Chapter 3.4, Problem 3.43P

A conducting sphere of radius a, at potential $V_0$ , is surrounded by athin concentric spherical shell of radius b. over which someone has glued a surfacecharge $\sigma \left(\theta \right)=k\cos \theta ,$ where k is a constant and $\theta$ is the usual spherical coordinate.
(a) Find the potential in each region: (I) $r>b$ , and (ii) $a<r<b$ .
(b) Find the induced surface charge ${\sigma }_i\left(\theta \right)$ on the conductor.
(c) What is the total charge of this system? Check that your answer is consistentwith the behavior of V at large r.
   $\left[AnswerV:\left(r,\theta \right)=\{\begin{array}{c}a{V}_0/r+\left(b^3-a^3\right)k\cos \theta /3r^2{\in }_0,\\ a{V}_0/r+\left(r^3-a^3\right)k\cos \theta /3r^2{\in }_0,\end{array}\begin{array}{c}r\ge b\\ r\le b\end{array}]\right]$