Introduction to Mathematical Physics/Electromagnetism/Electromagnetic interaction

Electromagnetic forces
Postulates of electromagnetism have to be completed by another postulate that deals with interactions:

This result can be generalized to continuous media using Poynting vector.\index{Poynting vector}

Electromagnetic energy, Poynting vector
Previous postulate using forces can be replaced by a "dual" postulate that uses energies:

Using Green's theorem, $${\mathcal P}$$ can be written as:

which yields, using Maxwell equations to:

Two last postulates are closely related. In fact we will show now that they basically say the same thing (even if Poynting vector form can be seen a bit more general).

Consider a point charge $$q$$ in a field $$E$$. Let us move this charge of $$dr$$. Previous postulated states that to this displacement corresponds a variation of internal energy:

where $$dD$$ is the variation of $$D$$ induced by the charge displacement.