Infinite array of charged wires

[Fr33Fa11]

In the Feynman Lectures, Volume II, Chapter 7, the final two pages (7-10 and 7-11), Feynman describes an infinite array of parallel charged wires and uses a Fourier series to solve for the field above them. He shows that the series can be expressed entirely in terms of cosine terms and that the coefficients have the value F(n)*e^(-z/k) where n is the order of the term, z is the distance from the plane of the wires, and k is a constant. What I don't understand is how one would then find the function F(n) (which he does not show). Later (in chapter 12 of the same volume) says that F(1) is twice the average field strength and that solving for the function F(n) is straightforward (in relation to another, related problem).
How would you go about solving for the function?
I've found the field via the coulomb interaction at several points where symmetry makes it easy but I'm not sure how this helps.
Any help would be appreciated.

 

[matonski]

The parallel charged wires can be represented as an array of delta functions spaced $a$ apart (times $\lambda$, the linear charge density of the wires). Then it would just be the Fourier Series of a delta function.

$$F_n = \frac{2}{a} \int_{-a/2}^{a/2} \lambda\delta(x) \cos \frac{2n\pi x}{a} \, dx = 2\lambda$$

 

[Fr33Fa11]

I just solved it last night. The delta function approach doesn't produce the correct answer; F(n)=lambda/(2pi*n*epsilon_0) (sorry I'm not sure how to embed latex into the forums). I found this by manipulating the infinite series for the field a distance a directly above one of the wires into a form which, when expanded, gave a function along the lines of e^(-n). (The field is related to the hyperbolic cotangent of pi*z/a).

 

[matonski]

You're right. My delta function approach doesn't work because the Fourier coefficients of the potential are what's needed, not the charge density.

 

[sardar]

To solve for the function F(n), we can use the boundary conditions and symmetry of the problem. The boundary conditions for this problem are that the electric field must go to zero as the distance from the plane of wires approaches infinity, and that the electric field must be continuous at the surface of the wires.
Using these boundary conditions, we can set up a system of equations for the coefficients F(n) by equating the field expression from the Fourier series to the known field values at various distances from the wires. This will give us a system of equations with an infinite number of unknowns (the F(n) coefficients) and we can solve this using mathematical techniques such as matrix inversion or least squares fitting.
Additionally, we can use the symmetry of the problem to simplify the system of equations. Since the infinite array of charged wires has translational symmetry, we can use this symmetry to reduce the number of unknowns in the system of equations and make it easier to solve.
In summary, to solve for the function F(n) in the Fourier series for the electric field above an infinite array of charged wires, we can use boundary conditions and symmetry to set up a system of equations and then solve for the coefficients using mathematical techniques.