Question

- State the principle of an ac generator and explain its working with the help of a labelled diagram. Obtain the expression of the emf induced in a coil having N turn each of cross-sectional area A, rotating with the constant angular speed in a magnetic field \[\omega \], directed perpendicular to the axis of rotation.

Answer 1

Hint: In this question we have been asked to state the general principle of ac generator and also explain its working. Therefore, we shall discuss about the ac generators and with the help of a labelled diagram explain its working. Ac generator is a machine that converts mechanical energy into electrical energy.
Formula used:
 \[E=-\dfrac{d\phi }{dt}\]

Complete answer:
AC generators work on the principle of Faraday’s law of electromagnetic induction. This law states that electro motive force is generated in a current carrying loop that is placed in a uniform magnetic field. Whenever a coil is rotated about its axis perpendicular to the uniform magnetic field, the magnetic flux in the coil changes and an induced emf is set up across its ends.

                

When the coil in the generator starts moving, the arm AB starts moving upwards and the arm CD moves downwards. These arms start cutting the magnetic lines of force therefore, by Fleming’s right hand rule, the induced current set up in these arms is along the direction of AB and CD.
Let us consider a coil of N turns and area A that is rotated at constant angular velocity in a magnetic field of B as shown in the figure. When the normal to the coil is at an angle \[\theta \], magnetic flux is generated in the coil. This flux is given by,
\[\phi =BAN\cos \theta \]
We know that,
\[E=-\dfrac{d\phi }{dt}\]
Substituting the value of \[\phi \]in above equation
We get,
\[\Rightarrow E=-\dfrac{dBAN\cos \theta }{dt}\]
Now, we know that
\[\theta =\omega t\]
Therefore,
\[\Rightarrow E=-\dfrac{dBAN\cos \omega t}{dt}\]
On solving,
\[\Rightarrow E=BAN\omega \sin \omega t\]
We know that, the maximum value for induced emf will be obtained when
\[\Rightarrow \theta =\omega t=90{}^\circ \]

Therefore, the maximum value of induced emf will be \[E=BAN\omega \].

Note:
The machine that generates electrical energy by converting the mechanical energy is known as an AC generator. The electrical energy that is generated is in the form of an alternating current as a sinusoidal waveform. The mechanical energy that is to be converted is usually supplied through combustion engines, turbines,etc.