Introduction to Mathematical Physics/Electromagnetism/Electromagnetic interaction

__toc__ Electromagnetic forces Postulates of electromagnetism have to be completed by another postulate that deals with interactions: 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 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: , thus Variation is finally . Moreover, we proved that: