Question #43107

A uniform electric field $\mathbf{E}(t) = E_0 \cos(\omega_0 t) \hat{k}$ is maintained in a cylindrical region of space of radius $R$. The region is filled with a non-magnetic, non-conducting dielectric medium whose permittivity is externally modulated such that it varies with time as $\epsilon(t) = \epsilon_0 (1 + \beta \cos(\Omega t))$. Here, $E_0 = 100 \text{ V m}^{-1}$, $\omega_0 = 10^6 \text{ rad s}^{-1}$, $\beta = 0.1$, and $\Omega = 2 \times 10^5 \text{ rad s}^{-1}$. Due to the Ampere-Maxwell law, an induced magnetic field $\mathbf{B}$ is generated inside the cylinder, containing components oscillating at multiple angular frequencies. What is the ratio of the amplitude of the magnetic field component oscillating at angular frequency $\omega_0 + \Omega$ to the amplitude of the component oscillating at angular frequency $\omega_0$?