Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 36B (Electrodynamics)

The Lorenz gauge is ${\partial }_{\mu }{A}^{\mu }=0$. Substituting the electromagnetic four-potential into the covariant Maxwell equations then removes the mixed derivative and givesEquivalently,

The outgoing Green function of the electromagnetic wave equation has support on the past light cone. Convolution with that retarded Green function yields the retarded electromagnetic potentialwhere the source is evaluated at the retarded timeThis is the causal solution with no incoming radiation.

The electric dipole moment of a localized charge density isDifferentiate and use the continuity equation $\dot{\rho }+\nabla \cdot \mathbf{J}=0$:Localization makes the boundary integral vanish. Component by component,This is the time derivative of the electric dipole moment.

Let $R=|\mathbf{x}|$, $\hat{\mathbf{R}}=\mathbf{x}/R$, and $t_R=t-R/c$. To leading nonzero multipole expansion order, the loop is a time-dependent magnetic dipole withExpanding the retarded electromagnetic potential across the small loop givesThe $R^{-2}$ term is the near magnetic-dipole field and the $R^{-1}$ term is the magnetic dipole radiation field. Substituting the Fourier series for the current givesThe $n=0$ summand vanishes for the stated sine series.