Question

$A$ and $B$ are two points on uniform ring of radius $r$. The resistance of the ring is $R$. $\angle AOB=\theta$ as shown in the figure. The equivalent resistance between $A$ and $B$ is ________ .

• A. $\frac{R\theta }{2\pi }$
• B. $\frac{R(2\pi -\theta )}{4\pi }$
• C. $R(1-\frac{\theta }{2\pi })$
• D. $\frac{R}{4{\pi }^2}(2\pi -\theta )\theta$

Answer 1

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

Length of the smaller arc $l=r\theta$
So, resistance of the smaller arc $R_1=\frac{r\theta }{2\pi r}R=\frac{\theta }{2\pi }R$
Resistance of bigger arc $R_2=\frac{2\pi -\theta }{2\pi }R$
Since $R_1$ and $R_2$ are connected in parallel across AB.
Thus, equivalent resistance between A and B $\frac{1}{R_{AB}}=\frac{1}{R_1}+\frac{1}{R_2}$
$\therefore \frac{1}{R_{AB}}=\frac{2\pi }{\theta R}+\frac{2\pi }{(2\pi -\theta )R}$
$⟹R_{AB}=\frac{R}{4{\pi }^2}(2\pi -\theta )\theta$
So, option D is correct.