Two air-cored solenoids, P and Q , are constructed using insulated copper wire. Solenoid P has a total of N turns, a radius of 2r , and a length L . Solenoid Q has a total of 3N turns, a radius of r , and a length 3L . The ratio of the self-inductance of solenoid P to that of solenoid Q ( L_P : L_Q ) is:

Step-by-step solution

The self-inductance of a solenoid is given by L_ self = ₀ N^2 A l , where N is the total number of turns, A is the cross-sectional area, and l is the length. For solenoid P : N_P = N A_P = (2r)^2 = 4 r^2 l_P = L L_P = ₀ N^2 (4 r^2) L = 4 ₀ N^2 r^2 L For solenoid Q : N_Q = 3N A_Q = r^2 l_Q = 3L L_Q = ₀ (3N)^2 ( r^2) 3L = 9 ₀ N^2 r^2 3L = 3 ₀ N^2 r^2 L The ratio of their self-inductances is: L_P L_Q = 4 3 Answer: 4 : 3