Charge/current density inside rectangular wave guide?

[Garrett Muersch]

Homework Statement

Given the Z component of E inside a rectangular wave guide, Find the ratio of the maximum charge density on the plane x = 0 to the maximum charge density on the plane y = 0.

Additionally, Find the ratio of the maximum surface current on the x = 0 plane to the maximum surface current on the y = 0 plane.

I don't need someone to work out the problem for me, i just need help with the general approach. I can be more specific with the exact electric field given if necessary.

Homework Equations

The Attempt at a Solution

since my z component of E is non zero, I know i must have a TM mode since rectangular wave guides do not support TEM mode waves.

therefore, my z component of B = 0, and i can then go ahead and calculate the different components of my electromagnetic fields from equations derived in my book.

At this point, i know what my E and B fields are inside the wave guide. My book also says we assume that wave guides are perfect conductors so that E=B=0 inside the walls and that free charges and currents will be induced onto the walls in order to maintain the boundary conditions at the inner wall that E parallel and B perpendicular must both be 0.

Do i try and use these boundary conditions to find out my charge and current densities?

 

[jasonRF]

You are on the right track - boundary conditions at a perfect conductor should allow you to compute the surface charges and currents on walls. There are boundary conditions for E and B both parallel to and perpendicular to the walls (so 4 conditions total). If you write out all 4 boundary conditions it should be clear which ones you want to use to solve your problem.

Jason

 

[Garrett Muersch]

thanks mate, me and my classmates solved it last night using the boundary conditions like you said.
a later part of the question was if the pressure on the walls of the wave guide would push them apart, or pull them in. Intuitively i thought the pressure would be greater on the inside and therefore push them out, but mathematically i got the other way around. I very easily could have made a sign mistake somewhere, as there were a ridiculous amount to keep track of, but there was some conflict about this between me and my classmates and we kept flip flopping on it.

It might very well be ridiculous to say the walls are pulled in, but we reasoned it is physically possible because the conductor cancels the inner charge, creating an electron configuration that induces a field on the outside which creates a force pushing back inwards. I also thought about how the field would pull electrons inwards to cancel the inner field. To me that seems indicative of an inward pressure. I am pretty conflicted on the reality of the situation, if you could shed some light that would be awesome!

 

[jasonRF]

Since you already know the fields, charges and currents, it should be fairly straightforward to work out the electromagnetic force on the charges/currents using the Lorentz force law.

EDIT: I neglected to account for the field momentum. Have you learned about the electromagnetic stress tensor?

 

[Garrett Muersch]

Yeah we calculated using the Lorentz law, but to be honest we got high towards the end and easily could have made sign mistakes as there were a ton of them to keep track of and finding the sign for our surface current was kinda weird. other people in the class were getting different answers so idk, i was just wondering if you knew weather or not an electromagnetic field within a conductor would push out or pull in on the sides. we already turned it in so i can't double check my work until i get the assignment back.

Yes we did learn about stress tensors, we just finished going through introduction to electrodynamics by Griffiths over the past 2 semesters. We are in the final chapter of relativity currently.

I am not the best with the tensors tbh, i can calculate with them but not always certain when i should implement them in a problem. We did not make any attempt to use a tensor in our solution, how were you thinking of applying it in this case?

 

[jasonRF]

Must be difficult to do physics while high (did I understand correctly?). I would not recommend getting high in general, and especially not while trying to learn anything.

Regarding the stress tensor, you can think of it as a linear operator that sends vectors to vectors. If $\mathbf{n}$ is a unit normal to a surface at a given location and $\mathbf{T}$ is the stress tensor evaluated at that same location, then $d\mathbf{F}=\mathbf{T \cdot n} \, da$ is the force on the small patch of area $da$ at that location. This is true of all stress tensors, not just the electromagnetic stress tensor. After that you just integrate.

 

[jasonRF]

I just wanted to add one point. The stress tensor tells us that field lines (either E or B) act like stretched rubber bands in the parallel direction, so exert negative pressure. in the transverse direction they exert a positive pressure. So for your waveguide mode, the electric fields that terminate on the walls will pull the walls in, and the magnetic field parallel to the walls will push the walls out. Some walls will have both of these, and the net effect depends on the relative magnitudes.