# Solved Problems-15

Problem-15

A long solenoid with length

*l*and a cross sectional area*A*consists of*N*_{1}turns of wire. An insulated coil of*N*_{2}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.

*M*to the self-inductances

*L*

_{1}and

*L*

_{2}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
where
Similarly, we have the self-inductance for the outer coil given as,
Thus, in terms of
Note: More generally, the mutual inductance is given as,
where

*N*_{1}turns is given by,*φ*_{11}is the magnetic flux through one turn of the solenoid due to the magnetic field produced by*I*_{1}.*L*_{1}and*L*_{2}, the mutual inductance can be expressed as,*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.