Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 4 / 5D / a / Solution

For charges $-Q,NQ,-MQ$ at $z=d,0,-d$, respectively,
$$
\Phi(r,\theta)=\frac{Q}{4\pi\epsilon_0}\left[
\frac{N-M-1}{r}+\frac{(M-1)d\cos\theta}{r^2}
-\frac{(M+1)d^2P_2(\cos\theta)}{r^3}+O(r^{-4})\right].
$$
The monopole vanishes when $N=M+1$, the dipole vanishes when $M=1$, and both vanish when $(N,M)=(2,1)$.


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 Incoming links: Multipole expansion of three collinear charges