= Solution
In SI units, the two curl <Maxwell equations> are
$$
\nabla\times\mathbf B=\mu_0\mathbf j+\mu_0\epsilon_0\mathbf E_t,
\qquad \nabla\times\mathbf E=-\mathbf B_t.
$$
Use the vector identity $\nabla\cdot(\mathbf E\times\mathbf B)=\mathbf B\cdot(\nabla\times\mathbf E)-\mathbf E\cdot(\nabla\times\mathbf B)$ to obtain
$$
\partial_t\left(\frac{\epsilon_0E^2}{2}+\frac{B^2}{2\mu_0}\right)
+\nabla\cdot\frac{\mathbf E\times\mathbf B}{\mu_0}
=-\mathbf j\cdot\mathbf E.
$$
Integration over the fixed volume and the <divergence theorem> give <Poynting's theorem>:
$$
\boxed{\frac{d}{dt}\int_V\left(\frac{\epsilon_0E^2}{2}+\frac{B^2}{2\mu_0}\right)dV
+\int_S\mathbf P\cdot\mathbf n\,dS=-\int_V\mathbf j\cdot\mathbf E\,dV,
\qquad \mathbf P=\frac{\mathbf E\times\mathbf B}{\mu_0}.}
$$
The first term is the rate of change of stored <electromagnetic energy>. The surface integral is the outward energy flux carried by the <Poynting vector>. The right side is minus the power delivered by the field to matter; for an <Ohmic conductor> it is minus the <Ohmic heating> rate. Thus energy lost from the field either flows out or does work on charges.
For the specified transverse <plane wave>, both divergence <Maxwell equations> hold because $E_y$ depends only on $x,t$. Faraday's equation requires
$$
\boxed{\mathbf B=\frac{kE_0}{\omega}(0,0,1)\cos(kx-\omega t).}
$$
Here we take the magnetic wave associated with the electric wave, without an additional static background field. Ampère's vacuum equation then gives $k^2=\mu_0\epsilon_0\omega^2$, equivalently
$$
\boxed{\omega^2=c^2k^2,\qquad c=(\mu_0\epsilon_0)^{-1/2}.}
$$
Thus all four vacuum <Maxwell equations> are satisfied. The <Poynting vector> is $\mathbf P=(kE_0^2/(\mu_0\omega))\cos^2(kx-\omega t)\mathbf e_x$, and averaging over a period gives
$$
\boxed{\langle\mathbf P\rangle=\frac{kE_0^2}{2\mu_0\omega}\mathbf e_x.}
$$
For propagation in the positive $x$ direction, $\omega=ck$ and this is $E_0^2\mathbf e_x/(2\mu_0c)$. It represents the mean energy transmitted per unit area per unit time, the wave's time-averaged <energy flux> per unit area, directed along its propagation.
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