Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2004 / Paper 2 / 2 / ii

For large $|z|$, expand in $a^2/z^2$ to obtainThe axisymmetric exterior multipole expansion replaces each $|z{|}^{-n-1}$ on the positive axis by $r^{-n-1}{P}_n(\cos \theta )$. Reflection symmetry eliminates the odd multipoles, soThe leading term is the potential of total charge $Q$ concentrated at the center; the next is its electric quadrupole correction. On either axis $P_2(\pm 1)=1$, recovering the same even axis potential.