Question 5 (4 pts)
A spherical shell of radius $R$, centered on the origin, carries a uniform surface density $\sigma_0$ on the
top hemisphere ($0 < \theta < \pi/2$), and a uniform surface density $-\sigma_0$ on the bottom hemisphere
($\pi/2 < \theta < \pi$). Show that the potential inside and outside the sphere is given by
$V(r, \theta) = \begin{cases} \frac{\sigma_0}{2\epsilon_0} \left[ \frac{r}{R} P_1(\cos\theta) - \frac{1}{4} \left(\frac{r}{R}\right)^2 P_3(\cos\theta) + \frac{1}{8} \left(\frac{r}{R}\right)^4 P_5(\cos\theta) + ... \right] & r \le R\\ \frac{\sigma_0}{2\epsilon_0} \frac{R^3}{r^2} \left[ P_1(\cos\theta) - \frac{1}{4} \left(\frac{R}{r}\right)^2 P_3(\cos\theta) + \frac{1}{8} \left(\frac{R}{r}\right)^4 P_5(\cos\theta) + ... \right] & r \ge R \end{cases}$
Hint: Look carefully at example 3.9 in the text, where a large part of the question is answered.
You need to calculate the expansion coefficients $A_\ell$ and $B_\ell$ explicitly, up to and including $\ell = 6$.
Since $P_\ell(-x) = (-1)^\ell P_\ell(x)$, more than half of them will be zero, and you only have to actually
compute 3 simple integrals.
