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) 
The other limit that is relatively easy is a good conductor, . In that case the imaginary term dominates and we see that
(11.77) 
(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) 