Question

In the figure, find the resistance between points $A$ and B. Both the circle and the diameter are made of uniform wire of resistance $1 \times 10^{-4}$ ohm per metre. The length $A B$ is 2 metre.

• A. $(2 / 3) \times 10^{-4}\, \Omega$
• B. $2 \pi \times 10^{-4}\, \Omega$
• C. $14.56 \times 10^{-4}\, \Omega$
• D. $0.88 \times 10^{-4}\, \Omega$

Answer 1

Answer: (D)

If $r$ is the radius of circle, then $2 r=2 m$ or $r=1 m$
Arc $A B=\pi r=\pi \times 1=\pi m$.
Here, three resistances of values $\pi \times 10^{-4} \Omega, 2 \times 10^{-4} \Omega$ and $\pi \times 10^{-4} \Omega$ are connected in parallel between points $A$ and $B$.
The resistance between points $A$ and $B$ is
$\frac{1}{R}=\frac{1}{\pi \times 10^{-4}}+\frac{1}{2 \times 10^{-4}}+\frac{1}{\pi \times 10^{-4}}$
$=\frac{2}{\pi \times 10^{-4}}+\frac{1}{2 \times 10^{-4}}$
$\frac{1}{R}=\frac{1}{10^{-4}}\left[\frac{4+\pi}{2 \pi}\right]$
or $R=\frac{2 \pi \times 10^{-4}}{4+\pi}=0.88 \times 10^{-4}\, \Omega$