Electrostatics: Coulomb, Gauss and Capacitance

Electrostatics deals with charges at rest. The field E and the potential V describe how they push on other charges.

Coulomb's law and electric field

• F = q₁q₂/(4πε r²), with ε = ε₀ε_r and ε₀ = 8.854 × 10⁻¹² F/m (1/(4πε₀) ≈ 9 × 10⁹).
• E = F/q (V/m). Fields of common charge distributions:

Gauss's law

∮ D·dS = Q_enclosed, with D = εE. In point form: ∇·D = ρ_v.

It gives the fields above quickly by choosing a surface where D is constant. Inside a conductor in equilibrium, E = 0 and charge sits on the surface.

Potential

• V = −∫E·dl; E = −∇V. For a point charge: V = Q/(4πε r).
• The electrostatic field is conservative: ∮E·dl = 0 (∇ × E = 0).
• Laplace's equation ∇²V = 0 in charge-free regions; Poisson's ∇²V = −ρ_v/ε.
• Energy stored: W = ½ ∫ ε E² dv; energy density ½εE².

Dielectrics and boundaries

• Polarisation increases D: D = ε₀E + P.
• At a boundary between two dielectrics: tangential E is continuous; normal D is continuous (no free surface charge).
• At a conductor surface: E is normal, E_n = ρ_S/ε.

Capacitance

Layered dielectrics in a parallel-plate capacitor act like capacitors in series (layers parallel to plates) or parallel (side by side).

Worked example

Two charges of 1 µC each are 0.3 m apart in air.

• F = 9 × 10⁹ × 10⁻¹²/0.09 = 0.1 N (repulsive).

Practice