# Introduction to Mathematical Physics/Electromagnetism/Electromagnetic interaction

## Electromagnetic forces[edit]

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

**Postulate:**

In the case of a charged particle of charge , Electromagnetic force applied to this particle is:

where is the electrical force (or Coulomb force) \index{Coulomb force} and is the Lorentz force. \index{Lorentz force}

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

secenergemag

## Electromagnetic energy, Poynting vector[edit]

Previous postulate using forces can be replaced by a "dual" postulate that uses energies:

**Postulate:**

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.

**Theorem:**

Internal energy variation is:

where is the electrical force applied to the charge.

**Proof:**

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:

,

thus

Variation is finally . Moreover, we prooved that: