Gauss's Law

Gauss's Law is one of the four Maxwell's equations and provides a powerful method for calculating electric fields in situations with high symmetry. It relates the electric flux through a closed surface to the charge enclosed within that surface.

Statement

The net electric flux through any closed surface equals the total charge enclosed divided by $\varepsilon_0$:

$$\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enclosed}}{\varepsilon_0}$$

Or in simpler form: $$\Phi_E = \frac{Q_{enclosed}}{\varepsilon_0}$$

Key Quantities

Symbol Meaning Value/Units
$\Phi_E$ Net electric flux through closed surface N·m²/C
$Q_{enclosed}$ Total charge inside the Gaussian surface C
$\varepsilon_0$ Permittivity of free space $8.85 \times 10^{-12}$ C²/(N·m²)
$\oint$ Integral over closed surface —

The Gaussian Surface

A Gaussian surface is an imaginary closed surface used to apply Gauss's Law.

Choosing a Gaussian Surface

The key to using Gauss's Law is choosing a surface where:

  1. $\vec{E}$ is constant in magnitude over parts of the surface
  2. $\vec{E}$ is parallel to the surface normal (or perpendicular)
  3. The geometry matches the symmetry of the charge distribution

Common Gaussian Surfaces

Charge Distribution Gaussian Surface Symmetry
Point charge Sphere Spherical
Line charge Cylinder Cylindrical
Plane sheet Cylinder (pillbox) Planar
Charged sphere Concentric sphere Spherical

Applications

1. Point Charge

Gaussian Surface: Sphere of radius $r$

$$E(4\pi r^2) = \frac{Q}{\varepsilon_0}$$

$$E = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r^2} = \frac{kQ}{r^2}$$

✓ Recovers Coulomb's Law

2. Infinite Line Charge

Setup: Line with linear charge density $\lambda$ (C/m)

Gaussian Surface: Cylinder of radius $r$, length $L$

$$E(2\pi r L) = \frac{\lambda L}{\varepsilon_0}$$

$$E = \frac{\lambda}{2\pi r \varepsilon_0}$$

Key points:

  • $E \propto \frac{1}{r}$ (not $\frac{1}{r^2}$)
  • No flux through end caps ($\vec{E} \perp \vec{A}$)

3. Infinite Plane Sheet

Setup: Plane with surface charge density $\sigma$ (C/m²)

Gaussian Surface: Cylinder (pillbox) cutting through plane

$$E(2A) = \frac{\sigma A}{\varepsilon_0}$$

$$E = \frac{\sigma}{2\varepsilon_0}$$

Key points:

  • $E$ is constant (independent of distance!)
  • Field perpendicular to plane
  • Equal magnitude on both sides

4. Parallel Conducting Plates

Setup: Two plates with $+\sigma$ and $-\sigma$

Region Result Explanation
Between plates $E = \frac{\sigma}{\varepsilon_0}$ Fields add
Outside plates $E = 0$ Fields cancel

5. Conducting Sphere

Setup: Sphere of radius $R$, total charge $Q$

Region Electric Field
Inside ($r < R$) $E = 0$
Outside ($r > R$) $E = \frac{kQ}{r^2}$

Key insight: All charge resides on the outer surface of a conductor.

Comparison: Coulomb's Law vs Gauss's Law

Aspect Coulomb's Law Gauss's Law
Calculates Force or field from point charges Field from symmetric distributions
Requires Vector addition Algebraic manipulation
Best for Discrete charges Continuous, symmetric distributions
Scope Electrostatics General (including dynamics)

Key Insights

  1. Flux depends only on enclosed charge — external charges don't contribute to net flux
  2. Shape doesn't matter — any closed surface enclosing the same charge has same flux
  3. Symmetry is crucial — choose Gaussian surface to exploit symmetry
  4. Conductors — $E = 0$ inside; all charge on surface

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