Intensity of EM waves in a conducting medium

[ptabor]

I'm attempting to calculate the above, but I'm stuck.
What I have thus far:

intensity is the time average of the poynting vector, where the poynting vector is the cross product of the E and B fields.
Using the given expressions, I get:
$$\frac{\sqrt{a^2 + b^2}}{\mu \omega} E_0^2 \exp(-2bz) \frac{\cos\phi}{2}$$

where a and b are the real and imaginary parts of the complex wave number, k.

$$\phi$$ is of course the phase angle between the E and B fields (since this is a conducting medium the B field lags).

I'm supposed to show that the intensity is
$$\frac{a}{2 \mu \omega} E_0^2 \exp(-2bz)$$
but I don't know how to proceed.

any help would be greatly appreciated.

 

[ptabor]

perhaps some clarification is in order.

The equations for the Electric and Magnetic fields are as follows:

$$E(z,t) = E_0 \exp (-kz) \cos(kz - \omega t + \delta_e) B(z,t) = \frac{K}{\omega} E_0 \exp(-kz) \cos(kz - \omega t + \delta_b)$$

I take their cross product (E is in x direction, B is in y, so poynting vector is in z) and integrate over a whole period (omega over two pi) to get the time average. This gives me the 1/2 cos phi term.

K is the modulous of the complex wave number, so it's the square root of the sum of the squared real and imaginary parts (the a and b)

 

[ptabor]

if anyone was wondering

This is problem 9.20 from Griffith's intro to E&M

What I missed, in case anybody was wondering, is that the modulous of the complex wave number times the cosine of phi is in fact the real part of the complex wave number, as required.

This can be deduced most easily by graphing k in the complex plane.