Energy conservation: electromagnetic wave in matter

[happyparticle]

Homework Statement

An electromagnetic wave passes through a material with conductivity $\sigma$. The wave is attenuated, and the amplitude exponentially decreases with the distance traveled. Show that the total energy is conserved and what happen with the energy lost by the wave.

Relevant Equations

$$u = u_e + e_m$$
$$\tilde{E} (z,t) = \hat{E}_0 e^{i(\tilde{k}z - \omega t)} \hat{x}$$
$$\tilde{B} (z,t) = \hat{E}_0 \frac{\tilde{k}}{\omega}e^{i(\tilde{k}z - \omega t)} \hat{y}$$

Hi,
I completely failed this homework. I mean I think I know what happen, but I don't know how to show it mathematically. The energy lost by the wave is used to oscillate the electrons inside the conductor. Thus, the electrons acts like some damped driven oscillators.
I guess I have to find $e_m, u_e$, but I don't know with what to compare. That's pretty all I know.
Any help will me appreciate, thanks.

 

[member 553083]

The energy lost by the wave is used to oscillate the electrons inside the conductor, so we can model the system as a damped driven oscillator. The equation of motion for a damped driven oscillator is given by:$$ \ddot{x} + 2 \beta \dot{x} + \omega_0^2 x = F_0 \cos(\omega t) $$ where $\beta$ is the damping coefficient, $\omega_0$ is the natural frequency of the oscillator and $F_0$ is the amplitude of the driving force. The solution to this equation is given by: $$ x(t) = A_1 e^{-(\beta + i \omega_d)t} + A_2 e^{-(\beta - i \omega_d)t} + \frac{F_0}{m(\omega_0^2 - \omega^2 + i 2 \beta \omega)} \cos(\omega t) $$where $\omega_d = \sqrt{\omega_0^2 - \beta^2}$. The constants $A_1$ and $A_2$ are determined by the initial conditions of the system. By solving this equation you can find the amplitude of oscillation of the electrons in the conductor. The maximum kinetic energy of the electrons is given by $e_m = \frac{1}{2}mv_e^2$, where $v_e$ is the peak velocity of the electrons. Hope this helps.