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 $q$ ), 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 ( $i$ ) is the rate $dq/dt$ at which charge passes a point:

i = \frac{dq}{dt}

Coulomb’s law describes the electrostatic force (or electric force) between two charged particles. If the particles have charges $q_1$ and $q_2$ , are separated by distance $r$ , 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 = \frac{1}{4\pi\epsilon_0} \frac{|q_1||q_2|}{r^2}

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

Units

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

1 \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 \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 $q$ that can be detected can be written as:

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

21.3 - Charge is Conserved

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

Annihilation Process

An annihilation process occurs when an electron $e^-$ (charge $-e$ ) and its antiparticle, the positron $e^+$ (charge $e$ ), 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^+ \rightarrow \gamma + \gamma