Introduction to Mathematical Physics/Energy in continuous media/Electromagnetic energy

Introduction

At section Electromagnetic energy, it has been postulated that the electromagnetic power given to a volume is the outgoing flow of the Poynting vector. \index{Poynting vector} If currents are zero, the energy density given to the system is: Template:IMP/eq

Multipolar distribution

It has been seen at section Electromagnetic interaction that energy for a volumic charge distribution $\rho$ is \index{multipole} Template:IMP/eq where $V$ is the electrical potential. Here are the energy expression for common charge distributions:

• for a point charge $q$, potential energy is: $U=qV(0)$.
• for a dipole \index{dipole} $P_i$ potential energy is: $U=\int V\text{ div }(P_i\delta )={\partial }_{i}V.P_i$.
• for a quadripole $Q_{i,j}$ potential energy is: $U=\int V({\partial }_i{\partial }_j{Q}_{i,j}\delta )={\partial }_i{\partial }_{j}V.Q_{i,j}$.

Consider a physical system constituted by a set of point charges $q_n$ located at $r_n$. Those charges can be for instance the electrons of an atom or a molecule. let us place this system in an external static electric field associated to an electrical potential $U_e$. Using linearity of Maxwell equations, potential $U_t(r)$ felt at position $r$ is the sum of external potential $U_e(r)$ and potential $U_c(r)$ created by the point charges. The expression of total potential energy of the system is: Template:IMP/eq In an atom,\index{atom} term associated to $V_c$ is supposed to be dominant because of the low small value of $r_n-r_m$. This term is used to compute atomic states. Second term is then considered as a perturbation. Let us look for the expression of the second term $U_e=\sum q_n{V}_e(r_n)$. For that, let us expand potential around $r=0$ position: Template:IMP/eq where $x_i^n$ labels position vector of charge number $n$. This sum can be written as: Template:IMP/eq the reader recognizes energies associated to multipoles. Template:IMP/rem

Field in matter

In vacuum electromagnetism, the following constitutive relation is exact: Template:IMP/label Template:IMP/eq Template:IMP/label Template:IMP/eq Those relations are included in Maxwell equations. Internal electrical energy variation is: Template:IMP/eq or, by using a Legendre transform and choosing the thermodynamical variable $E$: Template:IMP/eq We propose to treat here the problem of the modelization of the function $D(E)$. In other words, we look for the medium constitutive relation. This problem can be treated in two different ways. The first way is to propose {\it a priori} a relation $D(E)$ depending on the physical phenomena to describe. For instance, experimental measurements show that $D$ is proportional to $E$. So the constitutive relation adopted is: Template:IMP/eq Another point of view consist in starting from a microscopic level, that is to modelize the material as a charge distribution is vacuum. Maxwell equations in vacuum eqmaxwvideE and eqmaxwvideB can then be used to get a macroscopic model. Let us illustrate the first point of view by some examples: Template:IMP/exmp

Template:IMP/exmp The second point of view is now illustrated by the following two examples: Template:IMP/exmp Template:IMP/exmp