A wire of length $5 m$ and radius $1 mm$ has resistance of $1 Ω$ . What length of the same material at the same temperature, and of radius $2 mm$ will also have a resistance of $1 Ω$ ?

1.25 m

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2.5 m

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10 m

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20 m

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Solution

The correct option is D $20 m$
$R = ρ \frac{l}{A}$

$R_{1} = ρ_{1} \frac{l_{1}}{A_{1}} = ρ_{2} \frac{l_{2}}{A_{2}} = R_{2}$
Material is same,
$ρ_{1} = ρ_{2}$
$⟹ \frac{l_{1}}{π r_{1}^{2}} = \frac{l_{2}}{π r_{2}^{2}}$
$⟹ \frac{5}{(1)^{2}} = \frac{l_{2}}{(2)^{2}}$
$⟹ l_{2} = 20 m$

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