Recall from above that:

(9.136) |

Then:

(9.137) |

Also
so that

(9.138) |

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:

(9.139) | |||

(9.140) |

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

(9.141) |

(9.142) | |||

(9.143) |

Thus

(9.144) |

Recall that if we apply the
operator to
we get:

(9.145) |

and

(9.146) |

so and are

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:

(9.147) |