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:

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

where $$V$$ is the electrical potential. Here are the energy expression for common charge distributions: 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:
 * 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: $$U=\int   V(\partial_i\partial_jQ_{i,j}\delta)=\partial_i\partial_j V.Q_{i,j}$$.

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:

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

the reader recognizes energies associated to multipoles.

Field in matter
In vacuum electromagnetism, the following constitutive relation is exact:

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

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

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:

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:

The second point of view is now illustrated by the following two examples: {{IMP/exmp| 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: {{IMP/eq|$$\rho=\sum_V \mbox{ div } (p_i\delta (r_i))$$}} As the divergence operator is linear, it can also be written: {{IMP/eq|$$\rho=\mbox{ div } \sum_V (p_i\delta (r_i))$$}} Consider the vector: {{IMP/label|eqmoyP}} {{IMP/eq|$$ P(r)=\lim_{d\tau\rightarrow 0}\frac{\sum_{d\tau}p_i}{d\tau} $$}} This vector $$P$$ is called polarization vector\index{polarisation}. The evaluation of this vector $$P$$ is illustrated by figure figpolar. {{IMP/label|figpolar}} Maxwell--gauss equation in vacuum

can be written as:

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 $$E$$:

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

becomes:

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