potential - Concrete example of the application of complex analysis in electrostatics

I've heard complex analysis can be useful in solving electrostatics problems, but despite doing some research I was unable to find any concrete examples. Would anyone be able to provide a simple example of where complex analysis is useful in electrostatics?

Answer

Complex analysis is very useful in potential theory, the study of harmonic functions, which (by definition) satisfy Laplace's equation. One way to see this connection is to note that any harmonic function of two variables can be taken to be the real part of a complex analytic function, to which a conjugate harmonic function representing the imaginary part of the same analytic function can also be associated (using the Cauchy-Riemann equations).

The name of this field of mathematics actually comes from physics, because it originated from the study of potentials such as the electrostatic potential, which satisfies Laplace's equation with appropriate boundary conditions to account for charge distributions. As Wikipedia puts it:

one sees that the subject of two-dimensional potential theory is substantially the same as that of complex analysis

Where I've added italics for emphasis.

There is a very concrete class of applications of complex analysis to two-dimensional electrostatics. Given a setup which specifies some boundary conditions for a potential and asks to find the potential in the empty space between or around the charges, one uses (a chain of) conformal mappings, which preserve harmonicity and have a whole host of other nice properties, to bring the domain where the potential satisfies Laplace's equation to one of a small set of "standard domains" where one has, a priori, solved Laplace's equation. Then, one only needs to perform the inverse conformal transformations to obtain the solution one is looking for. As far as I can recall, a large class of two-dimensional problems can be reduced to one of the following three:

1. The potential between coaxial cylinders


2. The potential between parallel plates


3. The potential in an "angular region", i.e. $\bigg\{x\in \mathbb R^2 \cong \mathbb C\bigg| |\operatorname{Arg}(x)|\leq \theta, \theta<\pi\bigg\} $

For more details and examples, see e.g. this very practical book by Kreyszig (chapter 18). I will reproduce one example here:

Find the potential between the non-coaxial cylinders $C_1: |z|=1$ and $C_2: |z-\frac{2}{5}|=\frac 2 5$.

I leave it as an exercise to the reader to guess which of the three above standard situations applies here!

Essentially identical approaches also yield problems in other areas of physical interest, such as time-independent heat flow and fluid flow problems (where the relevant differential equations reduce to Laplace's equations as well).