Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 3 / 34A / ii / Solution

Use the mostly-plus metric, so $\square$ reduces to ${\nabla }^2$ for static fields. Seek a Green function satisfying $({\nabla }^2-{\lambda }^2)G={\delta }^{(3)}$. Convolving it with $-{\mu }_0\mathbf{J}$ gives the required solution.

For $R>0$, write $G=f(R)/R$. The homogeneous equation is $f^{\prime \prime }-{\lambda }^{2}f=0$. Decay discards the growing exponential, while the normalization at the origin follows from ${\nabla }^2(1/R)=-4\pi {\delta }^{(3)}$. Therefore for every $\lambda \ge 0$,The $1/R$ singularity fixes the delta coefficient; the extra factor is smooth to leading order at zero. For a conserved localized static current, integration by parts also gives $\nabla \cdot \mathbf{A}=0$, as required by the constraint.

At $\lambda =0$ the far field has algebraic multipole decay; for an ordinary localized steady current the vector potential starts at dipole order. At $\lambda >0$ the Yukawa potential gives exponential suppression on length scale ${\lambda }^{-1}$, with an overall bound of order $e^{-\lambda |\mathbf{x}|}/|\mathbf{x}|$ for bounded source support.