1. Consider a time-varying magnetic field in free space ($\varepsilon = \varepsilon_0$, $\mu = \mu_0$, $\sigma = 0$)
described by
$\vec{B} = e^{-\alpha x} \cos(\omega t - \beta z) \hat{a}_y$ Wb/m$^2$,
where $\alpha$ and $\beta$ are constant parameters. It is assumed that $\vec{J} = 0$, $\rho_v = 0$ (the medium
is source free).
(a) Using the curl Maxwell's equation $\nabla \times \vec{H} = \frac{\partial \vec{D}}{\partial t}$, determine the electric field $\vec{E}$.
(b) Using the result in part (a) in the Maxwell's equation $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$, determine a
relationship between $\alpha$ and $\beta$ in terms of $\omega$, $\varepsilon_0$, and $\mu_0$.
