# 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:

${\displaystyle dU=HdB+EdD}$

## Multipolar distribution

It has been seen at section Electromagnetic interaction that energy for a volumic charge distribution ${\displaystyle \rho }$ is \index{multipole}

${\displaystyle U=\int \rho Vd\tau }$

where ${\displaystyle V}$ is the electrical potential. Here are the energy expression for common charge distributions:

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

Consider a physical system constituted by a set of point charges ${\displaystyle q_{n}}$ located at ${\displaystyle r_{n}}$. 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 ${\displaystyle U_{e}}$. Using linearity of Maxwell equations, potential ${\displaystyle U_{t}(r)}$ felt at position ${\displaystyle r}$ is the sum of external potential ${\displaystyle U_{e}(r)}$ and potential ${\displaystyle U_{c}(r)}$ created by the point charges. The expression of total potential energy of the system is:

${\displaystyle U_{t}=\sum q_{n}(V_{c}(r_{n})+V_{e}(r_{n}))}$

In an atom,\index{atom} term associated to ${\displaystyle V_{c}}$ is supposed to be dominant because of the low small value of ${\displaystyle r_{n}-r_{m}}$. 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 ${\displaystyle U_{e}=\sum q_{n}V_{e}(r_{n})}$. For that, let us expand potential around ${\displaystyle r=0}$ position:

${\displaystyle U_{e}=\sum q_{n}V_{e}(r_{n})=\sum q_{n}(U(0)+x_{i}^{n}\partial _{i}(U)+{\frac {1}{2}}x_{i}^{n}x_{j}^{n}\partial _{i}\partial _{j}(U)+\dots )}$

where ${\displaystyle x_{i}^{n}}$ labels position vector of charge number ${\displaystyle n}$. This sum can be written as:

${\displaystyle U_{e}=\sum q_{n}U(0)+\sum q_{n}x_{i}^{n}\partial _{i}(U)+{\frac {1}{2}}\sum q_{n}x_{i}^{n}x_{j}^{n}\partial _{i}\partial _{j}(U)+\dots }$

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

${\displaystyle D=\epsilon _{0}E}$

eqmaxwvideB

${\displaystyle H={\frac {B}{\mu _{0}}}}$

Those relations are included in Maxwell equations. Internal electrical energy variation is:

${\displaystyle dU=EdD}$

or, by using a Legendre transform and choosing the thermodynamical variable ${\displaystyle E}$:

${\displaystyle dF=DdE}$

We propose to treat here the problem of the modelization of the function ${\displaystyle D(E)}$. 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 ${\displaystyle D(E)}$ depending on the physical phenomena to describe. For instance, experimental measurements show that ${\displaystyle D}$ is proportional to ${\displaystyle E}$. So the constitutive relation adopted is:

${\displaystyle D=\chi E}$

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:

${\displaystyle D_{i}=\epsilon _{ij}E_{j}}$

then medium is called dielectric .\index{dielectric} The expression of the energy is:

${\displaystyle F=F_{0}+\epsilon _{ij}E_{i}E_{j}}$

Example:

In the linear response theory \index{linear response}, ${\displaystyle D_{i}}$ at time ${\displaystyle t}$ is supposed to depend not only on the values of ${\displaystyle E}$ at the same time ${\displaystyle t}$, but also on values of ${\displaystyle E}$ at times anteriors. This dependence is assumed to be linear:

${\displaystyle D_{i}(t)=\epsilon _{ij}*E_{j}}$

where ${\displaystyle *}$ means time convolution.

Example:

To treat the optical activity [ph:elect:LandauEle], a tensor \index{optical activity} ${\displaystyle a_{ijk}}$ such that:

${\displaystyle D_{i}=\epsilon _{ij}E_{j}+a_{ijk}E_{j,k}}$

is introduced. Not that this law is still linear but that ${\displaystyle D_{i}}$ depends on the gradient of ${\displaystyle E_{i}}$.

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 ${\displaystyle r_{0}}$ can be modelized (see section Modelization of charge) by a charge distribution ${\displaystyle {\mbox{ div }}(p\delta (r_{0}))}$. Consider a uniform distribution of ${\displaystyle N}$ such dipoles in a volume ${\displaystyle V}$, dipoles being at position ${\displaystyle r_{i}}$. Function ${\displaystyle \rho }$ that modelizes this charge distribution is:

${\displaystyle \rho =\sum _{V}{\mbox{ div }}(p_{i}\delta (r_{i}))}$

As the divergence operator is linear, it can also be written:

${\displaystyle \rho ={\mbox{ div }}\sum _{V}(p_{i}\delta (r_{i}))}$

Consider the vector:

eqmoyP

${\displaystyle P(r)=\lim _{d\tau \rightarrow 0}{\frac {\sum _{d\tau }p_{i}}{d\tau }}}$

This vector ${\displaystyle P}$ is called polarization vector\index{polarisation}. The evaluation of this vector ${\displaystyle P}$ is illustrated by figure figpolar.

figpolar

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

Maxwell--gauss equation in vacuum

${\displaystyle {\mbox{ div }}\epsilon _{0}E=\rho }$

can be written as:

${\displaystyle {\mbox{ div }}(\epsilon _{0}E-P)=0}$

We thus have related the microscopic properties of the material (the ${\displaystyle p}$'s) to the macroscopic description of the material (by vector ${\displaystyle D=\epsilon _{0}E-P}$). We have now to provide a microscopic model for ${\displaystyle p}$. 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 ${\displaystyle E}$ field. But when there exist an non zero ${\displaystyle E}$ field, those molecules tend to orient their moment along the electric field lines. The mean ${\displaystyle P}$ of the ${\displaystyle p_{i}}$'s given by equation eqmoyP that is zero when ${\displaystyle E}$ is zero (due to the random orientation of the moments) becomes non zero in presence of a non zero ${\displaystyle E}$. A simple model can be proposed without entering into the details of a quantum description. It consist in saying that ${\displaystyle P}$ is proportional to ${\displaystyle E}$:

${\displaystyle P=\chi E}$

where ${\displaystyle \chi }$ is the polarisability of the medium. In this case relation:

${\displaystyle D=\epsilon _{0}E-P}$

becomes:

${\displaystyle D=(\epsilon _{0}+\chi )E}$

Example: A second model of susceptibility: Consider the Vlasov equation (see equation eqvlasov and reference [ph:physt:Diu89]. Function ${\displaystyle f}$ is the mean density of particles and ${\displaystyle n_{0}}$ represents the density of the positively charged background.

vlasdie

${\displaystyle {\frac {\partial f}{\partial t}}+{v}\partial _{x}f+{\frac {F}{m}}\partial _{v}f=0}$

let us assume that the force undergone by the particles is the electric force:

${\displaystyle {\vec {F}}=-eE(x,t)}$

Maxwell equations are reduced here to:

eqmaxsystpart

${\displaystyle {\mbox{ div }}E=\rho }$

where electrical charge ${\displaystyle \rho (x,t)}$ is the charge induced by the fluctuations of the electrons around the neutral equilibrium state:

${\displaystyle \rho =-e\int f(x,v,t)dv+en_{0}}$

Let us linearize this equation system with respect to the following equilibrium position:

${\displaystyle {\begin{matrix}f(x,v,t)&=&f^{0}(v)+f^{1}(x,v,t)\\F(x,v,t)&=&0+F_{1}\end{matrix}}}$

As the system is globally electrically neutral:

${\displaystyle \int f^{0}(v)=n_{0}}$

By a ${\displaystyle x}$ and ${\displaystyle t}$ Fourier transform of equations vlasdie and eqmaxsystpart one has:

${\displaystyle {\begin{matrix}\epsilon _{0}ik{\hat {E_{1}}}&=&-e\int {\hat {f_{1}}}dv\\-i\omega {\hat {f_{1}}}+ivk{\hat {f_{1}}}&=&e{\frac {\hat {E_{1}}}{m}}{\frac {\partial {\hat {f^{0}}}}{\partial v}}\end{matrix}}}$

Eliminating ${\displaystyle {\hat {f_{1}}}}$ from the previous system, we obtain:

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

The first term of the previous equation can be considered as the divergence of a vector that we note ${\displaystyle D}$ which is ${\displaystyle D=\epsilon *E_{1}}$, where ${\displaystyle *}$ is the convolution in ${\displaystyle x}$ and ${\displaystyle t}$:

eqmaxconvol

${\displaystyle {\mbox{ div }}(\epsilon *E_{1})=0}$

Vector ${\displaystyle D}$ is called electrical displacement. ${\displaystyle \epsilon }$ 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 ${\displaystyle \epsilon (k,\omega )}$:

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