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Q.
$A$ and $B$ are two points on a uniform ring of resistance R. The $\angle ACB = \theta$, where $C$ is the centre of the ring. The equivalent resistance between $A$ and $B$ is
Resistance per unit length = $\rho =\frac{R}{2\pi r}$
Lengths of sections $APB$ and $AQB$ are $r\theta$ and $r(2 \pi - \theta)$
Resistances of sections
$APB$ and $AQB$ are
$R_{1} =\rho r\theta =\frac{R}{2\pi r} r\theta =\frac{R}{2\pi}$
and $R_{2}=\frac{R}{2\pi r}r\left(2\pi-\theta\right)=\frac{R\left(2\pi-\theta\right)}{2\pi}$
As $R_1$ and $R_2$ are in parallel between A and B, their equivalent resistance is
$R_{eq}=\frac{R_{1}R_{2}}{R_{1}+R_{2}}=\frac{\frac{R\theta}{2\pi}. \frac{R\left(2\pi-\theta\right)}{2\pi}}{\frac{R\theta}{2\pi}+\frac{R\left(2\pi-\theta\right)}{2\pi}}=\frac{\frac{R^{2}\theta\left(2\pi-\theta\right)}{4\pi^{2}}}{\frac{R}{2\pi}\left[\theta+2\pi-\theta\right]} $
$=\frac{R\left(2\pi -\theta\right)\theta}{4\pi^{2}}$
