A small thin coil with N2 loops, each of area A2, is placed inside a long solenoid, near its center. The solenoid has N1 loops in its length l and has area A1. Determine the mutual inductance as a function of q, the angle between the plane of the small coil and the axis of the solenoid. Express your answer in terms of some or all of the variables A1, A2, N1, N2, l, q, and appropriate constants.

the mutual inductance as a function of q is  

Explanation:

The Magnetic field  that is being produced by the inner coil is mathematically given as

The magnetic flux is mathematically represented as

Where q is given in the question as the angle between the plane of the small coil and the axis of the solenoid

       The mutual inductance is mathematically represented as

Substituting the for in the equation

Where is permeability of free space

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snell's law

refractve index = sin incidence / sin refraction