Question

If the length of wire of resistance R is changed to n times, its mass remaining constant, then its new resistance will be

• A. $nR$
• B. $n^{2}R$
• C. $\frac{n^2}{R}$
• D. $\frac{R}{n^4}$

Answer 1

The correct option is B

$n^{2}R$

Step 1: Given data and assumptions

Resistance of the wire is $R$

Assigning the variables for the given resistor

Let $L$be length

$r$ be the radius of the resistor

$\rho$be the resistivity

$\sigma$ be the density of the material.

Step 2: Assigning variables for a new resistor

Let $L\text{'}$be the length, $L\text{'}=2L$

$r\text{'}$ be the radius of the resistor

$\rho$be the resistivity (No change )

$\sigma$ be the density of the material.

Step 3: Finding new resistance

The mass of the two resistors remains the same. So,

$\begin{array}{c}m=m\text{'}\\ \sigma AL=\sigma A\text{'}L\text{'}\\ \pi r^{2}L=\pi {(r\text{'})}^2\left(nL\right)\\ \Rightarrow {(r\text{'})}^2=\frac{r^2}{n}\end{array}$

We know that the formula for the resistance is $R=\frac{\rho L}{A}=\frac{\rho L}{\pi r^2}$. So,

$\begin{array}{c}R\text{'}=\frac{\rho L\text{'}}{\pi {(r\text{'})}^2}=\frac{\rho \left(nL\right)}{\pi \left(\frac{r^2}{n}\right)}\\ =\frac{\rho L}{\pi r^2}\left(n^2\right)\\ \boxed{R\text{'}=n^{2}R}\end{array}$

Hence, the correct option is B.