Current Density — Question 3

Question 3

Electron transit and signal transit occupy very different time scales

Use $e=1.602176634\times10^{-19}$ C, exact. A wire has $n=8.0\times10^{28}\,\mathrm{m^{-3}}$, area $1.0\,\mathrm{mm^2}$, length 2.0 m and current 1.60 A toward $+x$. A change in its electric field propagates at the supplied speed $2.0\times10^8$ m/s.

1. Find the electron drift velocity.

2. Compare the drift transit time with the field-change transit time.

Question 3 - Solution

Electron transit and signal transit occupy very different time scales

(a) Method and calculation

$$J=I/A=1.60\times10^6\,\mathrm{A/m^2},\quad\vec J=-ne\vec v_d.$$$$\vec v_d=-1.24830\times10^{-4}\hat x\,\mathrm{m/s}.$$

(b) Interpretation and further calculation

$$t_d=L/|v_d|=1.60218\times10^4\,\mathrm s=4.45049\,\mathrm h.$$$$t_s=L/u=1.00\times10^{-8}\,\mathrm s,\quad t_d/t_s=1.60218\times10^{12}.$$

Physical check

Establishing a field throughout the wire does not require one electron to drift its entire length.