We know that an alternating e.m.f. can be generated either by rotating a coil within stationary magnetic field or by rotating a magnetic field within a stationary coil. The magnitude of e.m.f. generated in the coil depends upon the number of turns on coil, strength of magnetic field and the speed at which the coil or magnetic field rotates.

Consider a coil of N turns rotating with angular velocity ω radians per second in a uniform magnetic field, as shown it.

Let the time be measured from the instant of coincidence of the plane of the instant of coincidence of the plane of the coil with the X-axis. At this instant maximum flux, φ_max  links with the coil. Let the coil assume the position as shown in after moving in counterclockwise direction for t seconds.


The angle θ through which the coil has rotated in t seconds = ωt

In this position, the component of flux along perpendicular to the plane of coil = φ_max cos ωt

Hence flux linkage of the coil at this instant = Number of turns on coil x linking flux = φ_max cos ωt

i.e. instantaneous flux linkage = N φ_{max cos} ωt

Since e.m.f. induced in a coil equal to the rate of change of flux linkage with minus sing.

so e.m.f. induced at any instant,  e = - d  [ N φ_max cos ωt ]
                                                       dt
                                                   =  φ_max N d   ( - cos ωt ) =  φ_max  Nω sin ωt
                                                                 dt

when ωt = 0, sin ωt = 0 therefore, induced e.m.f. is zero, when ωt = π , sin π  = 1,
                                                                                                  2        2

therefore induced e.m.f is maximum, which s denoted by E_max and is equal to Nω.

Substituting φmax Nω = E_max
Instantaneous e.m.f. e = Emax sin ωt

So the e.m.f. induced varies as the since function f the time angle ωt and if e.m.f. induced is plotted against time, a curve of sine wave shape is obtained, as shown in . Such an e.m.f. is called the sinusoidal e.m.f. The since curve is completed when the coil moves through an angel of 2π radians.



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