When can you use Gauss's law to find a field, instead of integrating?

Use Gauss's law whenever the charge distribution has spherical, cylindrical or planar symmetry, so the field is constant in magnitude and either perpendicular or parallel to a well-chosen Gaussian surface. Then the surface integral collapses to E times an area equal to the enclosed charge over epsilon-zero. If the symmetry is missing (a finite rod, a dipole), Gauss's law is still true but cannot isolate the field, so you fall back to integrating the Coulomb field of each element dq.

What is the standard recipe for integrating a charge distribution?

Write the element dq using the density (dq equals lambda dx, sigma dA or rho dV), find the distance r from dq to the field point in terms of the integration variable, use symmetry to keep only the surviving component (often writing it as dE cos theta), then integrate over the whole distribution with limits matching the coordinate. Always finish by checking a far-field limit reduces to a point charge.

How do you choose a Gaussian surface versus an Amperian loop?

A Gaussian surface is a closed surface for Gauss's law (a sphere for a point or ball of charge, a coaxial cylinder for a line, a pillbox for a sheet) chosen so the field is constant and perpendicular over the part with flux. An Amperian loop is a closed path for Ampere's law (a circle around a wire, a rectangle through a solenoid wall) chosen so the field is constant and parallel along the part that contributes. In both, the art is matching the shape to the symmetry so the field factors out of the integral.

How do you handle RC, LR and LC circuits on the exam?

Write Kirchhoff's loop rule, substitute the current as dq/dt (and the inductor voltage as L dI/dt), and you get a differential equation. RC and LR circuits give first-order equations with exponential solutions and a time constant (RC or L/R); LC circuits give a second-order equation with a sinusoidal solution and angular frequency 1 over root LC. Know the initial behavior (a capacitor acts like a wire, an inductor like an open switch) and the final behavior (capacitor open, inductor a wire).

How are calculus-based free-response questions marked?

Markers award points for the correct setup, not just the final number. Show the element and its field or the loop equation, the integral or differential equation with limits, the substitution, and the evaluated result with units. A correct Gaussian surface, a clearly stated symmetry argument, or a properly separated differential equation each earns marks, so always show the physics and the calculus steps, not just the answer.

United StatesPhysics C: Electricity and Magnetism

AP Physics C E&M applying Gauss's law and calculus: a complete skills guide to integrating charge distributions, choosing Gaussian and Amperian surfaces, and solving RC, LR and LC circuits

A deep-dive AP Physics C E&M skills guide to the calculus core: integrating charge distributions for fields and potentials, choosing Gaussian surfaces for Gauss's law and Amperian loops for Ampere's law, the Biot-Savart law, and solving the RC, LR and LC differential equations. With worked examples and full-mark exam technique.

Generated by Claude Opus 4.820 min read8.1-13.6Updated 2026-06-04

Reviewed by: AI editorial process; not yet individually human-reviewed

Jump to a section

Why this is the spine of AP Physics C E&M
Integrating a charge distribution
Gauss's law: when symmetry does the work
Ampere's law and the Biot-Savart law
Differential equations of circuits
Check your knowledge

Why this is the spine of AP Physics C E&M

What sets AP Physics C: Electricity and Magnetism apart from the algebra-based courses is calculus, and three calculus skills carry most of the exam: integrating a charge or current distribution, exploiting symmetry through Gauss's law and Ampere's law, and solving the differential equations of RC, LR and LC circuits. This guide ties together the matching dot-point pages, each with its own practice questions: Gauss's law, electric fields of charge distributions, Ampere's law, the Biot-Savart law, RC circuits, and LC circuits.

Integrating a charge distribution

The direct method for any continuous distribution treats each element $dq$ as a point charge. The recipe is fixed:

Write $dq$ with the density: $dq = \lambda\,dx$ (line), $\sigma\,dA$ (surface), or $\rho\,dV$ (volume).
Find $r$ , the distance from $dq$ to the field point, in terms of the integration variable.
Keep the surviving component using symmetry, writing it as $dE\cos\theta$ (or $dE\sin\theta$ ) and expressing $\cos\theta$ through the geometry.
Integrate with limits matching the coordinate.

For the field, $\vec{E} = \displaystyle\int \dfrac{k\,dq}{r^2}\hat{r}$ ; for the potential (a scalar, so no components), $V = \displaystyle\int \dfrac{k\,dq}{r}$ . Often the cleanest route is to find $V$ first and then take $\vec{E} = -\nabla V$ .

Gauss's law: when symmetry does the work

Gauss's law, $\oint\vec{E}\cdot d\vec{A} = \dfrac{q_{enc}}{\varepsilon_0}$ , is always true but only solves for the field when symmetry lets you pull $E$ out of the integral. Match the Gaussian surface to the symmetry:

Spherical (point, ball, shell): concentric sphere, giving $E(4\pi r^2) = \dfrac{q_{enc}}{\varepsilon_0}$ .
Cylindrical (line, long cylinder): coaxial cylinder, giving $E(2\pi r L) = \dfrac{q_{enc}}{\varepsilon_0}$ (the end caps carry no flux).
Planar (infinite sheet): a pillbox, giving $E = \dfrac{\sigma}{2\varepsilon_0}$ .

The crucial habit: count only the charge enclosed by the surface. Inside a uniformly charged ball that is the fraction $r^3/R^3$ ; inside a uniform wire (for Ampere's law) the fraction $r^2/a^2$ .

Ampere's law and the Biot-Savart law

Magnetism mirrors electrostatics. The Biot-Savart law is the direct integral, the magnetic counterpart of integrating the Coulomb field:

d\vec{B} = \frac{\mu_0}{4\pi}\frac{I\,d\vec{\ell}\times\hat{r}}{r^2}

Use it when there is no symmetry (a loop's central or axial field). Ampere's law, $\oint\vec{B}\cdot d\vec{\ell} = \mu_0 I_{enc}$ , is the symmetry shortcut, the magnetic Gauss's law: choose an Amperian loop on which $\vec{B}$ is constant and parallel (a circle around a wire, a rectangle through a solenoid wall) so the line integral becomes $B\times\text{length} = \mu_0 I_{enc}$ . The standard results are $B = \dfrac{\mu_0 I}{2\pi r}$ (wire) and $B = \mu_0 n I$ (solenoid).

Differential equations of circuits

Whenever a capacitor or inductor is present, Kirchhoff's loop rule becomes a differential equation, because $I = \dfrac{dq}{dt}$ and the inductor voltage is $L\dfrac{dI}{dt}$ :

RC charging: $\varepsilon = R\dfrac{dq}{dt} + \dfrac{q}{C}$ gives $q = C\varepsilon(1 - e^{-t/RC})$ , time constant $\tau = RC$ .
LR rise: $\varepsilon = IR + L\dfrac{dI}{dt}$ gives $I = \dfrac{\varepsilon}{R}(1 - e^{-Rt/L})$ , time constant $\tau = \dfrac{L}{R}$ .
LC oscillation: $L\dfrac{d^2 q}{dt^2} + \dfrac{q}{C} = 0$ gives $q = Q_0\cos(\omega t)$ , angular frequency $\omega = \dfrac{1}{\sqrt{LC}}$ .

Anchor every one with the limiting behavior: a capacitor starts as a wire and ends as an open switch; an inductor starts as an open switch and ends as a wire.

Field of an infinite line of charge with Gauss's law

An infinite straight line carries uniform linear charge density $\lambda$ . Find the field a distance $r$ away.

step 1 Identify the symmetry and choose the surface

Cylindrical symmetry: the field is radial and constant on a coaxial cylinder of radius $r$ and length $L$ .

step 2 Evaluate the flux

The curved side has $\vec{E}\parallel d\vec{A}$ , giving $E(2\pi r L)$ ; the flat end caps have $\vec{E}\perp d\vec{A}$ , contributing zero.

step 3 Apply Gauss's law

Enclosed charge $= \lambda L$ . So $E(2\pi r L) = \dfrac{\lambda L}{\varepsilon_0}$ .

step 4 Solve for the field

$E = \dfrac{\lambda}{2\pi\varepsilon_0 r}$ , falling off as $1/r$ , directed radially away from a positive line.

Charging an RC circuit

A capacitor $C$ , initially uncharged, charges through a resistor $R$ from a battery $\varepsilon$ . Find the charge as a function of time.

step 1 Write the loop rule

$\varepsilon - IR - \dfrac{q}{C} = 0$ , with $I = \dfrac{dq}{dt}$ , so $\varepsilon = R\dfrac{dq}{dt} + \dfrac{q}{C}$ .

step 2 Separate variables

Rearranging, $\dfrac{dq}{C\varepsilon - q} = \dfrac{dt}{RC}$ .

step 3 Integrate with $q(0) = 0$

$-\ln(C\varepsilon - q)\Big|_0^q = \dfrac{t}{RC}$ , giving $\ln\dfrac{C\varepsilon - q}{C\varepsilon} = -\dfrac{t}{RC}$ .

step 4 Solve and check limits

$q(t) = C\varepsilon\left(1 - e^{-t/RC}\right)$ . At $t = 0$ , $q = 0$ (capacitor uncharged, acts like a wire); as $t\to\infty$ , $q\to C\varepsilon$ (fully charged, acts like an open switch). The current is $I = \dfrac{dq}{dt} = \dfrac{\varepsilon}{R}e^{-t/RC}$ .

AP Physics C: Electricity and Magnetism rests on three calculus skills. (1) Integrate a distribution: write $dq$ with the density, find $r$ , keep the surviving component by symmetry, and integrate ( $\vec{E} = \int k\,dq/r^2\,\hat{r}$ ; $V = \int k\,dq/r$ as a scalar). (2) Exploit symmetry: Gauss's law $\oint\vec{E}\cdot d\vec{A} = q_{enc}/\varepsilon_0$ and Ampere's law $\oint\vec{B}\cdot d\vec{\ell} = \mu_0 I_{enc}$ collapse to one-line calculations for spherical, cylindrical or planar symmetry, counting only the enclosed charge or current; without symmetry, use the direct Coulomb or Biot-Savart integral. (3) Solve circuit differential equations: substitute $I = dq/dt$ (and $L\,dI/dt$ ) into the loop rule to get exponential RC and LR solutions (time constants $RC$ and $L/R$ ) or the sinusoidal LC solution ( $\omega = 1/\sqrt{LC}$ ), always anchored by the initial and final behavior of capacitors and inductors.

Check your knowledge

A mix of recall, derivation and calculation questions covering the calculus core. Attempt them under timed conditions, then check against the solutions.

State the recipe (four steps) for finding the field of a continuous charge distribution by integration. (2 marks)
When does Gauss's law let you solve for the field, and when must you integrate instead? (2 marks)
Write the field of an infinite sheet of surface charge density $\sigma$ . (1 mark)
A solid insulating sphere of radius $R$ has uniform charge $Q$ . State the field at $r < R$ . (2 marks)
Write the field a distance $r$ from a long straight wire carrying current $I$ . (1 mark)
State the magnetic field inside a long solenoid with $n$ turns per meter carrying current $I$ . (1 mark)
Write the differential equation for the charge on a capacitor charging through a resistor from a battery $\varepsilon$ . (2 marks)
State the time constant of an LR circuit and the final current. (2 marks)
State the angular frequency of an LC circuit. (1 mark)
State the initial and final behavior of an inductor in a DC circuit. (2 marks)

Solutions

Q1: Write $dq$ with the density; find $r$ from $dq$ to the field point; use symmetry to keep the surviving component ( $dE\cos\theta$ ); integrate over the distribution.
Q2: Gauss's law solves for the field when there is spherical, cylindrical or planar symmetry (so $E$ is constant and perpendicular on the Gaussian surface). Without that symmetry, integrate the Coulomb field of each $dq$ .
Q3: $E = \dfrac{\sigma}{2\varepsilon_0}$ , uniform and perpendicular to the sheet.
Q4: $E = \dfrac{kQr}{R^3}$ , growing linearly with $r$ (the Gaussian sphere encloses the fraction $r^3/R^3$ of the charge).
Q5: $B = \dfrac{\mu_0 I}{2\pi r}$ , circling the wire.
Q6: $B = \mu_0 n I$ , uniform inside.
Q7: $\varepsilon = R\dfrac{dq}{dt} + \dfrac{q}{C}$ (Kirchhoff's loop rule with $I = dq/dt$ ).
Q8: $\tau = \dfrac{L}{R}$ ; the final current is $\dfrac{\varepsilon}{R}$ .
Q9: $\omega = \dfrac{1}{\sqrt{LC}}$ .
Q10: At $t = 0$ an inductor acts like an open switch (current cannot jump, so it starts at zero); after a long time it acts like a plain wire (steady current, no back-EMF).

Sources & how we know this

AP Physics C: Electricity and Magnetism Course and Exam Description — College Board (2024)