Consider a rectangular coil m-n-o-p rotating in a uniform magnetic field B generated between the two poles of a magnet, with the axis of rotation lying in the plane of the coil but perpendicular to the field B. In the initial position, let it be assumed that the plane of the coil is perpendicular to the magnetic field as shown in the following figure. In this position, the angle between the normal to the plane of the coil and the magnetic field is zero.
At any other instant let the normal to the plane of the coil make an angle q with field B. lux passing through coil is given by,
f= B.A. cosq
Where A is the area of the coil. If the angular velocity of the coil is w and the time 't' is measured from the position of the coil when it is perpendicular to the magnetic field, i.e., when t = 0, q= 0 and at any instant of time 't',
q=w . t
The equation for the flux through the coil becomes,
f= B.A. cos w .t
If the coil contains 'N' turns, the flux passing through the coil will be
N.f .
N.f = B.A.N.cos w .t ----------- ( i)
The e.m.f induced in the coil is given by,
e= - d(N.f ) / dt
= d (B.A.N.cos w .t) / dt
= B.A.N. w . sin w . t ----------- (ii)
From the above equation, we can observe that when q= 0, initially, the e.m.f induced in the coil is also zero, because the instantaneous velocity of the coil is parallel to B. As q increases, the e.m.f also increases and reaches a maximum value for = 900. The maximum induced e.m.f is given by,
e(max) = B.A.N.w . ----------- (iii)
When q= 1800, the instantaneous velocity will be once again parallel to the field and the e.m.f will be zero. The e.m.f will be maximum again whenq = 2700 and becomes zero at q= 3600
Thus, the voltage or e.m.f generated will be alternating which can be expressed as,
e= em. sin w .t --------- (iv)
The variation of e.m.f with q is shown in the following graph: