A uniform wire of resistance 4 ohm is bent into the form of a circle of radius r

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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?

Options:

A . 44+π ohm

B . 33+π ohm

C . 22+π ohm

D . 11+π ohm

Answer: Option A
:
A
Circumference of the circle =

2 π r

Therefore the resistance per unit length of the wire =

\frac{R}{2 π 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 π r} \times 2 r = \frac{R}{π}

also the resistance of each semicircle is

R_{2} = \frac{R}{2}

equivalent resistance R' across the specimen is given by

\frac{1}{R^{'}} = \frac{2}{R} + \frac{2}{R} + \frac{π}{R} = \frac{4 + π}{R}

R^{'} = \frac{R}{4 + π} = \frac{4}{4 + π}

ohm

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