Question

If $\phi = 0.02\cos 100\pi t\;{\rm{weber}}$ and number of turns in 50 in the coil, the maximum induced emf is
(A) $314\;{\rm{V}}{\rm{olt}}$
(B) $100\;{\rm{Volt}}$
(C) $31.4\;{\rm{Volt}}$
(D) $6.28\;{\rm{Volt}}$

Answer 1

Hint: Here, we will use the formula for Faraday's law of induction. According to this law if we change the magnetic flux through a coil of number of turns, on induced emf appearing in every turn then the total induced emf in the coil is the sum of these individual induced emfs.

Complete step by step answer:
Given: The magnetic flux in the units of Weber is $\phi = 0.02\cos 100\pi t\;{\rm{weber}}$ and the number of turns in the coil is $N = 50$.

We write the relation between the magnetic flux, number of turns in the coil and induced emf.
\[\varepsilon = - N\dfrac{{d\phi }}{{dt}}\]
Here, $\varepsilon $ is the induced emf in the coil.
We differentiate the equation of magnetic flux with respect to t.
$
\dfrac{{d\phi }}{{dt}} = 0.02 \times \left( { - sin100\pi t} \right) \times 100\pi \\
\implies \dfrac{{d\phi }}{{dt}} = 2\pi \left( { - sin100\pi t} \right)
$
Now, we substitute the values in the Faraday's relation.
\[\begin{array}{l}
\varepsilon = - 50 \times 2\pi \left( { - \sin 100\pi t} \right)\\
\varepsilon = - 100\pi \left( { - \sin 100\pi t} \right)
\end{array}\]

Now, for the maximum value of induced emf, the value of sine function must be 1, as a result
$
\varepsilon = 100\pi \\
\implies \varepsilon = 100 \times 3.14\\
\implies \varepsilon = 314\;{\rm{Volt}}
$
The maximum induced emf in the coil is $314\;{\rm{Volt}}$.
Therefore, the maximum induced emf in the coil is $314\;{\rm{Volt}}$

So, the correct answer is “Option A”.

Note:
We know that the Faraday’s law of electromagnetic induction tells us that the emf induced in an electric circuit is equal to the time rate change of the magnetic flux through the electric circuit. As we know, the time rate change of the magnetic flux through the circuit is given by negative sign. The meaning of the negative sign is that the current always flows in such a direction that it is always against the change.