Question

A uniform wire of resistance $1\Omega m^{-1}$ is bent into a circle of radius $1m$. The same wire is connected across a diameter $AB$ as shown in the figure. The effective resistance between $A$ and $B$ is:

• A. $\frac{\pi }{\pi +4}\Omega$
• B. $\frac{2\pi }{\pi +2}\Omega$
• C. $\frac{\pi +4}{2\pi }\Omega$
• D. $\frac{2\pi }{\pi +4}\Omega$

Answer 1

The correct option is D $\frac{2\pi }{\pi +4}\Omega$

Length of each semicircular position of the wire $=\pi r=\pi$ metre $(t$ is $1\Omega$ per metre$)$
$\therefore R_1=R_2=11\Omega$
Length of $AB=2r=2m$.
$R_3=2\Omega$
$R_1,R_2$ and $R_3$ are in parallel
$\frac{1}{R_{AB}}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}=\frac{1}{\pi }+\frac{1}{\pi }+\frac{1}{2}$
$=\frac{2}{\pi }+\frac{1}{2}=\frac{4+\pi }{2\pi }$
or $R_{AB}=\frac{2\pi }{4+\pi }$