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Tekk
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Homework Statement
All charges in space are distributed on the xy-plane. The potential above the plane is known as
[itex]\phi = \phi_0 exp(-kz) cos(kx)[/itex]
What's the charge distribution on xy-plane?
Homework Equations
[itex]\vec E =- grad(\phi)[/itex]
The Attempt at a Solution
Applying the relationship between [itex]\vec E[/itex] and [itex]\phi[/itex], I have found:
[itex]E_x = k \phi_0 exp(-kz) sin(kx)[/itex]
[itex]E_z = k \phi_0 exp(-kz) cos(kx)[/itex]
I know that a charge distribution [itex]\sigma_z = \frac{E_z}{2\pi}[/itex] would produce [itex]E_z[/itex]. But how about [itex]E_x[/itex]?
I am thinking of a superposition of [itex]\sigma_z[/itex] and [itex]\sigma_x[/itex] to produce [itex]\vec E[/itex]. So the question now is to find [itex]\sigma_x[/itex]: what kind of charge distribution on the plane would produce [itex]E_x = k \phi_0 exp(-kz) sin(kx)[/itex]?
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