Question

Explain self-induction of a coil. Arrive at an expression for the induced emf in a coil and the rate of change of current in it.

Answer 1

Hint: In the given question, we must consider the case of self-induction which is stated by Faraday. Faraday’s law states that Induced EMF is equivalent to the rate of change of the magnetic flux. Keeping this in mind, we will proceed with the given question.

Complete step by step answer:
Inductance is defined as the property of a component that opposes the change of current flowing through it. Even if we consider a straight piece of wire, it will too consist of some inductance.
Inductors perform this by generating a self-induced emf within itself as a result of their changing magnetic field. In an electrical circuit, when the emf is induced in the same circuit in which the current is changing, then the effect will be known to us as self-induction.
Magnetic field gives rise to the magnetic flux, which is also denoted by $\phi$, which passes through each turn in the coil.
Now let us consider a coil having N number of turns. And the flux being ${{N}_{\phi }}$.
So, as we know that magnetic field is proportional to the current i flowing through the coil and so is the flux.
So, we can write that,
${{N}_{\phi }}$directly proportional to i
Now, ${{N}_{\phi }}$= Li
In this case, L is the self-inductance.
Now let us consider the current to change in the coil with time which changes the flux as well.
So this rate of change of flux induces an emf, which is denoted by e, by Faraday’s law of electromagnetic induction.
$e=-\dfrac{d({{N}_{\phi }})}{dt}=-\dfrac{d(Li)}{dt}=-L\dfrac{di}{dt}$
So if $\dfrac{di}{dt}$=1 A/s we have, e = - L
Hence it is proved that self-inductance is numerically equal to the emf induced in coil when current changes at a rate of 1 Ampere per second.

Note: In application, we can induce electromotive force by placing an electric conductor in a magnetic field at motion, and by involving the placement of a conductor carrying electricity that is constantly moving into a magnetic field being static in nature.