Previous postulate using forces can be replaced by a "dual" postulate that uses energies:
Consider a volume . The vector is called Poynting vector. It is postulated that flux of vector trough surface delimiting volume , oriented by a entering normal is equal to the Electromagnetic power given to this volume.
Using Green's theorem, 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 in a field . Let us move this charge of . Previous postulated states that to this displacement corresponds a variation of internal energy:
where is the variation of induced by the charge displacement.
Internal energy variation is:
where is the electrical force applied to the charge.
In the static case, field has conservative circulation () so it derives from a potential. \medskip Let us write energy conservation equation:
Flow associated to divergence of is zero in all the space, indeed decreases as and as and surface increases as . So:
Let us move charge of . Charge distribution goes from to where is Dirac distribution. We thus have . So: