= Solution
The <Poynting vector> is the <electromagnetic energy> flux density,
$$
\boxed{\mathbf S=\mathbf E\times\mathbf H,\qquad [\mathbf S]=\mathrm{W\,m^{-2}}.}
$$
In <vacuum> it equals $\mathbf E\times\mathbf B/\mu_0$. For a linear nondispersive medium with time-independent constitutive coefficients, <Maxwell's equations> and the identity $\nabla\cdot(\mathbf E\times\mathbf H)=\mathbf H\cdot\nabla\times\mathbf E-\mathbf E\cdot\nabla\times\mathbf H$ give <Poynting's theorem>,
$$
\frac{\partial u_{\rm EM}}{\partial t}+\nabla\cdot\mathbf S=-\mathbf J_f\cdot\mathbf E,
\qquad
u_{\rm EM}=\frac12\mathbf E\cdot\mathbf D+\frac12\mathbf B\cdot\mathbf H.
$$
The source term is <work> done by the fields on free charges; for an Ohmic conductor it is the positive <local Joule heating rate> $\sigma E^2$. Integrating over a <volume (thermodynamics)> states that loss of stored field <energy> plus outward <energy> flow equals <energy> transferred to matter. In dispersive media, such as a <plasma>, material motion contributes to stored <energy> and the simple nondispersive expression cannot be used indiscriminately.
For a <vacuum> <plane wave>, $B=E/c$ and $\mathbf E$, $\mathbf H$ and the propagation direction form a right-handed triad. With real peak field $E_0$, the time-averaged <wave intensity> is
$$
\langle S\rangle=\frac{E_0^2}{2\mu_0c}=\frac12\epsilon_0cE_0^2.
$$
More generally, complex peak amplitudes give $\langle\mathbf S\rangle=\tfrac12\operatorname{Re}(\mathbf E_0\times\mathbf H_0^*)$. Along a <transmission line>, the cross-section <integral> of the longitudinal <Poynting vector> is the transmitted electrical <power> $VI$. <Energy> flows through the electromagnetic fields in the surrounding <dielectric>, even though the conductors guide the currents.
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