Given a wire of length $l$ and resistance, $R$ is stretched. Now the length is doubled, and the area of the cross-section is halved. The new resistance and the resistivity will be

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quadrupled and same, respectively

quadrupled and increases, respectively

same and quadrupled, respectively

None of the above

answer is A.

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Detailed Solution

Given a wire of length

l

and resistance,

R

is stretched. Now the length is doubled, and the area of the cross-section is halved. The new resistance and the resistivity will be quadrupled and the same, respectively.
Given the length of the wire is doubled, that is

l^{'} = 2 l

, and the area of the cross-section is halved, that is

A^{'} = \frac{A}{2}

, then the resistance now will be

R' = \frac{ρ l^{'}}{A^{'}}

\Rightarrow R' = \frac{ρ (2 l)}{\frac{A}{2}}

\Rightarrow R' = \frac{4 ρl}{A}

\Rightarrow R' = 4 R

We know that resistivity depends on the material's temperature and nature. Hence when the length of the wire is doubled, there will be no change in the resistivity.
Resistivity does not depend on the shape or size and is independent of the dimensions of the material. (Using the relation

R = \frac{ρl}{A}

)

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