How Do Magnetic Planes Influence Current Density in Conductive Materials?

[carllacan]

Homework Statement

We have an infinite plane of width 2b made of a magnetic, conducting material (μ_r >> 1, σ >> 1). Two monochromatic electromagnetic plane waves, with magnetic excitation vector amplitude H_s approach it, each one traveling towards one of its two faces. Find the current density J generated.

Homework Equations

The Attempt at a Solution

I've taken the plane to be parallel to the XZ plane and centered on it, that is, it extends from y = -b to y = +b, for every z and x. The waves are traveling from y = -∞ and from y = ∞, respectively. I assumed that the frequencies and phases of both waves are equal.

Physically, here's what I think would happen: each wave will divide in a reflected and a transmitted wave upon reaching the interface. This transmitted wave will travel inside the material until it reaches the opposite interface, where it will again divide into a transmitted wave (into the air) and a reflected wave (into the material). This last reflected wave will repeat the process, infinitely.

Mathematically, what I've done so far is to set the equations for all the waves (only their H vector, since is what I'm given). For each region (y < -b; -b < y < b and y > b) I have a sum of two waves in opposite directions. I think it would suffice to use the boundary conditions for the H vector and find the six waves. The amplitudes of these waves will account for the "infinite" transmitted and reflected waves. Am I right?

Once I have the waves insides the material $\vec J$ can be found just using $\vec ∇ × \vec H = \vec J$

Now, what if the frequencies were different? I think I would set, (a) on y < b, three waves, an incident wave from y = -∞ with frequency ω and two waves traveling from y = -b to y = -∞ with freqs ω and ω' (b) on -< < y < b four waves, one for each combination of directions and frequency, and (c) on y > b three waves similar to on y < -b, and then use the B. C. as before.

 

[tech99]

How can an EM wave enter a conductive material? Would not the E-field be reduced to zero, and the wave not propagate?

 

[carllacan]

Now that you mention it I have completely overlooked that detail. The exercise is redacted exactly as in my post, so... the field inside will just be 0?

 

[carllacan]

Ok, the material is not a conductor to all frequencies, so if the waves have an appropriate ω they will enter the material.