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 $R\propto \frac{l^2}{Al}$
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 $R_1:R_2:R_3::27:6:1$