# Solved Problems-15

Problem-15
A long solenoid with length l and a cross sectional area A consists of N1 turns of wire. An insulated coil of N2 turns is wrapped around it, as shown in Fig.
A coil wrapped

(a) Calculate the mutual inductance M, assuming that all the flux from the solenoid passes through the outer coil.

(b) Relate the mutual inductance M to the self-inductances L1 and L2 of the solenoid and the coil.

Solution
(a) The magnetic flux through each turn of the outer coil due to the solenoid is,
where  is the uniform magnetic field inside the solenoid.

Hence, the mutual inductance between the solenoid and the coil is,

(b) We see that the self-inductance of the solenoid with N1 turns is given by,

where φ11 is the magnetic flux through one turn of the solenoid due to the magnetic field produced by I1.

Similarly, we have the self-inductance for the outer coil given as,

Thus, in terms of L1 and L2, the mutual inductance can be expressed as,

Note: More generally, the mutual inductance is given as,

where k is the coupling coefficient. In this example, we have k = 1 which means that all of the magnetic flux produced by the solenoid passes through the outer coil, and vice versa, in this idealisation.