Two square loops, P and Q , carry currents in the ratio 3 : 2 . If the side lengths of the loops are in the ratio 1 : 3 , what is the ratio of the magnetic dipole moment of loop P to that of loop Q ?

Step-by-step solution

The magnetic dipole moment of a current-carrying loop is given by M = I A , where I is the current and A is the area of the loop. For a square loop of side length a , the area is A = a^2 . The ratio of the magnetic dipole moments of loops P and Q is M_P M_Q = I_P A_P I_Q A_Q = ( I_P I_Q ) ( a_P a_Q )^2 Given I_P I_Q = 3 2 and a_P a_Q = 1 3 , substituting these values gives M_P M_Q = ( 3 2 ) ( 1 3 )^2 = 3 2 1 9 = 1 6 The ratio of the magnetic dipole moment of loop P to that of loop Q is 1 : 6 . Answer: 1 : 6