In a region of empty space, the magnetic field (in Gaussian units) is described by

$$
\mathrm{B}=B_{0} e^{a x} \hat{\mathbf{e}}_{z} \sin w
$$

where $w=k y-\omega t$.

Find the speed of propagation $v$ of this field.
Express $B$ as $\operatorname{Im}\left(B_{0} e^{a x} e^{i w}\right) \hat{\mathbf{e}}_{z}$.

Using Maxwell's equation

$$
\nabla \times \mathbf{B}=\frac{1}{c} \frac{\partial \mathrm{E}}{\partial t}
$$

and the definition $k=\frac{\omega}{c}$ for empty space, we obtain

$$
\mathbf{E}=\frac{i c}{\omega} \nabla \times \mathbf{B}=\frac{i}{k}\left|\begin{array}{ccc}
\mathbf{e}_{x} & \mathbf{e}_{y} & \mathbf{e}_{z} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & 0 \\
0 & 0 & B_{z}
\end{array}\right|
$$

where $\frac{\partial}{\partial z}=0$ as $\mathbf{B}$ does not depend on $z$.

Hence

$$
\begin{gathered}
E_{x}=\operatorname{Im}\left(\frac{i}{k} B_{0} e^{a x} i k e^{i w}\right)=-B_{0} e^{a x} \sin w \\
E_{y}=\operatorname{Im}\left(-\frac{i}{k} B_{0} a e^{a x} e^{i w}\right)=-\frac{a c}{\omega} B_{0} e^{a x} \cos w \\
E_{z}=0
\end{gathered}
$$If the wave form remains unchanged during propagation, we have

$$
d w=k d y-w d t=0,
$$

or $\frac{d y}{d t}=\frac{w}{k}=c$. Hence the wave propagates along the $y$ direction with a speed $v=c$.