Electrodynamics MOC

Poynting’s theorem

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

𝑃=∭Ω⃗𝐄⋅⃗𝐉𝑑𝜏=−𝑑𝑑𝑡∭Ω𝑢𝑑𝜏−⊂⊃∬𝜕Ω⃗𝐒⋅𝑑⃗𝐚

or equivalently by Острогра́дский’s divergence theorem

⃗𝐄⋅⃗𝐉=−𝜕𝑢𝜕𝑡−⃗∇⋅⃗𝐒

where

𝑢=12(1𝜇0𝐵2+𝜖0𝐸2)⃗𝐒=1𝜇0⃗𝐄×⃗𝐁

are the Electromagnetic energy density and Poynting vector respectively. In the case ⃗𝐄 ⋅⃗𝐉 =0 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 ⃗𝐟± =𝜌±(⃗𝐄 +⃗𝐯± ×⃗𝐁) whence the work done is

𝑑𝑊±=⃗𝐟±⋅𝑑⃗ℓ±=𝜌±(⃗𝐄+⃗𝐯±×⃗𝐁)⋅(⃗𝐯±𝑑𝑡)=𝜌±⃗𝐄⋅⃗𝐯±𝑑𝑡

and the power per unit volume is

𝑑𝑊𝑑𝑡=⃗𝐄⋅(𝜌+⃗𝐯++𝜌−⃗𝐯−)=⃗𝐄⋅⃗𝐉

Applying ^Differential to eliminate ⃗𝐉

⃗𝐄⋅⃗𝐉=1𝜇0⃗𝐄⋅(⃗∇×⃗𝐄)−𝜖0⃗𝐄⋅𝜕⃗𝐄𝜕𝑡

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

⃗∇⋅(⃗𝐄×⃗𝐁)=⃗𝐁⋅(⃗∇×⃗𝐄)−⃗𝐄⋅(⃗∇×⃗𝐁)=⃗𝐁⋅𝜕⃗𝐁𝜕𝑡−⃗𝐄⋅(⃗∇×⃗𝐁)

so

⃗𝐄⋅⃗𝐉=−1𝜇0⃗𝐁⋅𝜕⃗𝐁𝜕𝑡−𝜖0⃗𝐄⋅𝜕⃗𝐄𝜕𝑡−1𝜇0⃗∇⋅(⃗𝐄×⃗𝐁)=−12𝜕𝜕𝑡(1𝜇0𝐵2+𝜖0𝐸2)−1𝜇0⃗∇⋅(⃗𝐄×⃗𝐁)

giving the expressions above.

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Footnotes

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