Static Solutions

Lecture 49

Solutions to the wave equation which are static, are characterized by $\frac{\partial^2\psi}{\partial t^2} =0$ , and hence by $k^2=0$ . The governing equation becomes

$$\nabla^2\psi =0\,.$$

Relative to spherical coordinates this equation reads

$$\nabla^2\psi =\left\{\frac{1}{r}~\frac{\partial^2}{\partial r^2} ... ...{\sin^2\theta}~\frac{\partial^2} {\partial\varphi^2}\right)\right\} \psi =0\,.$$

Its solution is a superposition of solutions having the simple product form,

$$\psi_{\ell m}(t,\theta ,\varphi )=R_\ell (r)Y^m_\ell (\theta ,\varphi )\,.$$

They give rise to the two ordinary differential equations

$$\left\{\frac{1}{\sin\theta}~\frac{\partial}{\partial\theta}\sin\t... ...artial^2} {\partial\varphi^2} +\lambda \right\} Y^m_\ell (\theta ,\varphi )\,,$$

where

$$\lambda = \ell (\ell +1)\,,~~\qquad~~\qquad~~\ell =0,1,2,\dots$$

and hence

$$\frac{1}{r}\left\{\frac{d^2}{dr^2}-\frac{\ell (\ell +1)}{r^2}\right\} rR_\ell (r)=0\,.$$

This equation is Euler's differential equation whose solutions are $r^\ell$ and $r^{-(\ell +1)}$ . Thus, the static solution is a superposition

$$\psi = \sum^\infty_{\ell =0}~\sum^\ell_{m=-\ell} (A_{\ell m}r^\ell + B_{\ell m}r^{-(\ell +1)})Y^m_\ell (\theta ,\varphi )$$

whose coefficients, the $A$ 's and the $B$ 's, are determined by the given boundary conditions. These coefficients are called the multipole moments of the source of the field.