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

__toc__ 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: Multipolar distribution 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. 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 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: If one impose a relation of the following type: then medium is called dielectric .\index{dielectric} The expression of the energy is: 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. 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: 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 center | frame |Polarization vector at point is the limit of the ratio of the sum of elementary dipolar moments contained in the box over the volume 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: 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 :