Resistance and Resistivity — Question 4

Question 4

Radial leakage through a cylindrical insulator has logarithmic resistance

A material of uniform resistivity $\rho$ fills the gap between coaxial electrodes of radii a and b and length L. End effects are negligible; leakage flows radially.

1. Find the leakage resistance.

2. For a thin gap $d=b-a\ll a$, obtain the leading approximation and identify its effective area.

Question 4 - Solution

Radial leakage through a cylindrical insulator has logarithmic resistance

(a) Method and calculation

$$dR=\frac{\rho\,dr}{2\pi rL}.$$$$R=\frac{\rho}{2\pi L}\ln(b/a).$$

(b) Interpretation and further calculation

$$\ln(b/a)=\ln(1+d/a)=d/a+O((d/a)^2).$$$$R\simeq\rho d/(2\pi aL)=\rho d/A_{\mathrm{inner}}.$$

Physical check

Here the current path is radial, so the conducting area grows with radius.