Introduction to Mathematical Physics/Electromagnetism/Electromagnetic interaction

Electromagnetic forcesEdit

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 vectorEdit

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: