Generation of Alternating emf

Consider a rectangular coil of N turns rotating in a uniform magnetic field with an angular velocity ω radian/second about X-axis in its own plain as shown in fig.3.6. Let ф_m be the maximum flux perpendicular to the axis of rotation, when the plain of the coil coincides with the X-axis.

Let in t seconds, the coil rotates through an angle θ = ω t, and in this deflected position, the flux component perpendicular to the plane of the coil,

                                                            

Consequently, the flux linkage of the coil in this deflected position is given by:

                                                         

                                                                       

Now, according to faraday’s law of electromagnetic induction, the induced emf in the coil is equal to the rate of change of flux linkage of the coil. Consequently, the instantaneous induced emf ( e )at time, when θ =ω t, is given by:

                                            

                               

                                                                                                 

Now, the induced emf e has a maximum value of E_m, when sin θ = sin 90^o = 1, i.e.

                                                                                     

                                                           

Where f is the frequency of the rotation of the coil

 Therefore from equations above, we get:

                                                                                          

Now the current I at any time t in the coil is proportional to the induced emf e in the coil. Hence, the induced alternating current is given by:

                                                                                                                    

            Where I_m is the maximum value of current