A lot of people think that electrodynamics is all about the complicated problems – circuits, engines, waves and all that fun stuff. But my point is that most of the information is contained in the most basic problem – the electric field of a point charge on.
Yes, it is true, I am guilty of adoring simplicity! The thing is that if we understand the electric field of just a single point charge, we already know what the electric field for any arbitrary distribution of point charges is – the solution to the point charge problem is just a solution to the Poisson equation with a delta-like source. So, by definition, if we find this solution, we acquire the Green’s function of the problem.
Knowing the Green’s function of a problem gives us immediate control of any kind of solution. The Green’s function is nothing but the inverse operation to a differential equation and hence its complete solution. This approach is often used in more complicated situations if space is not just an isotropic, homogeneous something as for example for Dirichlet and Neumann boundary conditions in general boundary value problems. There, the field of a point charge may differ from the usual 1/r^2-law since mirror charges may occur. Nevertheless, if we know the field of just a single charge at an arbitrary placement, everything is known.
From a mathematical point of view, this is the case since the Poisson equation is a linear partial differential equation – a superposition of solutions remains a solution which is responsible for the famous superposition principle.
Anyways, the main message of my blog should be: Never underestimate the “easy” things as they may be a very fundamental piece in a more fundamental picture.