Question

A uniform wire of resistance R is uniformly compressed along its length, until its radius becomes n times the original radius. Now what is the resistance of the wire?​

Answer 1

The resistance is expressed as,

$\sf{\longrightarrow R=\dfrac{\rho L}{A}}$

where $\rho$ is resistivity, $\sf{L}$ is length of conductor and $\sf{A}$ is cross sectional area of conductor.

In case of a uniform wire $\rho$ should be constant. So,

$\sf{\longrightarrow R\propto\dfrac{L}{A}\quad\quad\dots(1)}$

The volume of the conductor remains constant even if it gets extended or compressed.

$\sf{\longrightarrow V=AL}$

$\sf{\Longrightarrow L\propto\dfrac{1}{A}}$

Then (1) becomes,

$\sf{\longrightarrow R\propto\dfrac{1}{A^2}\quad\quad\dots(2)}$

Relation between area and radius of conductor is,

$\sf{\longrightarrow A\propto r^2}$

So (2) becomes,

$\sf{\longrightarrow R\propto\dfrac{1}{r^4}}$

In the question the radius becomes n times the original value after getting compressed.

$\sf{\longrightarrow r\to nr}$

Hence the resistance becomes $\sf{\dfrac{1}{n^4}}$ times the original value.

$\sf{\longrightarrow\underline{\underline{R\to\dfrac{R}{n^4}}}}$