AP PHYSICS C E&M — physicsHard3 min read

Electrostatics

AP Physics C E&M guide to electrostatics: Coulomb's law, electric fields, Gauss's law, and electric field calculations for symmetric charge distributions.

Tutor AI Team2026-02-26T00:16:00.126Z

# Electrostatics — AP Physics C E&M

AP Physics C: E&M begins with electrostatics — the study of stationary charges and the electric fields they produce. You must apply Coulomb's law, calculate fields using integration, and use Gauss's law for symmetric charge distributions.

Key Concepts

Coulomb's Law (Vector Form)

$\vec{F} = \frac{kq_1 q_2}{r^2}\hat{r}$

Electric Field

$\vec{E} = \frac{\vec{F}}{q_0} = \frac{kQ}{r^2}\hat{r}$

Continuous charge distribution: $\vec{E} = k\int \frac{dq}{r^2}\hat{r}$

Common linear charge density: $\lambda = dq/dL$ ; surface: $\sigma = dq/dA$ ; volume: $\rho = dq/dV$ .

Gauss's Law

$\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\epsilon_0}$

Use when there is high symmetry (spherical, cylindrical, planar).

Common results:

Infinite line charge: $E = \frac{\lambda}{2\pi\epsilon_0 r}$
Infinite plane of charge: $E = \frac{\sigma}{2\epsilon_0}$
Uniformly charged sphere (outside): $E = \frac{Q}{4\pi\epsilon_0 r^2}$ (like a point charge)
Uniformly charged sphere (inside): $E = \frac{Qr}{4\pi\epsilon_0 R^3}$

Electric Field of a Charged Ring (on axis)

$E_x = \frac{kQx}{(x^2 + R^2)^{3/2}}$

Electric Field of a Charged Disk (on axis)

$E_x = \frac{\sigma}{2\epsilon_0}\left(1 - \frac{x}{\sqrt{x^2 + R^2}}\right)$

Worked Example

Problem: Use Gauss's law to find the electric field inside and outside a uniformly charged solid sphere of total charge $Q$ and radius $R$ .

Solution:

Outside ( $r > R$ ): Gaussian sphere of radius $r$ encloses all charge $Q$ . $E(4\pi r^2) = Q/\epsilon_0 \quad \Rightarrow \quad E = \frac{Q}{4\pi\epsilon_0 r^2}$

Inside ( $r < R$ ): Enclosed charge $Q_{\text{enc}} = Q(r/R)^3$ . $E(4\pi r^2) = \frac{Q(r/R)^3}{\epsilon_0} \quad \Rightarrow \quad E = \frac{Qr}{4\pi\epsilon_0 R^3}$

Inside, $E \propto r$ ; outside, $E \propto 1/r^2$ .

Practice Questions

1. Find the electric field at a distance $r$ from an infinite line of charge with linear charge density $\lambda$ using Gauss's law.

Cylindrical Gaussian surface of length $L$ and radius $r$ : $E(2\pi r L) = \lambda L/\epsilon_0$ . $E = \lambda/(2\pi\epsilon_0 r)$ .

2. A conducting sphere of radius $R$ carries charge $Q$ . What is the field at the surface?

$E = Q/(4\pi\epsilon_0 R^2) = \sigma/\epsilon_0$ (since all charge is on the surface).

3. Set up the integral for the electric field on the axis of a uniformly charged ring of radius $R$ and total charge $Q$ .

By symmetry, only the $x$ -component survives. $dE_x = \frac{k\,dq}{(x^2+R^2)} \cdot \frac{x}{\sqrt{x^2+R^2}}$ . Integrating over the ring: $E_x = kQx/(x^2+R^2)^{3/2}$ .

Want to check your answers and get step-by-step solutions?

Summary

Electric field from continuous distributions: $\vec{E} = k\int dq/r^2\,\hat{r}$ .
Gauss's law: $\oint \vec{E} \cdot d\vec{A} = Q_{\text{enc}}/\epsilon_0$ — powerful for symmetric distributions.
Inside a conductor: $E = 0$ ; charge resides on the surface.
Know standard results for sphere, cylinder (line), and plane.

Tags:

#ap#physics#electrostatics

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