Multipole Expansion — Charge Distribution

V(r) = (1/4πε₀)[Q/r + p·r̂/r² + (1/2)Qij r̂ir̂j/r³ + …] Monopole Q = Σqᵢ. Dipole p = Σqᵢrᵢ. Quadrupole Qij = Σqᵢ(3rᵢrⱼ−r²δᵢⱼ). Each term falls off faster — multipole expansion is exact for r→∞. Key to molecular electrostatics, antennas, gravitational waves.