Introduction to Electrodynamics/Chapter 8/3/3

8.3.3 Reflection and Transmission at a Conducting Surface

We've covered what EM waves do at boundaries of non-conducting linear media. Now let's see what happens for conducting linear media.

Hi, I'm Jonathan Gardner. We're covering 8.3.3 Reflection and Transmission at a Conducting Surface in Griffith's Introduction to Electrodynamics, 2nd Ed. I'm going to go fast, so be sure to rewind if you don't get it all on the first pass.

Setup

On the left side, we have a non-conducting linear media with constants ei, mu1, v1, and n1.

On the right side, we have a conducting linear media with constants e2, mu2, v2, n2, and sigma 2.

Waves

The three waves we are considering are the incident, reflected, and transmitted, as before. We're looking at waves that are normal to the surface, so we can simplify things a bit.

Note that the wave functions on the transmitted side are not described by the simple functions we use for non-conducting media. kappa is complex, and the B field is not simply 1/v.

Boundary Conditions

Now that we have free surface charges and currents, we are going to have to use a more complicated set of boundary conditions.

(i) The perpendicular D fields jump by the surface charge density.

(ii) No monopoles (yet), so no variation in the perpendicular B field.

(iii) Parallel E fields don't change.

(iv) Parallel H fields jump by the surface current.

For conducting materials, a surface current would imply an electric field parallel to the surface, at the surface. We already know that that can't be the case, so we note there is no surface current.

Applying the Boundary Conditions

As before, we eliminate (ii). There is no perp B field.

There are no perp E fields either, so we note that the free surface charge is 0.

(iii) and (iv) say that the omega must equal. This implies that kI and kR are the same.

(iii) says that E0I + E0R = E0T.

(iv) says that E0I - E0R = beta E0T. Except now our beta is more complicated. And complex!

Applying the above, we get the standard equations:

E0R = ( ) E0I

EOT = ( ) EOI

Conclusions

For a perfect conductor, conductivity sigma is infinite, so beta is infinite. This says there is no transmitted wave. This is a perfect mirror.

For a very good conductor, we can use series expansions to rewrite the reflection coefficient as this.

Example 7

Let's do air-to-silver, what you have, roughly, in mirrors.

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