# Question on a simple caluclation of electric susceptibility

1. Jan 1, 2014

### ShayanJ

The simplest model for calculating electric susceptibility of dielectrics,is a driven oscillation of an electric dipole with a resistive force proportional to its velocity.Its equation is like below:
$m\ddot{x}+\gamma \dot{x}+kx=-eE$
We know that the restoring force is the same force binding the electron to its nucleus and the driving force is due to an external field.Now,for a bound charge,the only interpretation of the resistive force can be the radiation reaction due to the electron's radiation because of its acceleration.My question is this:
The radiation reaction is given(classically) by the Abraham–Lorentz formula which says its proportional to the time derivative of the acceleration.How can such a force be approximated by a term proportional to velocity???
Thanks

2. Jan 1, 2014

### Meir Achuz

The resistive force proportional to v is just a parametrization used to make the equations soluble in a much oversimplified model. Although it is not physically realistic, it does give a mathematically reasonable description of the complex structure.

3. Jan 2, 2014

### Jano L.

That is one source of damping, and Lorentz and Planck used it that way. But proper description of action of molecule's own field would really require more than by viscous force term. Viscous term is more appropriate for friction due to other particles/fields. For example, Einstein in his paper on Quantum theory of radiation considers neutral molecule moving in a system of molecules and EM radiation in equilibrium state. The force molecule feels is divided into average damping force (given by viscous term $-\gamma \mathbf v$) and random fluctuating force (describable by fluctuating E(t) in your equation). So the term $-\gamma \mathbf v$ may be regarded as approximate description of action of other bodies, not just due to "radiation reaction" of the molecule's own field. Similarly, in the Drude model of conductivity of metals, this term is due to randomization of motion after collision of the electron with some scatterer in the metal.