Question

A uniform wire of resistance 4 ohm is bent into the form of a circle of radius r. A specimen of the same wire is connected along the diameter of the circle. What is the equivalent resistance across the ends of this wire?

• A. $\frac{4}{4+\pi }$ ohm
• B. $\frac{3}{3+\pi }$ ohm
• C. $\frac{2}{2+\pi }$ ohm
• D. $\frac{1}{1+\pi }$ ohm

Answer 1

The correct option is A $\frac{4}{4+\pi }$ ohm

Circumference of the circle = $2\pi r$
Therefore the resistance per unit length of the wire = $\frac{R}{2\pi r}$
where r= 4 ohm is the resistance of wire
Now the length of the specimen connected along the diameter = 2r
Therefore the resistance of this specimen is
$R_1=\frac{R}{2\pi r}\times 2r=\frac{R}{\pi }$
also the resistance of each semicircle is
$R_2=\frac{R}{2}$
equivalent resistance R' across the specimen is given by
$\frac{1}{R^{\prime }}=\frac{2}{R}+\frac{2}{R}+\frac{\pi }{R}=\frac{4+\pi }{R}$
$R^{\prime }=\frac{R}{4+\pi }=\frac{4}{4+\pi }$ ohm