Question

An a.c. generator consists of a coil of 10,000 turns and of area 100 $cm^2$. The coil rotates at an angular speed of 140 rpm in a uniform magnetic field of \[3.6 \times {10^{ - 2}}\;T\]. Find the maximum value of the emf induced.

Answer 1

Hint: We can apply the concept of Faraday’s law which basically summarizes the methods a voltage or emf can be produced via a changing magnetic environment. The induced voltage or emf in a coil is mainly equal to the negative of the rate of changing magnetic flux times the number of turns in the coil. It majorly involves the interaction of charge with the magnetic field.

Complete Step by Step Answer :

From Faraday’s law, the expression of induced emf in the rotating coil is:

$\varepsilon  =  - \dfrac{{d{\phi _m}}}{{dt}}$

And magnetic flux through the loop can be expressed as ${\phi _m} = NBA\cos \omega t$

As a result, the induced emf in the rotating coil is: 

$\varepsilon  = NBA\omega (\sin \omega t)$

The maximum emf is induced when $\left| {\sin \omega t} \right| = 1$

Thus, the induced emf can be expressed as:

${\varepsilon _{\max }} = NBA\omega  \\$

$\text{where}{\text{ }}\omega  = 2\pi f $

$N$ = number of turns, $B$ = magnetic field, $A$ = area ($m^2$), $f$ = frequency of rotation, $\omega$ = angular speed

In the question, we are provided with the following information:

$N = 10,000$

$A = 100 \,cm^2 = 0.01\,m^2$

$f = 140\,rpm = 140/60\,rps$

\[B = 3.6 \times {10^{ - 2}}\;T\]

First of all, we will calculate the value of angular speed:

$\omega  = 2 \times \pi  \times \dfrac{{140}}{{60}} = 14.7\,rad/s$

Now we will calculate the maximum value of induced emf by substituting all the values in emf formula:

${\varepsilon _{\max }} = 10000 \times 3.6 \times {10^{ - 2}} \times 0.01 \times 14.7 = 52.9\,volts$

Therefore, the maximum value of the emf induced is 52.9 volts.

Note: It should be noted that the action of moving a loop or coil of wire through a magnetic field actually induces a voltage in coil having the magnitude of the induced voltage being related to or proportional to the velocity or speed of the movement.