= Solution
In a linear, nondispersive, lossless medium, the field <energy density> and energy flux are
$$
\boxed{u_{\rm em}=\tfrac12(\mathbf E\cdot\mathbf D+\mathbf B\cdot\mathbf H),\qquad\mathbf S=\mathbf E\times\mathbf H.}
$$
The <Poynting theorem> follows by dotting the two curl <Maxwell equations> with the other field and subtracting:
$$
\frac{\partial u_{\rm em}}{\partial t}+\nabla\cdot\mathbf S=-\mathbf J_f\cdot\mathbf E.
$$
The right-hand side is power transferred from the field to free charges. In vacuum the <energy density> is $\epsilon_0E^2/2+B^2/(2\mu_0)$.
Static electric or magnetic fields store energy even though their time derivatives vanish. <Static Poynting flow> occurs because static crossed fields can also have a nonzero Poynting flux: for example, a steady current-carrying coaxial line has radial electric and azimuthal magnetic fields, carrying energy along the line toward the load. Time independence therefore does not imply zero energy flow.
For phasors whose real parts represent the physical fields,
$$
\langle\mathbf S\rangle=\tfrac12\operatorname{Re}(\mathbf E_0\times\mathbf H_0^*),\qquad\langle u_{\rm em}\rangle=\tfrac14(\epsilon|E_0|^2+\mu|H_0|^2).
$$
For a travelling <plane wave> in vacuum the electric and magnetic contributions are equal and $\langle S\rangle=c\langle u_{\rm em}\rangle$. In dispersive matter the simple macroscopic energy formula needs modification to include the material response; it is not a universal formula for plasma energy.<Electromagnetism by David Tong>[Chapter 4: conservation of electromagnetic energy and the Poynting vector]{ref}
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