You are given a wire of length 100 cm and linear resistance of 1 ohm/cm. If it is cut into two parts, so that when they are in parallel, the effective resistance is 24 ohm. The lengths of the two parts are

30cm & 70cm

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60cm & 40cm

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70cm & 30cm

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Solution

The correct option is A 60cm & 40cm
The resistance of a wire of length $l$ is $R = ρ (\frac{l}{A})$ .
Suppose the two piece of wire has lengths and resistances $l_{1}, l_{2}$ and $R_{1}, R_{2}$ respectively.
So according to the question:
$l_{1} + l_{2} = 100$ ...................................................................(1)

and $\frac{R_{1} R_{2}}{R_{1} + R_{2}} = 24$

Substituting $R = \frac{ρ}{l} l$ , we get,

$\frac{(\frac{ρ}{A}) l_{1} (\frac{ρ}{A}) l_{2}}{(\frac{ρ}{A}) l_{1} + (\frac{ρ}{A}) l_{2}} = 24$
But $\frac{ρ}{A} = 1$
$\frac{l_{1} l_{2}}{l_{1} + l_{2}} = 24$ ...........................(2)

Solving equation (1) and (2), we will get $l_{1} = 60$ cm $, l_{2} = 40$ cm

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