Question

The masses of three wires of copper are in the ratio $1 : 2 : 3$ and their lengths are in the ratio $3 : 2 : 1.$ The ratio of their resistances is:

• A. 3 : 2 : 1
• B. 1 : 2 : 3
• C. 27 : 6 : 1
• D. 1 : 6 : 27

Answer 1

Answer: (C)

Resistance of wire
$R=\frac{\rho l}{A}$
or $R \propto \frac{l^{2}}{A l}$
or $R \propto \frac{l^{2}}{V}$
or $R \propto \frac{l^{2} d}{m}$
(where $d$ is density and $m$ is mass)
or $R \propto \frac{l^{2}}{m}$
Here, $l_{1}: l_{2}: l_{3}=3: 2: 1$
$m_{1}: m_{2}: m_{3}=1: 2: 3$
$\therefore R_{1}: R_{2}: R_{3}:: \frac{l_{1}{ }^{2}}{m_{1}}: \frac{l_{2}^{2}}{m_{2}}: \frac{l_{3}{ }^{2}}{m_{3}}$
or $R_{1}: R_{2}: R_{3}:: \frac{9}{1}: \frac{4}{2}: \frac{1}{3}$
or $R_{1}: R_{2}: R_{3}:: 27: 6: 1$