# 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:
$dU=HdB+EdD$

## Multipolar distribution

It has been seen at section Electromagnetic interaction that energy for a volumic charge distribution $\rho$ is \index{multipole}
$U=\int \rho Vd\tau$
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: $U=\int V{\mbox{ div }}(P_{i}\delta )=\partial _{i}V.P_{i}$.
* 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:
$U_{t}=\sum q_{n}(V_{c}(r_{n})+V_{e}(r_{n}))$
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:
$U_{e}=\sum q_{n}V_{e}(r_{n})=\sum q_{n}(U(0)+x_{i}^{n}\partial _{i}(U)+{\frac {1}{2}}x_{i}^{n}x_{j}^{n}\partial _{i}\partial _{j}(U)+\dots )$
where $x_{i}^{n}$ labels position vector of charge number $n$. This sum can be written as:
$U_{e}=\sum q_{n}U(0)+\sum q_{n}x_{i}^{n}\partial _{i}(U)+{\frac {1}{2}}\sum q_{n}x_{i}^{n}x_{j}^{n}\partial _{i}\partial _{j}(U)+\dots$
the reader recognizes energies associated to multipoles.
Remark: In quantum mechanics, passage laws from classical to quantum mechanics allow to define tensorial operators (see chapter Groups) associated to multipolar momenta.

## Field in matter

In vacuum electromagnetism, the following constitutive relation is exact:
eqmaxwvideE
$D=\epsilon _{0}E$
eqmaxwvideB
$H={\frac {B}{\mu _{0}}}$
Those relations are included in Maxwell equations. Internal electrical energy variation is:
$dU=EdD$
or, by using a Legendre transform and choosing the thermodynamical variable $E$:
$dF=DdE$
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:
$D=\chi E$
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:
Example:
If one impose a relation of the following type:
$D_{i}=\epsilon _{ij}E_{j}$
then medium is called dielectric .\index{dielectric} The expression of the energy is:
$F=F_{0}+\epsilon _{ij}E_{i}E_{j}$
Example:
In the linear response theory \index{linear response}, $D_{i}$ at time $t$ is supposed to depend not only on the values of $E$ at the same time $t$, but also on values of $E$ at times anteriors. This dependence is assumed to be linear:
$D_{i}(t)=\epsilon _{ij}*E_{j}$
where $*$ means time convolution.
Example:
To treat the optical activity [ph:elect:LandauEle], a tensor \index{optical activity} $a_{ijk}$ such that:
$D_{i}=\epsilon _{ij}E_{j}+a_{ijk}E_{j,k}$
is introduced. Not that this law is still linear but that $D_{i}$ depends on the gradient of $E_{i}$.
The second point of view is now illustrated by the following two examples:
Example:
A simple model for the susceptibility: \index{susceptibility} An elementary electric dipole located at $r_{0}$ can be modelized (see section Modelization of charge) by a charge distribution ${\mbox{ div }}(p\delta (r_{0}))$. Consider a uniform distribution of $N$ such dipoles in a volume $V$, dipoles being at position $r_{i}$. Function $\rho$ that modelizes this charge distribution is:
$\rho =\sum _{V}{\mbox{ div }}(p_{i}\delta (r_{i}))$
As the divergence operator is linear, it can also be written:
$\rho ={\mbox{ div }}\sum _{V}(p_{i}\delta (r_{i}))$
Consider the vector:
eqmoyP
$P(r)=\lim _{d\tau \rightarrow 0}{\frac {\sum _{d\tau }p_{i}}{d\tau }}$
This vector $P$ is called polarization vector\index{polarisation}. The evaluation of this vector $P$ is illustrated by figure figpolar.
figpolar
Maxwell--gauss equation in vacuum
${\mbox{ div }}\epsilon _{0}E=\rho$
can be written as:
${\mbox{ div }}(\epsilon _{0}E-P)=0$
We thus have related the microscopic properties of the material (the $p$'s) to the macroscopic description of the material (by vector $D=\epsilon _{0}E-P$). We have now to provide a microscopic model for $p$. Several models can be proposed. A material can be constituted by small dipoles all oriented in the same direction. Other materials, like oil, are constituted by molecules carrying a small dipole, their orientation being random when there is no $E$ field. But when there exist an non zero $E$ field, those molecules tend to orient their moment along the electric field lines. The mean $P$ of the $p_{i}$'s given by equation eqmoyP that is zero when $E$ is zero (due to the random orientation of the moments) becomes non zero in presence of a non zero $E$. A simple model can be proposed without entering into the details of a quantum description. It consist in saying that $P$ is proportional to $E$:
$P=\chi E$
where $\chi$ is the polarisability of the medium. In this case relation:
$D=\epsilon _{0}E-P$
becomes:
$D=(\epsilon _{0}+\chi )E$
Example: A second model of susceptibility: Consider the Vlasov equation (see equation eqvlasov and reference [ph:physt:Diu89]. Function $f$ is the mean density of particles and $n_{0}$ represents the density of the positively charged background.
vlasdie
${\frac {\partial f}{\partial t}}+{v}\partial _{x}f+{\frac {F}{m}}\partial _{v}f=0$
let us assume that the force undergone by the particles is the electric force:
${\vec {F}}=-eE(x,t)$
Maxwell equations are reduced here to:
eqmaxsystpart
${\mbox{ div }}E=\rho$
where electrical charge $\rho (x,t)$ is the charge induced by the fluctuations of the electrons around the neutral equilibrium state:
$\rho =-e\int f(x,v,t)dv+en_{0}$
Let us linearize this equation system with respect to the following equilibrium position:
${\begin{matrix}f(x,v,t)&=&f^{0}(v)+f^{1}(x,v,t)\\F(x,v,t)&=&0+F_{1}\end{matrix}}$
As the system is globally electrically neutral:
$\int f^{0}(v)=n_{0}$
By a $x$ and $t$ Fourier transform of equations vlasdie and eqmaxsystpart one has:
${\begin{matrix}\epsilon _{0}ik{\hat {E_{1}}}&=&-e\int {\hat {f_{1}}}dv\\-i\omega {\hat {f_{1}}}+ivk{\hat {f_{1}}}&=&e{\frac {\hat {E_{1}}}{m}}{\frac {\partial {\hat {f^{0}}}}{\partial v}}\end{matrix}}$
Eliminating ${\hat {f_{1}}}$ from the previous system, we obtain:
$ik{\hat {E_{1}}}(\epsilon _{0}-{\frac {e^{2}}{km}}\int {\frac {1}{vk-\omega }}{\frac {\partial {\hat {f}}^{0}}{\partial v}}dv)=0$
The first term of the previous equation can be considered as the divergence of a vector that we note $D$ which is $D=\epsilon *E_{1}$, where $*$ is the convolution in $x$ and $t$:
eqmaxconvol
${\mbox{ div }}(\epsilon *E_{1})=0$
Vector $D$ is called electrical displacement. $\epsilon$ is the susceptibility of the medium. Maxwell equations eqmaxsystpart describing a system of charges in vacuum has thus been transformed to equation eqmaxconvol that described the field in matter. Previous equation provides $\epsilon (k,\omega )$:
$\epsilon (k,\omega )=(\epsilon _{0}-{\frac {e^{2}}{km}}\int {\frac {1}{vk-\omega }}{\frac {\partial {\hat {f}}^{0}}{\partial v}}dv)$