Question

Discuss the temperature dependence and frequency dependence during various polarisation

Answer 1

The easiest way to look at relaxation phenomena is to consider what happens if the driving force - the electrical field in our case - is suddenly switched off, after it has been constant for a sufficiently long time so that an equilibrium distribution of dipoles could be obtained. We expect then that the dipoles will randomize, i.e. their dipole moment or their polarization will go to zero. However, that cannot happen instantaneously. A specific dipole will have a certain orientation at the time the field will be switched off, and it will change that orientation only by some interaction with other dipoles (or, in a solid, with phonons), in other words upon collisions or other "violent" encounters. It will take a characteristic time, roughly the time between collisions, before the dipole moment will have disappeared. Since we are discussing statistical events in this case, the individual characteristic time for a given dipole will be small for some, and large for others. But there will be an averagevalue which we will call the relaxation timet of the system. We thus expect a smooth change over from the polarization with field to zero within the relaxation time t, or a behavior as shown below        In formulas, we expect that P decays starting at the time of the switch-off according to     P(t)  =  P0 · exp –tt   This simple equation describes the behavior of a simple system like our "ideal" dipoles very well. It is, however, not easy to derive from first principles, because we would have to look at the development of an ensemble of interacting particles in time, a classical task of non-classical, i.e. statistical mechanics, but beyond our ken at this point. Nevertheless, we know that a relation like that comes up whenever we look at the decay of some ensemble of particles or objects, where some have more (or less) energy than required by equilibrium conditions, and the change-over from the excited state to the base state needs "help", i.e. has to overcome some energy barrier. All we have to assume is that the number of particles or objects decaying from the excited to the base state is proportional to the number of excited objects. In other words, we have a relation as follows:     dndt  µ  n =   –  1t · n    n  =   n0 · exp –  tt       This covers for example radioactive decay, cooling of any material, and the decay of the foam or froth on top of your beer: Bubbles are an energetically excited state of beer because of the additional surface energy as compared to a droplet. If you measure the height of the head on your beer as a function of time, you will find the exponential law.When we turn on an electrical field, our dipole system with random distribution of orientations has too much energy relative to what it could have for a better ortientation distribution. The "decay" to the lower (free) energy state and the concomitant built-up of polarization when we switch on the field, will follow our universal law from above, and so will the decay of the polarization when we turn it off.We are, however, not so interested in thetime dependence P(t) of the polarization when we apply some disturbance or inputto the system (the switching on or off of the electrical field). We rather would like to know its frequency dependence P(w)with w = 2pn = angular frequency, i.e. the output to a periodic harmonic input, i.e. to a field like E = Eo · sinwt. Since any signal can be