# 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}

$U=\int \rho V d\tau$

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: $U=\int V(\partial_i\partial_jQ_{i,j}\delta)=\partial_i\partial_j V.Q_{i,j}$.

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:

$U_e=\sum q_n V_e(r_n)=\sum q_n (U(0) +x_i^n\partial_i(U)+ \frac{1}{2}x_i^nx_j^n\partial_i\partial_j(U)+\dots)$

where $x_i^n$ labels position vector of charge number $n$. This sum can be written as:

$U_e=\sum q_n U(0)+\sum q_nx_i^n\partial_i(U)+\frac{1}{2}\sum q_nx_i^nx_j^n\partial_i\partial_j(U)+\dots$

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

$D=\epsilon_0E$

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:

$F=F_0+\epsilon_{ij}E_iE_j$

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

$\mbox{ div } \epsilon_0E=\rho$

can be written as:

$\mbox{ div } (\epsilon_0E-P)=0$

We thus have related the microscopic properties of the material (the $p$'s) to the macroscopic description of the material (by vector $D=\epsilon_0E-P$). 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:

$D=\epsilon_0E-P$

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 $\epsilon(k,\omega)$:

$\epsilon (k,\omega)=(\epsilon_0 - \frac{e^2}{km}\int \frac{1}{vk-\omega}\frac{\partial \hat{f}^0}{\partial v}dv)$