Let $r, \theta, \phi$ be spherical polar coordinates, and let $P_{n}$ denote the $n$th Legendre polynomial. Write down the most general solution for $r>0$ of Laplace's equation $\nabla^{2} \Phi=0$ that takes the form $\Phi(r, \theta, \phi)=f(r) P_{n}(\cos \theta)$.

Solve Laplace's equation in the spherical shell $1 \leqslant r \leqslant 2$ subject to the boundary conditions

$$\begin{aligned} &\Phi=3 \cos 2 \theta \text { at } r=1 \\ &\Phi=0 \quad \text { at } r=2 \end{aligned}$$

[The first three Legendre polynomials are

$$\left.P_{0}(x)=1, \quad P_{1}(x)=x \quad \text { and } \quad P_{2}(x)=\frac{3}{2} x^{2}-\frac{1}{2} .\right]$$