# 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

Polarization vector at point $r$  is the limit of the ratio of the sum of elementary dipolar moments contained in the box $d\tau$  over the volume d\tau[/itex] as it tends towards zero.}

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 $\epsilon (k,\omega )$ :

$\epsilon (k,\omega )=(\epsilon _{0}-{\frac {e^{2}}{km}}\int {\frac {1}{vk-\omega }}{\frac {\partial {\hat {f}}^{0}}{\partial v}}dv)$