations to finally get:

xcom=1V∫x dVx_{com} = \frac{1}{V} \int x \space dV
ycom=1V∫y dVy_{com} = \frac{1}{V} \int y \space dV
zcom=1V∫z dVz_{com} = \frac{1}{V} \int z \space dV

where VV is the total volume of the object.

💡
Symmetry as a Shortcut: You can bypass one or more of these integrals if +an object has a point, a line, or a plane of symmetry. The center of mass of such +an object then lies at that point, on that line, or in that plane.

9.2 - Newton's Second Law for a System of Particles

The motion of the center of mass of any system of particles is governed by Newton’s second law for a system of particles, which is:

F⃗net=Ma⃗com\vec F_{net} = M\vec a_{com}

where F⃗net\vec F_{net} is the net force of all external forces acting on the system, MM is the total mass of the system, and a⃗com\vec a_{com} is the acceleration of the system's center of mass.

9.3 - Linear Momentum

For a single particle, we can define a quantity p⃗\vec p called its linear momentum as:

p⃗=mv⃗\vec p = m \vec v

which is a vector quantity in the same direction as the particle's velocity.

Newton's second law can also be written in terms of momentum:

F⃗net=dp⃗dt\vec F_{net} = \frac{d \vec p}{dt}

Systems of Particles

For a system of particles, these relations can be written as the following:

P⃗=Mv⃗com\vec P = M \vec v_{com}
F⃗net=dP⃗dt\vec F_{net} = \frac{d \vec P}{dt}

9.4 - Collision and Impulse

Impulse-Linear Momentum Theorem

The change in a body's linear momentum over time during a collision is the impulse, J⃗\vec J.

p⃗f−p⃗i=Δp⃗=J⃗\vec p_f - \vec p_i = \Delta \vec p = \vec J

The impulse can also be written as the integral of the force F⃗(t)\vec F(t) exerted on the body in the collision:

J⃗=∫titfF⃗(t) dt\vec J = \int_{t_i}^{t_f} \vec F(t) \space dt

One-Dimensional Motion

If FavgF_{avg} is the average magnitude of F⃗(t)\vec F(t) during the collision and Δt\Delta t is the duration of the collision, then for one-dimensional motion the impulse can be simplified as

J=FavgΔtJ = F_{avg} \Delta t

Steady Stream of Collisions with a Fixed Body

When a steady stream of bodies, each with mass mm and speed vv, collides with a body whose position is fixed, the average force on the fixed body is:

Favg=−nΔtΔp=−nΔtmΔvF_{avg} = -\frac{n}{\Delta t} \Delta p = -\frac{n}{\Delta t} m \Delta v

where n/Δtn/\Delta t is the rate at which the bodies collide with the fixed body, and Δv\Delta v is the change in velocity of each colliding body. The average force can also be written as:

Favg=−ΔmΔtΔvF_{avg} = -\frac{\Delta m}{\Delta t} \Delta v

where Δm/Δt\Delta m / \Delta t is the rate at which mass collides with the fixed body. The change in velocity is Δv=−v\Delta v = -v if the bodies stop upon impact and Δv=−2v\Delta v = -2v if they bounce backward with no change in their speed.

9.5 - Conservation of Linear Momentum

If a system is closed and isolated so that no net external force acts on it, then the linear momentum P must be constant even if there are internal changes:

P⃗i=P⃗f\vec P_i = \vec P_f

9.6 - Momentum and Kinetic Energy in Collisions

In an inelastic collision of two bodies, the kinetic energy of the two-body system is not conserved. If the system is closed and isolated, the total linear momentum of the system must be conserved, which we can write in vector form as:

p⃗1i+p⃗2i=P⃗2i+p⃗2f\vec p_{1i} + \vec p_{2i} = \vec P_{2i} + \vec p_{2f}

where the ii subscripts are before the collision and the ff subscripts are the values right after the collision.

If the motion of the bodies is along a single axis, the collision is one-dimensional and we can write the equation in terms of velocity components along that axis:

m1v1i+m2v2i=m1v1f+m2v2fm_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}

Bodies Sticking Together

If the bodies colliding stick together after the collision, then the collision is completely inelastic and the bodies have the same final velocity VV.

Center of Mass

The center of mass of a closed, isolated system of two colliding bodies is not affected by a collision. In particular, the velocity v⃗com\vec v_{com} of the center of mass cannot be changed by the collision.

9.7 - Elastic Collisions in One Dimension

An elastic collision is a special type of collision in which the kinetic energy of a system of colliding bodies is conserved. If the system is closed and isolated, its linear momentum is also conserved. For a one-dimensional collision in which body 2 is a target and body 1 is an incoming projectile, conservation of kinetic energy and linear momentum yield the following expressions for the velocities immediately after the collision:

v1f=m1−m2m1+m2v1iv_{1f} = \frac{m_1-m_2}{m_1+m_2} v_{1i}
v2f=2m1m1+m2v1iv_{2f} = \frac{2m_1}{m_1+m_2} v_{1i}
💡
You don't necessarily have to memorize these, they are simply the result of the conservation of the kinetic energy of the system during the collision.

9.8 - Collisions in Two Dimensions

If two bodies collide and their motion is not along single axis (the collision is not straight on), the collision is two-dimensional. If the two-body system is closed and isolated, the law of conservation of momentum applies and can be written as:

P⃗1i+P⃗2i=P⃗1f+P⃗2f\vec P_{1i} + \vec P_{2i} = \vec P_{1f} + \vec P_{2f}

In the case that the collision is also elastic, then the total kinetic energy is also conserved:

K1i+K2i=K1f+K2fK_{1i} + K_{2i} = K_{1f} + K_{2f}

9.9 - Systems with Varying Mass: A Rocket

+

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Video 1: Motion

Definitions

  • Kinematics: The study of describing motion.
  • Dynamics: The study of the causes of motion.

Terms to describe Linear Motion

  • Distance (dd)
  • Position (xx)
  • Displacement (Δx\Delta x)
  • Average Speed (SavgS_{avg})
  • Average Velocity (VavgV_{avg})
  • Instantaneous Speed
  • Instantaneous Velocity
  • Average Acceleration
  • Instantaneous Acceleration

Basic Formulas

Savg=distancetimeS_{avg} = \frac{distance}{time}
V⃗avg=ΔxΔt\vec V_{avg} = \frac{\Delta x}{\Delta t}

Instantaneous speed is a very, very small distance divided by a very very small time.

V=lim⁡Δt→0ΔxΔt=dxdtV = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}

In other words, velocity is the derivative of position (x) with respect to time (t).

a⃗avg=ΔVΔt\vec a_{avg} = \frac{\Delta V}{\Delta t}
a⃗=lim⁡Δt→0ΔvΔt=dvdt\vec a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt}

In other words, acceleration is the derivative of velocity (v⃗\vec v) with respect to time (t).

+

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Video 2: Now with Calculus

Deriving Kinematics Equations

Change in Velocity

a⃗=dvdt\vec a = \frac{dv}{dt}
a⃗∗dt=dv\vec a * dt = dv
∫0tadt=∫vivfdv\int_0^t a dt = \int_{v_i}^{v_f} dv
at∣0t=v∣vivfat |_0^t = v|_{v_i}^{v_f}
at−a(0)=vf−viat - a(0) = v_f - v_i
vf=vi+atv_f = v_i + at

Change in Position

v=vi+atv = v_i + at
dxdt=vi+at\frac{dx}{dt} = v_i + at
dx=(vi+at)dtdx = (v_i + at)dt
∫xixfdx=∫0t(vi+at)dt\int_{x_i}^{x_f} dx = \int_0^t (v_i + at)dt
x∣xixf=(vit+12at2)∣0tx|_{x_i}^{x_f} = (v_it + \frac{1}{2} a t^2)|_{0}^{t}
xf−xi=(vit+12at2)−0x_f - x_i = (v_i t + \frac{1}{2}at^2) - 0
Δx=vit+12at2\Delta x = v_i t + \frac{1}{2}at^2

Bonus Kinematics Equation

a⃗=dvdt\vec a = \frac{dv}{dt}
a⃗=dvdxdxdt\vec a = \frac{dv}{dx} \frac{dx}{dt}
a⃗=v⃗dvdx\vec a = \vec v \frac{dv}{dx}
∫xixfa⃗dx=∫v0vfvdv\int_{x_i}^{x_f} \vec a dx = \int_{v_0}^{v_f} v dv
ax∣xixf=12v2∣v0vfax |_{x_i}^{x_f} = \frac{1}{2} v^2 |_{v_0}^{v_f}
2aΔx=vf2−vi22 a \Delta x = v_f^2 - v_i^2
vf2=vi2+2aΔxv_f^2 = v_i^2 + 2a\Delta{x}
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