TABLE OF CONTENTS In the following we investigate further extensions of electrostatics as well as more complicated applications, thereby also emphasising the methodical differences. Green's theorems and Fourier transforms are introduced, and finally the multipole expansion of the potential of a charge distribution is derived.
The problems we considered thus far were mostly treated with the help of the integral form of the Gauss law, and so with
We now investigate other methods.
In the preceding we encountered the Poisson equation as the equation
In solving this equation we require boundary conditions. These are:
(1) ? = const. on conducting surfaces (in an ideal conductor the electrons do not perform work) with the discontinuity for the derivatives as for the electric field strength, i.e. (cf. Eq. (2.16))
(2) otherwise ? continuous (this is no contradiction with the difference of the potential on both sides of a dipole layer; [*] the potential of a dipole p is, as we shall see later, p r/ r 3, and this is continuous in r (no step function ?( x)). If ? were discontinuous anywhere, i.e. if ? contained a step-function ?( x ? x 0), then at this point E = ? ? ? would be proportional to ?( x ? x 0), i.e. infinite, which would be meaningless and...
