Electric flux, Gaussian surfaces, and conductor shielding
Wrap any closed surface around a charge and count the field passing through it. The total depends only on the charge inside, never on the surface's shape.
Switch between a sphere, a cylinder, and a cube, and change Q. The flux stays Q/ε₀ whichever surface you choose.
The total electric flux through any closed surface equals Q/ε₀, regardless of the surface shape. Purple arrows show E-field; gray arrows show outward normals.
Gauss's law relates the electric flux through a closed surface to the enclosed charge: Φ = ∮ E · dA = Q/ε₀. The flux depends only on the enclosed charge, not on the surface shape. When combined with symmetry, Gauss's law determines the field from highly symmetric charge distributions in a single step.
Choose a Gaussian surface -- sphere, cylinder, or cube -- and verify that the flux equals Q/ε₀ regardless of shape. The surface only needs to enclose the charge.
Key insight: Gauss's law is a consequence of the inverse-square law. The 1/r² dependence of E exactly cancels the r² growth of the surface area, making the flux independent of radius.
Pick a sphere, a line, or a plane, then step through the derivation with Back and Next. Compare how fast each field falls off: 1/r², 1/r, or not at all.
Step through the derivation to see how symmetry simplifies Gauss's law for each geometry. The key insight: when E is constant on the Gaussian surface, the flux integral becomes trivial.
For a sphere of charge, an infinite line, or an infinite plane, symmetry forces E to be constant on the Gaussian surface. This reduces the flux integral to E × Area = Q/ε₀, giving E directly.
Key insight: Sphere → E ∝ 1/r². Line → E ∝ 1/r. Plane → E = constant. The geometry determines how fast the field falls off.
Switch between an internal charge, an external charge, and both. Watch charge gather on the conductor's surfaces so that no field survives inside the metal.
The conducting shell (gray region) has E = 0 inside the material. An internal charge induces opposite charge on the inner surface and equal charge on the outer surface. External fields cannot penetrate the cavity.
Inside a conductor, E = 0 in electrostatic equilibrium. Charges redistribute on the surfaces to cancel any internal field. A hollow conductor shields its interior from external fields -- the Faraday cage effect.
Key insight: The Faraday cage works because charges on the conductor rearrange to exactly cancel the external field inside. This is why your car protects you from lightning.
Spot a mistake or have an idea?