Microscopic Description of Current Derivation

[mateomy]

Listening to lecture today in my E&M class, the professor was giving us the derivation of the micro description of current.

$$I=neAv_d$$

Somewhere along the line he comes to $v_d=at$ that "t" might be a tau, I can't decipher my scribble. He said he isn't really sure how they get that particular expression and, "When in doubt, we turn to Feynman…unfortunately Feynman too, seems to do a lot of arm-waiving in this area. I'm not quite sure how this expression arises."

I know it has to do with the drift velocity of a charge carrier and that it relates to temperature. Can someone lead me to a reference or explain how this comes about? Super curious.

Thanks.

 

[jbsteele]

The expression you mentioned is known as the Einstein relation and it describes the relation between the drift velocity of a charge carrier and its temperature. This relation can be derived from the Boltzmann transport equation, which describes the motion of particles in an electric field. Essentially, the drift velocity is proportional to the electric field, and the temperature determines how strongly the particles will be affected by the field. You can find more information about this relation and how to derive it in any standard textbook on statistical physics or condensed matter physics.