Consider a rectangular coil having N turns and rotating in a uniform magnetic field with an angular velocity of w radian/second. Maximum flux Øm is linked with the coil when its plane coincides with the X-axis. In time t seconds, this coil rotates through an angle q = wt. In this deflected position , the component of the flux which is perpendicular to the plane of the coil is Ø= Øm cos wt. Hence flux linkage at any time are NØ=NØm cos wt.
According to Faraday's Laws of Electromagnetic Induction, the e.m.f. induced in the coil is given by the rate of change of flux linkage of the coil. Hence the value of the induced e.m.f. is
e = - d(NØ)/dt volt
= - N d(Øm cos wt) / dt volt
= - NØm w(-sin wt) volt
= wNØm sin wt volt
= w NØm sin q volt ----------------------- (i)
When the coil turned through 90º i.e. when q = 90º, then sin q = 1, hence e has maximum value, say Em. Therefore from Eq(i) we get
Em = wNØm
= w NBmA = 2pfNBmA volt
where Bm = maximum flux density in Wb/m2.
A = Area of the coil in m2.
f = frequency of rotation of the coil in rev/second.
Substituting this value of Em in Eq(i), we get
e = Em sin q = Em sin wt
Similarly, the equation of the induced alternating current is
i = Im sin wt