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</math> 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 $\epsilon (k,\omega )$:

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