Question

A cylindrical metal rod has a resistance . If both its length and its diameter are quadrupled, its new resistance will be:________.
A. 16R
B. R/4
C. R
D. 4R

Answer 1

Answer:

R' = R/4

Explanation:

The resistance of a metal rod is R. It is given by the relation as follows :

$R=\rho\dfrac{l}{A}$

Where

l is the length and A is the area of cross-section

$A=\pi r^2=\pi (\dfrac{d}{2})^2$

If both its length and its diameter are quadrupled, it means,

l' = 4l

and d'= 4d

It means,

$A'=\pi (\dfrac{4d}{2})^2$

Let new resistance be R'. So,

$R'=\rho\dfrac{l'}{A'}\\\\R'=\rho\dfrac{4l}{\pi (\dfrac{4d}{2})^2}\\\\=\rho \dfrac{4l}{\pi \dfrac{16d^2}{2}}\\\\=\dfrac{4}{16}\times \dfrac{\rho l}{\pi \dfrac{d^2}{2}}\\\\=\dfrac{1}{4}\times \dfrac{\rho l}{\pi \dfrac{d^2}{2}}\\\\R'=\dfrac{R}{4}$

So, the correct option is (B) "R/4".