Question

What are the factors on which the magnitude of induced emf depends?
$\text{A}\text{.}$ Change in magnetic flux
$\text{B}\text{.}$ The time in which the magnetic flux changes
$\text{C}\text{.}$ Resistance of the coil
$\text{D}\text{.}$ Both A and B

Answer 1

- Hint- Here, we will proceed by stating Michael Faraday's law of electromagnetic induction and then, we will express this law into mathematical form. Then, we will use the formula $\varphi =N\times B\times A\times \cos \theta$ to deduce it furthermore.

Complete step-by-step solution -

Michael Faraday’s famous law of electromagnetic induction states that a voltage is induced in a circuit whenever relative motion exists between a conductor and a magnetic field and that the magnitude of this voltage is proportional to the rate of change of the flux.
According to Michael Faraday’s law,
Induced emf in the coil, e = $-{\displaystyle \frac{d\varphi }{dt}}\to \text{(1)}$ where $\varphi$ represents the magnetic flux in the coil
From equation (1), we can say that the induced emf is directly proportional to the change in magnetic flux.
As we know that
Magnetic flux in a coil, $\varphi =N\times B\times A\times \cos \theta \to \text{(2)}$ where N represents the number of turns in the coil, B represents the magnetic field, A represents the area of the coil and $\theta$ represents the angle between the magnetic field and the normal to the area of the coil.
Also, $\theta =\omega t\to \text{(3)}$ where $\omega$ represents the angular speed and t represents the time.
By substituting equation (3) in equation (2), we get
$\varphi =N\times B\times A\times \cos \left(\omega t\right)\to \text{(4)}$
By substituting equation (4) in equation (1), we get
$$\begin{gathered} e=-{\displaystyle \frac{d}{dt}}\left(NBA\cos \left(\omega t\right)\right)=-NBA\left[-\sin \left(\omega t\right)\right]{\displaystyle \frac{d}{dt}}\left(\omega t\right)=NBA\omega \sin \left(\omega t\right){\displaystyle \frac{dt}{dt}} \\ \Rightarrow e=NBA\omega \sin \left(\omega t\right) \end{gathered}$$
Clearly from the above equation, we can write
The induced emf is directly proportional to the number of turns, magnetic field, area and the time in which the magnetic flux changes.
Therefore, the magnitude of the induced emf depends on the change in magnetic flux and also on the time in which the magnetic flux changes.
Hence, option D is correct.

Note- By increasing the amount of individual conductors cutting through the magnetic field, the amount of induced emf produced will be the sum of all the individual loops of the coil. If the same coil of wire passed through the same magnetic field but its speed or velocity is increased, the wire will cut the lines of flux at a faster rate so more induced emf would be produced.