Chapter 2 — Electrostatics (Concise Mathematical Notes)
Electric Field:
E(r) = (1/4π■■) ∫ ρ(r′) (r − r′)/|r − r′|³ dτ′
E = −∇V
Gauss’s Law:
■ E·da = Q_enc/■■
∇·E = ρ/■■
Curl-free:
∇×E = 0 → existence of scalar potential V
Poisson/Laplace:
∇²V = −ρ/■■
∇²V = 0 (charge-free regions)
Boundary Conditions:
E_⊥ above − E_⊥ below = σ/■■
E_■ continuous
Energy:
U = (■■/2) ∫ E² dτ
Chapter 3 — Special Techniques
Laplace Solutions:
Cartesian: V(x,y,z) = X(x)Y(y)Z(z)
X″/X + Y″/Y + Z″/Z = 0
Spherical:
V(r,θ) = Σ (A_l r^l + B_l r^{−l−1}) P_l(cosθ)
Uniqueness Theorems:
Dirichlet: potential fixed on boundary → unique solution.
Neumann: normal derivative fixed → unique up to constant.
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