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If $\varphi = 0.02\cos 100\pi t$ $weber$ and the number of turns is $50$ in the coil. The maximum emf induced is –
A. $314$volt
B. $100$volt
C. $31.4$volt
D. $6.28$volt

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Last updated date: 13th Jun 2024
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Answer
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Hint:The total number of magnetic lines of force passing normally through an area placed in a magnetic field is equal to the magnetic flux linked with that area. The process by which an emf is induced in a circuit by the virtue of changing the magnetic field around it is known as electromagnetic induction.

Complete step by step answer:
Magnetic Field$\left( B \right)$: A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to the magnetic field. This magnetic field can be seen as imaginary lines known as the magnetic field lines.

Magnetic Flux $\left( \varphi \right)$: The total number of magnetic lines of force passing normally through an area$\left( A \right)$ placed in a magnetic field$\left( B \right)$ is equal to the magnetic flux linked with that area. That is,
$\varphi = \oint {\overrightarrow B .d\overrightarrow A } $
The SI unit of magnetic flux is weber$\left( {Wb} \right)$.

Faraday’s Laws of Electromagnetic Induction:
First Law: Whenever the number of magnetic lines of force (magnetic flux) passing through a circuit changes, an emf called induced emf is produced in the circuit. The induced emf persists as long as there is change of flux.
Second Law: The induced emf $\left( \varepsilon \right)$ is given by the rate of change of magnetic flux linked with the circuit. That is,
$\varepsilon = - \dfrac{{d\varphi }}{{dt}}$
For $N$ turns, $\varepsilon = - \dfrac{{Nd\varphi }}{{dt}}$.
So, in the above case,
$\varepsilon = - \dfrac{{50 \times d\left( {0.02\cos 100\pi t} \right)}}{{dt}}$
$ \Rightarrow \varepsilon = - 50 \times \left( {0.02} \right) \times \left\{ { - \sin 100\pi t} \right\} \times 100\pi $
$ \Rightarrow \varepsilon = 100\pi \sin 100\pi t$
For maximum emf, $\sin 100\pi t = 1$. So,
$ \Rightarrow \varepsilon = 100 \times 3.14$
$\therefore \varepsilon = 314volt$

Thus, the correct answer is option A.

Note:Since magnetic flux is the dot product of magnetic field vector and areal vector, therefore magnetic flux is a scalar quantity. As soon as the magnetic flux stops changing the induced emf returns to zero.