A Microscopic View of Ohm's Law — Question 7

Question 7

Residual drift transports finite charge after the field is removed

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. Before t=0 the field is $E_0\hat x$ and drift is steady. At t=0 the field is set to zero. A cross-section has area A.

1. Find the current for t>0.

2. Integrate the residual transported charge and find the mean extra electron displacement.

Question 7 - Solution

Residual drift transports finite charge after the field is removed

(a) Method and calculation

$$v_x(0)=-eE_0\tau/m,\quad v_x(t)=v_x(0)e^{-t/\tau}.$$$$I(t)=I_0e^{-t/\tau},\quad I_0=ne^2\tau AE_0/m.$$

(b) Interpretation and further calculation

$$Q_{\mathrm{residual}}=\int_0^\infty I(t)dt=I_0\tau.$$$$\Delta x=\int_0^\infty v_x(t)dt=-eE_0\tau^2/m,\quad-neA\Delta x=Q_{\mathrm{residual}}.$$

Physical check

The finite tail follows carrier inertia in the prescribed model; external circuit transients are excluded.