Question

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

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

Answer 1

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

Consider the ring as two parts as two resistances joined in parallel between two points $A$ and $B$, then two resistance would be
$R_1=\frac{2\pi r}{\cdot }r\theta =\frac{R}{2\pi }\theta$
and $R_2=\frac{R}{2\pi r}r(2\pi -\theta )$
$=\frac{R}{2\pi }(2\pi -\theta )$
Now, equivalent or effective resistance between $A$ and $B$
$R_{eq}=\frac{R_1\times R_2}{R_1+R_2}$
$\Rightarrow R_{eq}=\frac{\frac{R}{2\pi }\theta \times \frac{R}{2\pi }(2\pi -\theta )}{\frac{R}{2\pi }[\theta +2\pi -\theta ]}$
$=\left[\frac{\frac{R^2\theta (2\pi -\theta )}{4{\pi }^2}}{\frac{2\pi R}{2\pi }}\right]$
$=\frac{R^2\theta (2\pi -\theta )}{4{\pi }^2}\times \frac{2\pi }{2\pi R}$
$=\frac{R\theta (2\pi -\theta )}{4{\pi }^2}$.