course |
course_year |
question_number |
tags |
title |
year |
Methods |
IB |
30 |
|
3.I.2G |
2001 |
Laplace's equation in the plane is given in terms of plane polar coordinates $r$ and $\theta$ in the form
$$\nabla^{2} \phi \equiv \frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial \phi}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} \phi}{\partial \theta^{2}}=0$$
In each of the cases
$$\text { (i) } 0 \leqslant r \leqslant 1, \text { and (ii) } 1 \leqslant r<\infty \text {, }$$
find the general solution of Laplace's equation which is single-valued and finite.
Solve also Laplace's equation in the annulus $a \leqslant r \leqslant b$ with the boundary conditions
$$\begin{aligned}
&\phi=1 \quad \text { on } \quad r=a \text { for all } \theta \\
&\phi=2 \quad \text { on } \quad r=b \text { for all } \theta .
\end{aligned}$$