derive the formula for mutual inductance in terms of coefficient of coupling and self-inductance

The fraction of magnetic flux produced by the current in one coil that links with the other coil is called the coefficient of coupling between the two coils. It is denoted by (k).
Two coils are taken coil A and coil B, when current flows through one coil it produces flux; the whole flux may not link with the other coil coupled, and this is because of leakage flux by a fraction (k) known as Coefficient Of Coupling.
k=1, when the flux produced by one coil, completely links with the other coil and is called magnetically tightly coupled.
k=0, when the flux produced by one coil, does not link at all with the other coil and thus the coils are said to be magnetically isolated.
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