Principles of dielectric relaxation spectroscopy

Neglecting higher electric multipoles, the resulting electric polarization P of the homogeneous system can be assumed equal to the dipole density  

The proportionality factor x is called the dielectric susceptibility. Equation (4.2) can be rewritten in terms of the dielectric displacement D  

If a dielectric is anisotropic, like liquid crystals and most solids, the scalar susceptibility in eqn (4.2) must be replaced by a tensor. Hence, the permittivity must also be a tensor quantity. 

For static electric fields the polarization is in equilibrium with the field. However, if the applied field changes in time, then, in general, the polarization is not in equilibrium with it, since the charged particles and permanent dipole moments may not follow the changes. 

Let the electric field vary harmonically, i.e. in the complex notation let it be given by E (t) = E exp(icot), where E is the amplitude and co the angular frequency of variation. For an isotropic system, the time dependence of the electric displacement is also harmonic D (t) = DSexp(icot- eo), although and the phase difference are frequency dependent. Equation (4.3) now becomes 

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