Question

The length and radius of a wire are $l$ and $r$ respectively, and its resistance is $R$. If the length and radius are doubled, then its new resistance _____

• A. remains same
• B. is doubled
• C. is $\frac{R}{2}$
• D. is $\frac{R}{4}$

Answer 1

The correct option is C

is $\frac{R}{2}$

Step 1: Recall the formula of resistivity-

Resistivity $\rho =\frac{RA}{l}=\frac{R\pi r^2}{l}$ $\because A={\mathrm{\pi r}}^2$ is the area of cross-section of the wire

where $r$ is the radius of the cross-section of the wire and $l$ is its length

Therefore, resistance $R=\frac{\rho l}{A}=\frac{\rho l}{\pi r^2}$

Step 2: Calculate new resistance-

Given the length and radius are doubled.

then new resistance $R\text{'}=\frac{\rho \left(2l\right)}{\pi {\left(2r\right)}^2}=\frac{2\rho l}{4\pi r^2}=\frac{\rho l}{2\pi r^2}=\frac{R}{2}$

So, the correct option is (C)$\frac{R}{2}$.