Rectangular Waveguide: Boundary Conditions for Electromagnetic Waves

[Observer Two]

Homework Statement

I have a rectangular, hollow, conductor. Something like this:

The length in z direction should be infinite. The propagation of electromagnetic waves in the conductor are given via the equations:

Homework Equations

$\Delta \vec{E} = \frac{1}{c^2} \frac{\partial^2 \vec{E}}{\partial t^2}$

$\Delta \vec{B} = \frac{1}{c^2} \frac{\partial^2 \vec{B}}{\partial t^2}$

Question 1: Which boundary conditions follow for the E field? And what does that mean for the B field?

The Attempt at a Solution

Well. Since there are free charges in the conductor, the tangential component of the E field should be 0 (otherwise the charges would start to move). Which in return means the normal component of the B field must also be 0.

Is there more I could derive from this or is that it? My answer seems a bit too simplistic ...

 

[mark726]

Dear student,

Your answer is correct, but there are a few more details that can be derived from this situation. First, because the conductor is hollow, the E field inside the conductor must also be zero. This is because if there were an E field inside the conductor, it would induce a current in the free charges and thus violate the condition of zero tangential E field at the boundary.

Additionally, the B field inside the conductor must be constant. This is because the B field is related to the current density, and in a steady state situation (such as an infinite conductor), the current density must also be constant.

Furthermore, the boundary conditions for the B field are the opposite of those for the E field. This means that the tangential component of the B field must be continuous at the boundary, while the normal component must be zero. This can be derived from Maxwell's equations and the fact that there are no free currents in the conductor.

I hope this helps clarify the situation for you. Keep up the good work!