Gauss's Law

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.

Worked example

A solid nonconducting sphere of radius R carries charge Q spread uniformly through its volume. Find the field inside, at r < R.

1. Charge density: ρ = Q / (4/3 πR³).
2. Charge inside a Gaussian sphere of radius r: Q_enc = ρ × 4/3 πr³ = Q r³/R³.
3. Gauss's law: E × 4πr² = Q r³ / (ε₀ R³).
4. 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.