Question

The formula for induced EMF in a coil due to change in magnetic flux through the coil is:
(A is the area of the coil, B is the magnetic field)
A) $e = - A\dfrac{{dB}}{{dt}}$
B) $e = - B\dfrac{{dA}}{{dt}}$
C) $e = - \dfrac{{dB.A}}{{dt}}$
D) $e = - \dfrac{{dB \times A}}{{dt}}$

Answer 1

Hint: Induced EMF concept is given by Faraday Law which states;
Any change in the magnetic field or magnetic flux of a coil will cause an induced EMF.
$e = - N\dfrac{{d\phi }}{{dt}}$( e is the induced EMF, N is the number of turns of the coil and $\phi $ is magnetic flux)
Minus sign of the above equation is because of Lenz law.
We will discuss the above two laws in more detail and solve the given problem.

Complete step by step solution:
Let's explain Lenz Law and Faraday Law in more detail first.
Faraday Law states that: Induced EMF is directly proportional to the rate of change of magnetic flux produced in the coil due to the flowing current in the conductor having a finite number of turns in it.
In mathematical terms Faraday Law is given as:
$e = - N\dfrac{{d\phi }}{{dt}}$
Minus sign of the equation is given by Lenz law which states that;
Direction of the electric current which is induced in a conductor by a changing magnetic field is such that the magnetic field is such that the magnetic field created by the induced current opposes the initial changing magnetic field.
In lay man's language we can state Lenz law as; Current or voltage which is induced in a circuit opposes the change that produces it.
Now , we will calculate the formula of induced EMF.
Magnetic field is given as:
$B = \dfrac{\phi }{A}$ (B is the magnetic field, A is the area and $\phi $ is the flux)
Therefore, flux is given as, $\phi $ = B.A
As per the definition of induced EMF, rate of change of flux is equal to the induced EMF, thus the formula becomes;
$e = - \dfrac{{dB.A}}{{dt}}$

Hence, Option C is correct.

Note: Faraday Law is applicable in many electrical equipment such as Transformers where step up or step down process of current or voltage takes place because of Electromagnetic Induction, Alternators which produce rotating magnetic fields, DC motors, measuring meters such as energy meters etc.