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State and explain Gauss's law

Short answer

Gauss's law: ∮ D·dS = Q_enc; total electric flux through a closed surface equals enclosed free charge. It simplifies field calculation for symmetric distributions; differential form is ∇·D = ρ_v. Example: for a point charge, D = Q/(4πr²) a_r.

State and explain Gauss's law.

Statement. The total electric flux passing through any closed surface equals the total free charge enclosed by that surface.

Ψ = ∮ D·dS = Q_enclosed

Here D = εE is the electric flux density (C/m²).

Explanation

  • The closed surface is called a Gaussian surface; it can be any shape, real or imaginary.
  • Flux out of the surface counts as positive; charge outside the surface contributes no net flux, because every line that enters also leaves.
  • If the enclosed charge is a volume distribution, Q = ∫ρ_v dv, giving ∮D·dS = ∫ρ_v dv.
  • Applying the divergence theorem gives the point (differential) form, ∇·D = ρ_v, the first Maxwell equation for statics.

Use

For symmetric charge distributions (point, line, sheet, sphere, cylinder), choose a Gaussian surface on which D is constant and normal (or tangential), so D comes out of the integral. Example for a point charge Q: take a sphere of radius r, so D(4πr²) = Q, giving D = Q/(4πr²) a_r and E = Q/(4πε₀r²) a_r (Coulomb's law).

✓ Answer checked by Claude Opus 5.5 (high effort)

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