Question

Resistance of a wire of uniform diameter and length $l$ is $R.$ The resistance of another wire of the same material but diameter $2d$ and length $4l$ will be

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

Answer 1

Answer: (B)

$R$

Given:

Resistance of a wire $=R$

Length of wire $=l$

Diameter $=d$

Radius $r=\frac{d}{2}$

Area $A=\pi r^2=\pi {\left(\frac{d}{2}\right)}^2=\pi \frac{d^2}{4}$

Let resistance of another wire $=R^{\prime }$

Length of wire $=4l$

Diameter$=2d$

Radius $r^{\prime }=d$

Area $A^{\prime }=\pi d^2$

Let the resistivity be $\rho$. It will be same for $R$ as well as for $R^{\prime }$ because the material is same.

We know $R=\rho \frac{l}{A}$

For $1^{st}$ wire, $R=\rho \frac{l}{\left(\pi \frac{d^2}{4}\right)}=\rho \frac{4l}{\pi d^2}$ . . . . $\left(i\right)$

For $2^{nd}$ wire, $R^{\prime }=\rho \frac{4l}{A^{\prime }}=\rho \frac{4l}{\pi d^2}$ . . . . . . . . $\left(ii\right)$

From $\left(i\right)$ and $\left(ii\right)$, we have $R=R^{\prime }.$ So option B is correct.