DIELECTRIC PROPERTIES OF INSULATING MATERIALS 515 



equilibrium position Si of the ion corresponds to the static value Pi of 

 the polarization of the model. When the voltage is varying with the 

 time according to V = Vo cos cot, the greatest amplitude which the 

 displacement can have is 5i, and in general the amplitude will fall short 

 of this value by an amount which increases with increasing frequency. 

 The value Si is then closely approached only when the frequency is 

 low as compared with the reciprocal of the relaxation-time, because at 

 high frequencies the applied field reverses its direction before the ion 

 has had time to reach Si. At sufficiently low frequencies, namely 

 where w is negligible by comparison with 1/r, the frictional dissipation 

 of energy by the moving ion is so small that there is practically no 

 difference between the instantaneous position of the ion when the 

 voltage has any given value and the position it would finally attain 

 upon reaching equilibrium for that voltage. The ion then moves 

 through a succession of near-equilibrium positions, as in a reversible 

 process in thermodynamics. The dielectric constant has its static 

 value and the conductivity is zero unless there is a d-c conduction 

 component in the total conductivity. 



At the high-frequency extremity of a dispersion region we see that 

 the situation is simply reversed : the alternations in the direction of the 

 applied field are so rapid that the bound ion does not have time to 

 move an appreciable distance from its equilibrium position before the 

 direction of the applied field is reversed (C, Fig. 3). The amplitude 

 of the displacement of the ion by the applied field is then small as 

 compared with Si and the dielectric constant of the material receives 

 practically no contribution from the bound ion of this model in these 

 circumstances. However, though the amplitude of motion of the ion 

 shrinks to a small fraction of Si, its velocity is comparatively high and 

 independent of frequency. The conductivity jx is proportional to the 

 average velocity of the bound ion of the model under these conditions. 



As the restoring force is proportional to the displacement, its effect 

 upon the motion of the ion is negligible by comparison with that of the 

 applied force when the displacement 5 is small as compared with Si. 

 On this basis, the fact that the conductivity is an increasing function 

 of frequency may be attributed to the decrease in the influence of the 

 restoring forces as the frequency increases. In fact, when the ampli- 

 tude of displacement is very small as compared with Si, the ion moves 

 as if the only force opposing the applied force were the frictional force; 

 that is, its average velocity is the same as that of a free ion subjected 

 to the same applied field and the same friction. 



For many of the purposes of this discussion we could use a model of 

 the dielectric consisting of an air capacity Cs in series with'a resistance 



