Gauss's Law
AP Physics C: Electricity and Magnetism, Unit 8. Electric flux, Gauss's law, choosing a Gaussian surface for spherical, cylindrical and planar symmetry, a worked example and the free response approach.
Gauss's law is the center of Unit 8 in AP Physics C: Electricity and Magnetism, Electric Charges, Fields, and Gauss's Law. It turns a hard integral over a charge distribution into one line of algebra, but only when the charge has enough symmetry.
Electric flux
Flux measures how much field passes through a surface:
Φ = ∮ E · dA
- A field parallel to the surface gives zero flux through it.
- For a closed surface, outward flux is positive and inward flux is negative.
Gauss's law
∮ E · dA = Q_enc / ε₀
Only the charge inside the closed surface counts. A charge outside sends as much flux in as it sends out.
Choosing the Gaussian surface
The surface is a tool you invent. Choose it so that on each part, E is either constant and perpendicular to the surface, or parallel to it.
| Symmetry | Surface | Result |
|---|---|---|
| Point or sphere | concentric sphere | E = kQ/r² outside |
| Long line or cylinder | coaxial cylinder | E = λ/(2πε₀r) |
| Infinite sheet | pillbox through the sheet | E = σ/(2ε₀) |
Worked example
A solid nonconducting sphere of radius R carries charge Q spread uniformly through its volume. Find the field inside, at r < R.
- Charge density:
ρ = Q / (4/3 πR³). - Charge inside a Gaussian sphere of radius r:
Q_enc = ρ × 4/3 πr³ = Q r³/R³. - Gauss's law:
E × 4πr² = Q r³ / (ε₀ R³). - So
E = Q r / (4πε₀ R³) = kQr/R³.
The field grows linearly from zero at the center and matches kQ/R² at the surface. A common mistake is to use the total charge Q for a surface inside the sphere.
Conductors
In electrostatic equilibrium the field inside a conductor is zero, so any excess charge sits on its surface. Just outside the surface, E = σ/ε₀, twice the value for a nonconducting sheet, because all the flux leaves on one side.
The free response approach
- State the symmetry and name your Gaussian surface before writing any equation.
- Write Q_enc as a function of r when the charge is spread through a volume, often with an integral when the density is not uniform.
- Sketch E against r when asked, marking where the function changes form.
Short Lesson Video
Mock Exam
Practice Quiz
Test yourself: instant results and explanations.
1. A closed surface encloses a charge of +4 μC. A charge of −2 μC lies outside the surface. What is the net flux through the surface?
2. A thin spherical shell of radius R carries charge Q on its surface. What is the field at r = R/2?
3. Which Gaussian surface is best for finding the field of a long, uniformly charged wire?
4. How does the field outside a long charged wire depend on the distance r from it?
5. A solid conductor carries excess charge and is in electrostatic equilibrium. Where is the excess charge?
Need support with this topic?
In a free 45-minute intro call we assess your level and build a study plan tailored to you.
Free intro call