Poisson's equation

We know that in steady-state we can write

$${\bf E} = -\nabla \phi,$$ (663)

with the scalar potential satisfying Poisson's equation:

$$\nabla^2 \phi = -\frac{\rho}{\epsilon_0}.$$ (664)

We even know the general solution to this equation:

$$\phi({\bf r}) = \frac{1}{4\pi \epsilon_0} \int \frac{\rho({\bf r}')} {\vert{\bf r} - {\bf r'}\vert} d^3{\bf r}'.$$ (665)

So, what else is there to say about Poisson's equation? Well, consider a positive (say) point charge in the vicinity of an uncharged, insulated, conducting sphere. The charge attracts negative charges to the near side of the sphere, and repels positive charges to the far side. The surface charge distribution induced on the sphere is such that it is maintained at a constant electrical potential. We now have a problem. We cannot use formula (665) to work out the potential $\phi({\bf r})$ around the sphere, since we do not know how the charges induced on the conducting surface are distributed. The only things which we know about the surface of the sphere are that it is an equipotential surface, and carries zero net charge. Clearly, in the presence of conducting surfaces the solution (665) to Poisson's equation is completely useless. Let us now try to develop some techniques for solving Poisson's equation which allow us to solve real problems (which invariably involve conductors).