Example 5: Consider the cylinder $p = a$ and the cylinder $p = b$ ($b > a$ and both are coaxial). Cylinder $p = a$ is held at $V = 0$, and cylinder $p = b$ is held at $V = V_0$. The region in between has a conductivity $\sigma$. The potential in the region $a < p < b$ is given by
$V(p) = \frac{V_0}{\ln(b/a)} \ln(p/a)$.
Find the resistance for a length $d$ of this configuration.
Ans:
$\nabla V = \frac{V_0}{\ln(b/a)} \frac{1}{p} \hat{a}_p \implies |\nabla V|^2 = \left(\frac{V_0}{\ln(b/a)} \frac{1}{p}\right)^2$
$P = \int_V \sigma |\nabla V|^2 dV = 2\pi d \int_a^b \sigma \left(\frac{V_0}{\ln(b/a)} \frac{1}{p}\right)^2 p dp = \frac{2\pi \sigma d}{\ln(b/a)} V_0^2$
$\implies R = \frac{V_0^2}{P} = \frac{\ln(b/a)}{2\pi \sigma d}$
