A Microscopic View of Ohm's Law — Question 1

Question 1

A step field builds drift toward a finite steady velocity

Use the relaxation model $m\,d\vec v_d/dt=-e\vec E-m\vec v_d/\tau$, with electron charge $-e$, mass m and constant positive relaxation time tau. At t=0 a uniform field $E_0\hat x$ is switched on in an initially zero-drift electron gas of density n.

1. Solve for drift velocity and current density.

2. Find when the current reaches 95 percent of its steady value and state the steady conductivity.

Question 1 - Solution

A step field builds drift toward a finite steady velocity

(a) Method and calculation

$$v_x(t)=-\frac{eE_0\tau}{m}(1-e^{-t/\tau}).$$$$J_x=-nev_x=\frac{ne^2\tau}{m}E_0(1-e^{-t/\tau}).$$

(b) Interpretation and further calculation

$$J_x/J_\infty=.95\Rightarrow t_{95}=\tau\ln20=2.99573\tau.$$$$\sigma_0=J_\infty/E_0=ne^2\tau/m.$$

Physical check

Scattering balances the electric force in steady drift; it does not stop all microscopic motion.