Derive the expression for motional emf induced in a conductor moving in a uniform magnetic field

Suppose, imagine a simple AC generator that has a coil of ‘N’ turns and is rotating in a counter-clockwise direction with the uniform angular velocity of ‘w’ radian/second in a uniform magnetic field or flux of φmaxm weber. The time is counted when the coil lies with the ‘X-axis’. At this position of the coil when the value of ‘θ’= 0°, the maximum value of flux ‘φ’ maxm cutting by the coil.

Now, the magnetic flux linking with the coil will be φ maxm, xcosθ or φ maxm = coswt at any angular position between the coil and the reference line. As per Faraday’s laws of electromagnetic induction, we know that the magnetic of the induced emf is equal to the flux linkage of the coil occurs.