Question #43170

An electromagnetic wave of angular frequency $\omega$ propagates in the $+x$ direction through a non-magnetic dispersive medium ($\mu = \mu_0$). The medium's dielectric response is modeled by an equivalent series LCR circuit. The applied electric field $\vec{E}$ drives a polarization current density $\vec{J}_p$ such that $\vec{J}_p = \vec{E} / Z$, where $Z$ is the complex specific impedance of the medium.

The specific impedance is given by $Z = \rho + i\left(\omega L - \frac{1}{\omega C}\right)$, where $\rho$, $L$, and $C$ are the effective resistivity, specific inductance, and specific capacitance of the medium. The wave frequency is tuned to the resonant frequency of the medium ($\omega = 1/\sqrt{LC}$). At this frequency, the effective resistivity is $\rho = \frac{1}{\sqrt{3} \epsilon_0 \omega}$, where $\epsilon_0$ is the vacuum permittivity.

If the electric field of the wave is $\vec{E}(x,t) = E_0 e^{-\alpha x} \cos(\beta x - \omega t) \hat{y}$, which of the following represents the corresponding magnetic field $\vec{B}(x,t)$? (Here $c = 1/\sqrt{\mu_0 \epsilon_0}$ is the speed of light in vacuum)