**Working
:**

Consider a rectangular coil having N turns and rotating in a uniform magnetic
field with an angular velocity of w
radian/second. Maximum flux Ø_{m}
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=
Ø_{m
}cos wt.
Hence flux linkage at any time are NØ=NØ_{m
}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(Ø _{m
}cos wt)
/ dt volt**

**
= - NØ _{m
}w(-sin
wt)
volt**

**
= wNØ _{m
}sin
wt
volt**

**
= w
NØ _{m
}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 E_{m}.
Therefore from Eq(i) we get

**E _{m}
=_{ }w**

** _{
}= _{
}w NB_{m}A
= 2pfNB_{m}A
volt**

**
where B _{m}
= maximum flux density in Wb/m^{2}.**

**
A = Area of the coil in m ^{2}.**

**
f = frequency of rotation of the coil in rev/second. **

**
Substituting
this value of E _{m}
in Eq(i), we get**

**
e = E _{m
} sin q
= E_{m
}sin
wt**

**
Similarly,
the equation of the induced alternating current is**

**
i = I _{m
}sin
wt
**