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Solved Problems-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.
(a) The magnetic flux through each turn of the outer coil due to the solenoid is,
Description: Description: 104434.png
where Description: Description: 104427.png is the uniform magnetic field inside the solenoid.
Hence, the mutual inductance between the solenoid and the coil is,
Description: Description: 104418.png
Description: Description: 104411.png
(b) We see that the self-inductance of the solenoid with N1 turns is given by,
Description: Description: 104402.png
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,
Description: Description: 108928.png
Thus, in terms of L1 and L2, the mutual inductance can be expressed as,
Description: Description: 108920.png
Description: Description: 108912.png
Note: More generally, the mutual inductance is given as,
Description: Description: 108903.png
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.

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