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 $\mathscr{E}$ , 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 The term emf comes from the outdated phrase electromotive force, which was adopted before scientists clearly understood the function of an emf device.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 $dW$ is the work the device does to force positive charge $dq$ from the negative to the positive terminal, then the emf (work per unit charge) of the device is:

\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 $R$ in the direction of the current is $-iR$ ; in the opposite direction it is $+iR$ .

EMF Rule

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

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 $\mathscr{E}$ and internal resistance $r$ does work on the charge carriers in an current $i$ through the battery, the rate $P$ of energy transfer to the charge carriers is:

P = iV

where $V$ is the potential across the terminals of the battery.

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

P_r = i^2r

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

P_{\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:

R_{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 ( $V$ ). The equivalent resistance that can replace a parallel combination of resistances is given by:

\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 $\mathscr{E}$ is applied to a resistance $R$ and capacitance $C$ in series, the charge on the capacitor increases according to:

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

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

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

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

q = q_0 e^{-t/RC}

During the discharging, the current is equal to:

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