Introduction to Electrodynamics · Problem 3.44

Potentials — David J. Griffiths

Physics · multipole-expansion

Problem

A charge

+Q

is distributed uniformly along the

z

axis from

z = -a

z = +a

. Show that the electric potential at a point

\mathbf{r}

is given by

V(r,\theta) = \frac{Q}{4\pi\epsilon_0}\frac{1}{r}\left[1 + \frac{1}{3}\left(\frac{a}{r}\right)^2 P_2(\cos\theta) + \frac{1}{5}\left(\frac{a}{r}\right)^4 P_4(\cos\theta) + \ldots\right],

for

r > a