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**1. Homework Statement**

Consider a vacuum filled infinitely long metal cylindrical waveguide of radius, a. Suppose the fields in the waveguide are as follows:

[tex] \vec{E}=\vec{E_{0}}\left(s,\phi\right)\exp i(kz-\omega t)[/tex]

[tex] \vec{B}=\vec{B_{0}}\left(s,\phi\right)\exp i(kz-\omega t)[/tex]

Find [itex]E_{0z}[/itex]

**2. The attempt at a solution**

Usually when i post my questions ill have a clue as to what to do. But this time around i have no clue whatsoever.

Since it is a metal waveguide we can assume that hte parallel component of E and the perpedicular component of B is zero.

The E0 given to us depends on s and phi. But this doesn't mean that it only has s and phi components (?)

[tex] \vec{E_{0}}=E_{s}\hat{s}+E_{\phi}\hat{\phi}+E_{z}\hat{z}[/tex]

But what now? How could the Laplacian be useful? Since there is no charge

[tex] \nabla \cdot E = 0 [/tex] so does that imply

[tex] \nabla^2 E=0 [/tex]?

So if we did that we would get

[tex] \frac{1}{s}\frac{\partial }{\partial s}\left(s\frac{\partial E_{s}}{\partial s}\right)+\frac{1}{s^2}\frac{\partial^2 E_{\phi}}{\partial \phi^2}+\frac{\partial^2 E_{z}}{\partial z^2} = 0[/tex]

Since they are all equal to zero should be use separation of variables to solve this? I think we hav to use Bessel functions? But the Laplacian would solve for the potential, not the electric field?

Please help!

Thanks in advance!