Current Density — Question 8

Question 8

A radial steady current spreads over growing spherical surfaces

An inner spherical electrode injects total current I radially outward into a medium between radii a and b. Assume spherical symmetry and no charge accumulation.

1. Find $\vec J(r)$ and the ratio of its magnitudes at the two electrodes.

2. For carriers of charge $q>0$ and constant number density n, find their radial drift time from a to b.

Question 8 - Solution

A radial steady current spreads over growing spherical surfaces

(a) Method and calculation

$$I=\oint\vec J\cdot d\vec A=4\pi r^2J_r(r).$$$$\vec J=\frac{I}{4\pi r^2}\hat r,\quad J(b)/J(a)=(a/b)^2.$$

(b) Interpretation and further calculation

$$v_r=J_r/(nq)=I/(4\pi nq r^2).$$$$t=\int_a^b\frac{dr}{v_r}=\frac{4\pi nq}{3I}(b^3-a^3).$$

Physical check

Constant total current is compatible with a position-dependent local density.