Next: Kramers-Kronig Relations Up: Penetration of Waves Into Previous: Penetration of Waves Into   Contents

## Wave Attenuation in Two Limits

Recall from above that:

 (11.72)

Then:

 (11.73)

Also so that

 (11.74)

Oops. To determine and , we have to take the square root of a complex number. How does that work again? See the appendix on Complex Numbers...

In many cases we can pick the right branch by selecting the one with the right (desired) behavior on physical grounds. If we restrict ourselves to the two simple cases where is large or is large, it is the one in the principle branch (upper half plane, above a branch cut along the real axis. From the last equation above, if we have a poor conductor (or if the frequency is much higher than the plasma frequency) and , then:

 (11.75) (11.76)

and the attenuation (recall that ) is independent of frequency.

The other limit that is relatively easy is a good conductor, . In that case the imaginary term dominates and we see that

 (11.77)

or
 (11.78) (11.79)

Thus

 (11.80)

Recall that if we apply the operator to we get: · & = & 0
ik _0· & = & 0
_0· & = & 0 and - t & = & ×E
i&omega#omega;&mu#mu; _0 & = & i( × _0)(1 + i)&mu#mu;&sigma#sigma;&omega#omega;2
_0 & = & 1&omega#omega;&sigma#sigma;&omega#omega;&mu#mu; ( × _0)12(1 + i)
& = & 1&omega#omega;&sigma#sigma;&omega#omega;&mu#mu; ( × _0) e^i&pi#pi;/4 so and are not in phase (using the fact that ).

In the case of superconductors, and the phase angle between them is . In this case (show this!) and the energy is mostly magnetic.

Finally, note well that the quantity is an exponential damping length that describes how rapidly the wave attenuates as it moves into the conducting medium. is called the skin depth and we see that:

 (11.81)

We will examine this quantity in some detail in the sections on waveguides and optical cavities, where it plays an important role.

Next: Kramers-Kronig Relations Up: Penetration of Waves Into Previous: Penetration of Waves Into   Contents
Robert G. Brown 2017-07-11