Question

A conductor is in the shape of a quarter of annulus (angular ring) of inner and outer radii a and b, and thickness h. Resistance between the two curved surfaces would be ($\sigma$ = conductance)

Answer 1

Take strip of length $dx$ at a distance $a+x$ from the centre.

$R=\frac{l}{\sigma A}$ [where A is area of cross section]

$A=\frac{\pi rh}{2}$

for a strip at distance $a+x$

$\begin{array}{c}A=\frac{\pi \left(a+x\right)h}{2}\\ R_{total}={\int }_a^b\frac{2dx}{\sigma \pi \left(a+x\right)h}\\ =\frac{2\log \left(a+x\right)}{\sigma \pi h}{\int }_a^b\\ =\frac{2}{\sigma \pi h}\left[\log \left(a+b\right)-\log \left(2a\right)\right]\\ =\frac{2}{\sigma \pi h}\log \left(\frac{a+b}{2a}\right)\end{array}$