# Are electrodynamics problems in the complex plane relevant to real life?

This is a question I asked in Maths SE, and it was suggested I ask it here. This is a direct copy of that question.

I have been reading Tristan Needham's excellent Visual Complex Analysis. The end of the book deals almost entirely with physics, using symmetries of conformal mappings to generalise the famous method of images technique in electrodynamics. The method of images is used in finding the electric field due to a charge when a grounded surface (such as a sphere or plane) is nearby. (See e.g. Wikipedia.)

However, the problems seem to have very little "real life" applications to me, the main problem being that the complex plane is two dimensional, whereas we live in a 3 dimensional world.

To see this problem concretely, the electrostatic force is goes like $F\sim \frac{1}{r^2}$ because the surface area of a ball of radius $r$ centred at the charge is proportional to $r^2$. However since the complex plane is 2 dimensional, a charge in the complex plane produces a field which goes like $\frac1r$. So any solution we find to a problem of this kind in the complex plane isn't relevant in 3d.

And this is my question, is there any physical application of this technique? Or is it completely irrelevant?

As Muphrid wrote in math.se, 2D problems in "real life" are really 3D problems where one dimension has been ignored because the system has translational symmetry along it. Thus a single point charge in 2D corresponds to an infinitely long line charge in three dimensions - which has indeed got a force scaling as $1/r$.