How it works
Coulomb's law gives the force between two point charges q₁ and q₂ separated by r as F = q₁q₂/(4πε₀r²) in free space. The electric field E = F/q = Q/(4πε₀r²) r̂. Gauss's law in integral form: ∮E·dS = Q_enclosed/ε₀. This is most powerful for symmetric geometries — a uniformly charged sphere of radius R with total charge Q gives E = Q/(4πε₀r²) outside (r > R) and E = Qr/(4πε₀R³) inside. Electric potential V = −∫E·dl; for a point charge, V = Q/(4πε₀r). Poisson's equation is ∇²V = −ρ_v/ε; Laplace's equation is ∇²V = 0 where ρ_v = 0.
Key points to remember
The boundary conditions at a dielectric interface: normal component of D is continuous if no surface charge (D₁ₙ = D₂ₙ); tangential component of E is always continuous (E₁ₜ = E₂ₜ). For a conductor surface, E is entirely normal and E_tangential = 0 inside. Energy density in the electric field is w_e = ½ε|E|² J/m³. The capacitance of a spherical capacitor with inner radius a and outer radius b is C = 4πε₀εr·ab/(b−a). Dipole moment p = Qd determines the torque in an external field as τ = p × E. Electric flux density D = ε₀εr·E.
Exam tip
The examiner always asks you to apply Gauss's law to a coaxial cable or spherical charge distribution and find E at various radii — draw the Gaussian surface, write the integral form, and state the enclosed charge explicitly before evaluating.