Question

We have two wires $A$ and $B$ of the same mass and the same material. The diameter of the wire $A$ is half of that $B$. If the resistance of wire $A$ is $24\text{ohm}$ then the resistance of wire $B$ will be

• A. $1.5\text{ohm}$
• B. None of the above
• C. $12\text{ohm}$
• D. $3.0\text{ohm}$

Answer 1

The correct option is A $1.5\text{ohm}$

Given :
Same mass, same material, i.e., volume is the same $Al=$ constant
$R_1=24\text{ohm}$
Let diameter of $B$ is $d$ and diameter of $A$ is $\frac{d}{2}$.
Resistance is given by
$R=\rho \frac{l}{A}$
$R=\rho \frac{l}{A}\times \frac{A}{A}=\frac{\rho V}{A^2}$
$R\propto \frac{1}{A^2}\propto \frac{1}{d^4}$
$\Rightarrow \frac{R_1}{R_2}={\left(\frac{d_2}{d_1}\right)}^4$
$\Rightarrow \frac{24}{R_2}={\left(\frac{d}{\frac{d}{2}}\right)}^4=16$
$R_2=1.5\text{ohm}$
Hence, option $\left(C\right)$ is correct.