Question

A wire of length l and resistance R is bent in the form of the ring. The resistance between two points which are separated by an angle $\theta$ is

• A. $\frac{R}{4{\pi }^2}\left(2\pi -\theta \right)\theta$
• B. $\frac{Rl}{4{\pi }^2}\left(2\pi -\theta \right)\theta$
• C. $R\left(2\pi -\theta \right)$
• D. $R\theta$

Answer 1

The correct option is A $\frac{R}{4{\pi }^2}\left(2\pi -\theta \right)\theta$

Let, $R=\frac{\rho l}{A}$

LEngth of wire segment $l=r\theta$

Resistance of first wire segment,

$R_1=\frac{\rho r\theta }{A}{R}_2=\frac{\rho r(2\pi .\theta )}{A}$

These two segments are in parallel. Therefore net $R$

$\frac{1}{R_{net}}=\frac{A}{\rho R\theta }+\frac{A}{\rho r(2\pi -\theta )}$

On solving

$R_{net}=\frac{(2\pi -\theta )\theta \rho r}{2\pi A}\therefore R_{net}=\frac{(2\pi -\theta )\theta R}{4{\pi }^2}$