Electrodynamics MOC

Poynting's theorem

Poynting's theorem states the rate of energy going into charged matter in a volume is equal to the decrease in energy stored in the fields in minus the rate of energy transported out of by the fields1

or equivalently by Острогра́дский's divergence theorem

where

are the Electromagnetic energy density and Poynting vector respectively. In the case everywhere we get a continuity equation for electromagnetic energy.

Derivation from Maxwell's equations

First note that by the Lorentz force law, the force density acting on the charge density it whence the work done is

and the power per unit volume is

Applying ^Differential to eliminate

Now by a Product rule and Faraday's law of induction

so

giving the expressions above.

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Footnotes

  1. 2023. Electromagnetism and special relativity, p. 73 ↩