A wire of resistance R 1 R 1 is drawn out so that its length is increased by twice of its original length. The ratio of new resistance to original resistance is:

Question

A wire of resistance R 1 ​ R 1 ​ is drawn out so that its length is increased by twice of its original length. The ratio of new resistance to original resistance is:

TestHub · Accepted Answer

As per the given question, the change in the length is $Δ l = 2 l$ .
Resistance of a wire is given by, $R = \frac{ρl}{A}$
Therefore, $R_{1} = ρ \frac{l _{1}}{A _{1}}$
When the length is starched by $2 l_{1}$ , final length will be $l_{2} = l_{1} + 2 l_{1} = 3 l_{1}$ . As the volume remains constant $l_{2} A_{2} = l_{1} A_{1} \Rightarrow A_{2} = \frac{A _{1}}{3}$ . Therefore,
$R_{2} = ρ (\frac{3 l _{1}}{\frac{A _{1}}{3}}) = 9 ρ \frac{l _{1}}{A _{1}}$
$∴ \frac{R _{2}}{R _{1}} = 9$

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