Q. A uniform wire is resistance 4Ω 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) $\displaystyle \frac{4}{(4 + \pi)} $

(b) $ \displaystyle \frac{3}{(3 + \pi)} $

(c) $ \displaystyle \frac{2}{(2 + \pi)} $

(d) $\displaystyle \frac{1}{(1 + \pi)} $

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Ans: (a)

Sol: Resistance per unit length = 4/2πr

Resistance of diameter is

Resistance of diameter is

$\displaystyle = \frac{4}{2\pi r}\times 2r = \frac{4}{\pi}$

Now, Resistance of half circle = 2 ohm

Hence , circuit has three resistances.

The resistances 2 ohm , 2 ohm & 4/π are in parallel .

$ \displaystyle \frac{1}{R}= \frac{1}{2}+\frac{1}{2} + \frac{\pi}{4}$