Review of Electrostatics & Magnetostatics: Recall Coulomb’s Law, electric fields (E), Gauss’s Law (∫E⋅dA = Q_enc/ε₀), electric potential (V, E = -∇V), Biot-Savart Law, magnetic fields (B), Ampere’s Law (∫B⋅dl = μ₀I_enc).
Vector Calculus Recap: Mastery of gradient (∇V), divergence (∇⋅F), curl (∇×F), the divergence theorem (∫(∇⋅F)dV = ∫F⋅dA), and Stokes’ theorem (∫(∇×F)⋅dA = ∫F⋅dl) is essential. These tools allow us to express E&M laws in their p/details
owerful local (differential) forms.
Laplace’s and Poisson’s Equations:
Poisson’s Equation: ∇²V = -ρ/ε₀. Relates the potential V to the volume charge density ρ at any point. ∇² is the Laplacian operator (∂²/∂x² + ∂²/∂y² + ∂²/∂z² in Cartesians).
Laplace’s Equation: ∇²V = 0. A special case of Poisson’s equation that holds in regions free of charge (ρ=0). Solutions to Laplace’s equation are called harmonic functions.
Electric Dipole Moment (p): For a pair of charges +q and -q separated by d, p = qd (vector from -q to +q). For a general distribution, p = ∫r’ ρ(r’) dV’. The potential far from a dipole falls off as 1/r², and the field as 1/r³.