Q: Masses of the three wires of same material are in the ratio of 1 : 2 : 3 and their lengths in the ratio of 3: 2: 1, Electrical resistance of these wires will be in the ratio of:

(a) 1 : 1: 1

(b) 1: 2: 3

(c) 9: 4: 1

(d) 27: 6: 1

Ans: (d)

Sol : Mass, M = volume × density = A l × d

A = M/ld

Resistance $ \displaystyle R = \rho \frac{l}{A} = \frac{\rho l}{M/ld} $

$ \displaystyle R = \rho \frac{l^2 d}{M} $

$ \displaystyle R = \propto \frac{l^2 }{M} $

$ \displaystyle R_1 : R_2 : R_3 = \frac{l_1^2 }{M_1} : \frac{l_2^2 }{M_2} : \frac{l_3^2 }{M_3} $

$ \displaystyle = \frac{3^2 }{1} : \frac{2^2 }{2} : \frac{l^2 }{3} $

= 27 : 6 : 1