Recall from elementary physics that the rate at which work is done on
an electric charge by an electromagnetic field is:

(8.81) |

(8.82) |

If we use AL to eliminate

(8.83) |

(8.84) |

Using:

(8.85) |

(8.86) |

It is easy to see that:

(8.87) | |||

(8.88) |

from which we see that these terms are the time derivative of the

(8.89) |

(8.90) |

Equating the terms under the integral:

(8.91) |

(8.92) |

This has the precise appearance of conservation law. If we apply the
divergence theorem to the integral form to change the volume integral of
the divergence of
into a surface integral of its flux:

(8.93) |

In words, the sum of the work done by all fields on charges in the
volume, plus the changes in the field energy within the volume, plus the
energy that flows out of the volume carried by the field must *balance* - this is a version of the *work-energy theorem*, but one
expressed in terms of the *fields*.

In this interpretation, we see that
must be the *vector
intensity* of the electromagnetic field - the energy per unit area per
unit time - since the flux of the Poynting vector through the surface
is the power passing through it. It's magnitude is the intensity
proper, but it also tells us the *direction* of energy flow.

With this said, there is at least one assumption in the equations above
that is not strictly justified, as we are assuming that the medium is
dispersionless and has no resistance. We do not allow for energy to
appear as heat, in other words, which surely would happen if we drive
currents with the electric field. We also used the *macroscopic*
field equations and energy densities, which involve a coarse-grained
average over the microscopic particles that matter is actually made up
of - it is their random motion that *is* the missing heat.

It seems, then, that Poynting's theorem is likely to be applicable in a
*microscopic* description of particles moving in a vacuum, where
their individual energies can be tracked and tallied:

(8.94) | |||

(8.95) | |||

(8.96) |

but not necessarily so useful in macroscopic media with dynamical dispersion that we do not yet understand. There we can identify the term as the rate at which the mechanical energy of the charged particles that make up changes and write:

(8.97) |

Momentum can similarly be considered, again in a microscopic
description. There we start with Newton's second law and the Lorentz
force law:

(8.98) |

(8.99) |

(8.100) |

(8.101) |

Again, we distribute:

(8.102) |

(8.103) |

(8.104) |

Finally, substituting in FL:

(8.105) |

Reassembling and rearranging:

The quantity under the integral on the left has units of momentum density. We define:

(8.106) |

(8.107) |

(8.108) |

I wish that I could do better with this, but analyzing the Maxwell Stress Tensor termwise to understand how it is related to field momentum flow is simply difficult. It will actually make more sense, and be easier to derive, when we formulate electrodynamics relativistically so we will wait until then to discuss this further.