Gauss's Law & Electric Flux Visualizer

Select a charge distribution, adjust parameters, and see how Gauss's Law determines the electric field. Interactive 2D cross-sections, step-by-step KaTeX derivations, and E(r) field plots for five fundamental geometries.

Parameters

Charge Distribution

Charge Q

Gaussian Surface Radius r

Presets

2D Cross-Section

E(r) Field Magnitude vs Distance

Results for Point Charge

Enclosed Charge Q_enc

1.00e-6 C

Electric Flux Φ

1.13e+5 N·m²/C

E-field |E|

3.60e+6 N/C

Step-by-Step Gauss's Law Derivation

Step 1: Symmetry Argument

Step 2: Gaussian Surface Area

Step 3: Enclosed Charge

Step 4: Gauss's Law

Step 5: By Symmetry, E is Constant on Surface

Step 6: Solve for E

Step 7: Electric Flux

Reference Guide

Gauss's Law

The total electric flux through any closed surface equals the enclosed charge divided by the permittivity of free space.

\Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}

where $\varepsilon_0 = 8.854 \times 10^{-12} \text{ C}^2/(\text{N}\cdot\text{m}^2)$ is the permittivity of free space. The integral is over the entire closed Gaussian surface.

Symmetry Arguments

Gauss's Law is most useful when the charge distribution has high symmetry, letting you pull E out of the integral.

Spherical (point charge, charged sphere) uses a spherical Gaussian surface
Cylindrical (line charge, charged cylinder) uses a cylindrical surface
Planar (infinite plane) uses a pillbox surface

In each case, the symmetry ensures E is constant on the Gaussian surface and either parallel or perpendicular to the area element.

Common Distributions

\text{Point: } E = \frac{Q}{4\pi\varepsilon_0 r^2}

\text{Line: } E = \frac{\lambda}{2\pi\varepsilon_0 r}

\text{Plane: } E = \frac{\sigma}{2\varepsilon_0}

Electric Flux

Electric flux measures how much electric field passes through a surface. For a uniform field passing through a flat area at angle $\theta$ ,

\Phi_E = E \cdot A \cos\theta = \vec{E} \cdot \vec{A}

Flux is positive when field lines exit the surface and negative when they enter. The net flux through a closed surface depends only on the enclosed charge, not on charges outside.