Question

The resistance if a wire is 'R' ohm. If it is melted and stretched to 'n' times its original length, its new resistance will be:
(A) $\dfrac{R}{{{n^2}}}$
(B) $nR$
(C) $\dfrac{R}{n}$
(D) ${n^2}R$

Answer 1

Hint Given that the resistance of wire R has an initial length ${l_1}$ and is stretched n times to new length ${l_2}$. It is observed that there is no change in radius. This means, assume volume is constant. Find a relation between areas of the wire before and after extension and substitute it in the resistance formula $R = \dfrac{{\rho l}}{A}$.

Complete Step By Step Solution
It is given that there is a wire of radius r, length ${l_1}$ having a resistance R. Now this wire is said to be extended to a new length ${l_2}$, which is n times ${l_1}$. Since , there is no change in radius of the wire, let us assume that the volume before and after stretching are constant.
${V_1} = {V_2}$, where ${V_1}$ is volume before stretching and ${V_2}$is volume after stretching.
Now we know that volume is a product of area and height. In our case, height is the length of the wire
${A_1} \times {l_1} = {A_2} \times {l_2}$
We know that , ${l_2} = n \times {l_1}$. Substituting this , we get
$\Rightarrow {A_1} \times {l_1} = {A_2} \times n \times {l_1}$
$\Rightarrow \dfrac{{{A_1}}}{n} = {A_2}$
We know that resistance R of a material is calculated by using the formula$R = \dfrac{{\rho l}}{A}$, where $\rho$is represented as resistivity of the material , $A$ is area and $l$is length of the wire.
Now , before extension , resistivity is given as , ${R_1} = \dfrac{{\rho {l_1}}}{{{A_1}}}$
After extension, resistivity is given as, ${R_2} = \dfrac{{\rho {l_2}}}{{{A_2}}}$
Substituting for ${A_2}$in the equation for${R_2}$, we get
$\Rightarrow {R_2} = \dfrac{{\rho {l_2} \times n}}{{{A_1}}}$
Now substituting${l_2} = n \times {l_1}$, we get
$\Rightarrow {R_2} = \dfrac{{\rho {l_1} \times {n^2}}}{{{A_1}}}$
(The term $\dfrac{{\rho {l_1}}}{{{A_1}}}$is equal to${R_1}$ and ${R_1}$ value given in the question is $R$)
$\Rightarrow {R_2} = R \times {n^2}$

Thus, Option (d) is the right answer for the given question.

Note Electrical resistivity is defined as the electrical property of a material which defines its strength to oppose electric current.