19.3.6.1 Solution pnifd-a

Question:

Find a suitable solution $u_{\rm {out}}$ outside the sphere in three dimensions. Show that it satisfies the Laplace equation.

Answer:

In three dimensions, you will have to take

$$u_{\rm out}(r,\vartheta ,\varphi) = A \frac{1}{r} u(\bar r,\vartheta ,\varphi) \qquad\mbox{where}\qquad\bar r = \frac{1}{r}.$$

The Laplacian in spherical coordinates is readily available in table books. Taking derivatives goes in the same way as in two dimensions. However, note that you will need to use the product rule immediately.