Question #42776

A uniform plane electromagnetic wave propagates along the $+z$-direction in a homogeneous, non-magnetic, lossy dielectric medium. The electric field of the wave is given by $\vec{E}(z,t) = E_0 \cos(\omega t - kz) \hat{i}$, where $E_0 = 200 \text{ V/m}$ and $\omega = 10^8 \text{ rad/s}$. Due to the finite relaxation time of the medium, the dielectric polarization $\vec{P}$ lags behind the electric field by a phase angle $\delta = \pi/6 \text{ rad}$, and is given by $\vec{P}(z,t) = P_0 \cos(\omega t - kz - \delta) \hat{i}$, where $P_0 = 10^{-8} \text{ C/m}^2$. Assuming no free currents exist in the medium, what is the magnitude of the spatial attenuation rate of the time-averaged Poynting vector, $\left| \frac{\partial \langle S_z \rangle}{\partial z} \right|$, in the medium?