Given potential, ask plane charge distribution

[Tekk]

Homework Statement

All charges in space are distributed on the xy-plane. The potential above the plane is known as
$\phi = \phi_0 exp(-kz) cos(kx)$

What's the charge distribution on xy-plane?

Homework Equations

$\vec E =- grad(\phi)$

The Attempt at a Solution

Applying the relationship between $\vec E$ and $\phi$, I have found:

$E_x = k \phi_0 exp(-kz) sin(kx)$
$E_z = k \phi_0 exp(-kz) cos(kx)$

I know that a charge distribution $\sigma_z = \frac{E_z}{2\pi}$ would produce $E_z$. But how about $E_x$?
I am thinking of a superposition of $\sigma_z$ and $\sigma_x$ to produce $\vec E$. So the question now is to find $\sigma_x$: what kind of charge distribution on the plane would produce $E_x = k \phi_0 exp(-kz) sin(kx)$?

 

[mfb]

I don't see how a constant charge density would lead to the cosine term (or even the exponential).

Your potential is periodic in x and does not depend on y, what do you expect for the charge distribution?
Does this problem appear in the context of Fourier transformations?

 

[Tekk]

I don't see how a constant charge density would lead to the cosine term (or even the exponential).

Your potential is periodic in x and does not depend on y, what do you expect for the charge distribution?
Does this problem appear in the context of Fourier transformations?

No, the problem is not appeared in the context of Fourier transformation. It is the problem 31(c) from Chapter 2: Electric Potential of Purcell 2ed E&M textbook. Purcell asks us to describe the charge distribution on the non-conducting flat sheet.

I neither see how a charge distribution would lead to an exponential decay along the z-axis. But what if the charge is not uniformly distributed in the plane? Specifically, let the charge only distributed on a finite circle, then the electric field at least decay along the z-axis. My reason is: in the very far, we could treat this circle as a point charge, so the electric field it produce would decay.

If we temporary don't care the electrical field far away and focus only on the location very near to the xy-plane, in that case, z approaches to zero, $E_z = kϕ_0cos(kx)$. I can find a charge distribution $\sigma_z = \frac{E_z}{2\pi} = \frac{kϕ_0}{2\pi}cos(kx)$ to produce $E_z$, at least when z is very small. But I have no idea how to find $\sigma_x$ to produce $E_x$. Do you have any suggestion. A qualitative approach is welcome.

 

[mfb]

so the electric field it produce would decay.

But not exponentially.
The approach with very small z could work. Can you calculate E_z for all z based on this charge distribution?

I think if one fits, the other one will fit as well, so I would not worry about E_x for now.