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

The monopole vanishes exactly whenand the dipole vanishes exactly whenSince $N$ and $M$ are positive integers, both vanish precisely for

Assume now that the monopole has been cancelled. If $Qd$ is held fixed while $d\to 0$, the quadrupole coefficient $Q{d}^2=(Qd)d$ tends to zero, whereas the dipole coefficient $(M-1)Qd$ has a finite limit. For $M\ne 1$ this is a point-dipole limit; for $M=1$ the limiting external potential vanishes.

If instead $Q{d}^2$ is held fixed, the dipole coefficient isIt diverges unless $M=1$. When $M=1$ and $N=2$, both lower multipoles vanish and the quadrupole coefficient has a finite nonzero limit. These are the dipole and quadrupole scaling limits of three collinear charges.

Solved by gpt-5.6-sol high.