Required math: calculus
Required physics: electrostatics
Reference: Griffiths, David J. (2007) Introduction to Electrodynamics, 3rd Edition; Prentice Hall – Problems 3.12 – 3.13.
Here are a few examples of calculating the Fourier coefficients for some special cases.
Example 1. Consider the infinite slot problem with the boundary at consisting of a conducting strip with a constant potential of . In this case we get
The coefficients are thus zero for even and for odd :
The potential is thus
Example 2. Now suppose the boundary at consists of two conducting strips, insulated from each other and from the infinite sheets. The first strip, from to has a constant potential while the other strip, from to is held at potential .
Here, the coefficients are given by
If is odd, this comes out to zero. If is even, there are two cases. First, if the term in brackets is 4. If the term in brackets is zero. Thus we get
Thus the potential is
where in the last line we’ve changed the index of summation since the non-zero terms in the first sum are just those with starting at .
Example 3. The infinite slot with the strip at held at potential has the solution
For a conductor, the surface charge density can be found from the derivative taken normal to the surface:
In this case, the normal to the surface is the direction, so we get
This looks fine except for the problem that the series doesn’t converge. Consider . The series is then a sum of an alternating sequence of and . The original series for the potential does converge at due to the in the denominator. Not sure what the solution to this is.