## Physics Friday 131

Part 11: Electromagnetic Energy and Poynting’s Theorem

We previously noted that the rate of work done per unit times by electric fields (magnetic fields do no work) in a volume V with current density J is
.
This represents the power being transfered out of the electromagnetic fields and into mechanical or thermal energy. Now, let us examine further the conservation of energy.

First, we can use AmpĂ¨re’s law to eliminate J in the above, working only in terms of the fields instead. In SI units, we have
,
so
,
and so

Now, ,
so
,
and we recall from here that the term in the parentheses above is the energy density of the electric field UE,
so
.

Next, for the dot product with the curl, we use the product rule for divergence of a cross product (similar to our usage here, albeit with E instead of A):
.
Now, Faraday’s Law gives us the curl of the electric field:
; thus
,
and .
Plugging these in to our work equation,
;
and
,
where  is the energy density of the magnetic field, which we found here. Thus, we have
;
assuming that the total electromagnetic energy density U is the sum of the electric and magnetic energy densities, this is
;
defining , this is written
.
This has the form of a general continuity equation for electromagnetic energy:  represents the rate of removal of energy from the fields. The vector S, then, describes the flux of electromagnetic energy from one location to another. This equation of energy conservation is called Poynting’s theorem, and S is called the Poynting vector, and has units of intensity (or power per unit area). Since only the divergence of the Poynting vector appears in the theorem, one might think that it is arbitrary in that adding the curl of any vector field will not change that divergence; however, relativistic considerations confine the Poynting vector to the specific value .