Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 4 / 5D (Electromagnetism)

Let $\theta$ be the angle between $r$ and the positive $z$-axis. By superposition of the electric potential of point charges,For $r>d$, the Legendre polynomial expansion givesThereforeThe three displayed terms are respectively the monopole, dipole, and quadrupole terms of the electric multipole expansion. In particular, the total charge is $Q(N-M-1)$ and the electric dipole moment is $Qd(M-1)\hat{z}$. This is the multipole expansion of three collinear charges.

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.

Here $N=2$ and $M=1$. At $(x,0,0)$, the $z$-components of the fields of the two outer charges cancel by reflection symmetry. By Coulomb's law, the remaining field isMultiplying by the test charge $-Q$ givesThe quantity in parentheses is positive. Thus for $x>0$ the force points in the negative $x$ direction, while for $x<0$ it points in the positive $x$ direction. It is always attractive toward the origin, as in the transverse force from a collinear electric quadrupole.