Introduction to Mathematical Physics/Energy in continuous media/Electromagnetic energy

IntroductionEdit

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:

 

Multipolar distributionEdit

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

 

where   is the electrical potential. Here are the energy expression for common charge distributions:

  • for a point charge  , potential energy is:  .
  • for a dipole \index{dipole}   potential energy is:  .
  • for a quadripole   potential energy is:  .

Consider a physical system constituted by a set of point charges   located at  . 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  . Using linearity of Maxwell equations, potential   felt at position   is the sum of external potential   and potential   created by the point charges. The expression of total potential energy of the system is:

 

In an atom,\index{atom} term associated to   is supposed to be dominant because of the low small value of  . 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  . For that, let us expand potential around   position:

 

where   labels position vector of charge number  . This sum can be written as:

 

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 matterEdit

In vacuum electromagnetism, the following constitutive relation is exact:

eqmaxwvideE

 

eqmaxwvideB

 

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

 

or, by using a Legendre transform and choosing the thermodynamical variable  :

 

We propose to treat here the problem of the modelization of the function  . 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   depending on the physical phenomena to describe. For instance, experimental measurements show that   is proportional to  . So the constitutive relation adopted is:

 

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:

 

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

 

Example:

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

 

where   means time convolution.

Example:

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

 

is introduced. Not that this law is still linear but that   depends on the gradient of  .

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   can be modelized (see section Modelization of charge) by a charge distribution  . Consider a uniform distribution of   such dipoles in a volume  , dipoles being at position  . Function   that modelizes this charge distribution is:

 

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

 

Consider the vector:

eqmoyP

 

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

figpolar

 
Polarization vector at point   is the limit of the ratio of the sum of elementary dipolar moments contained in the box   over the volume d\tau</math> as it tends towards zero.}

Maxwell--gauss equation in vacuum

 

can be written as:

 

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

 

where   is the polarisability of the medium. In this case relation:

 

becomes:

 

Example: A second model of susceptibility: Consider the Vlasov equation (see equation eqvlasov and reference [ph:physt:Diu89]. Function   is the mean density of particles and   represents the density of the positively charged background.

vlasdie

 

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

 

Maxwell equations are reduced here to:

eqmaxsystpart

 

where electrical charge   is the charge induced by the fluctuations of the electrons around the neutral equilibrium state:

 

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

 

As the system is globally electrically neutral:

 

By a   and   Fourier transform of equations vlasdie and eqmaxsystpart one has:

 

Eliminating   from the previous system, we obtain:

 

The first term of the previous equation can be considered as the divergence of a vector that we note   which is  , where   is the convolution in   and  :

eqmaxconvol

 

Vector   is called electrical displacement.   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  :