A wire of resistor R is bent into a circular ring of radius r. Equivalent resistance between two points X and Y on its circumference, when angle XOY is α α , can be given by
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Here $R_{xwy} = \frac{R}{2 \pi r} \times (r \alpha) = \frac{R \alpha}{2 \pi}$ $\left(\boxed{\times}\alpha = \frac{1}{r}\right)$
and $R_{XZY} = \frac{R}{2 \pi r} \times r(2 \pi - \alpha) = \frac{R}{2 \pi} (2 \pi - \alpha)$
$R_{eq} = \frac{R_{XWY} R_{XZY}}{R_{XWY} + R_{XZY}} = \frac{\frac{R \alpha}{2 \pi} \times \frac{R}{2 \pi} (2 \pi - \alpha)}{\frac{R \alpha}{2 \pi} + \frac{R (2 \pi - \alpha)}{2 \pi}}$ $= \frac{R \alpha}{4 \pi^{2}} \left( 2 \pi - \alpha \right)$
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