Electromagnetic Induction

# Self Induction

## Inductor and Inductance

An inductor is a conducting wire shaped as a coil which can store energy in a magnetic field created by the electric current passing through it. The ability of an inductor to store magnetic energy due to electric current is called its inductance denoted by $L$. The circuit symbol of an inductor is given below.

An inductor having no resistance but having only inductance is an ideal inductor. It does not dissipate or radiate energy. A real inductor should have the combination of inductance and resistance.

There are two types of electromagnetic induction: self induction and mutual induction.

## Self Induction

When the current or magnetic flux through a coil is changed, an emf is induced in it. The phenomenon of the production of an opposing emf in the coil as a result of change in current or magnetic flux linked with the coil itself is called self induction.

Consider an electric circuit containing a coil, a rheostat and a cell. When the current flowing through the coil is varied using a rheostat, the flux linked with the coil changes.

When the current is increased by adjusting rheostat, magnetic lines of force linked with the coil also increase and an emf is induced in the circuit according to Faraday’s law. The direction of induced emf is such that it opposes its cause (current) according to Lenz’s law. Hence, the induced emf is in opposite direction of current.

If the current in the circuit is decreased by adjusting rheostat, magnetic flux linked with the coil decreases and an induced emf is produced but according to Lenz’s law, it will oppose the decay of current i.e. the induced emf and the current will have the same direction.

In this way, we see that whenever there is a change in the current flowing through a coil, an induced emf is set up in the coil itself. The produced emf is called self induced emf or back emf (because of its opposing nature to its cause) and the corresponding current is called self induced current. This phenomenon is called self induction.

Self induction opposes both the growth (when the circuit is switched on) and decay (when the circuit is switched off) of the current in a coil and thus it may be treated as analogous to the inertia of material body. Therefore, self induction is also known as inertia of electricity.

### Coefficient of Self Induction or Self Inductance

The property of an electric circuit by virtue of which any change in the magnetic flux linked with it induces an emf is called inductance.

Let $I$ be the current flowing through a coil then the magnetic flux $\phi$ linked with the coil is found to be proportional to the current $I$.

$\text{i.e. }\phi∝I$

$\phi=LI$

where $L$ is a proportionality constant called coefficient of self induction or self inductance or simply inductance. Its value depends upon the number of turns, size, shape, material of the coil and permeability of the medium inside the coil.

$L=\frac{\phi}{I}$

Therefore, self inductance of the circuit (coil) is defined as the total magnetic flux linkage of the coil per unit current.

According to Faraday’s laws of electromagnetic induction, the induced emf in the coil is given by

$\varepsilon=-\frac{d\phi}{dt}=-L\frac{dI}{dt}$

$\therefore\varepsilon=-L\frac{dI}{dt}$

If $\frac{dI}{dt}=1$, then $L=-\varepsilon$. Hence, self inductance can also be defined as the induced emf in the circuit produced due to unit rate of change of current in the circuit.

The SI unit of inductance is $\text{Wb A}^{-1}$ or $\text{V A}^{-1}\text{s}$ which is given the name henry in the memory of Joseph Henry.

$\therefore 1\text{ henry (H)}=1\text{ V A}^{-1}\text{s}=1\text{ Wb A}^{-1}$

The smaller units of inductance are millihenry $(\text{mH})$, microhenry $(\mu\text{H})$, etc.

$1\text{ mH}=10^{-3}\text{ H}$

$1\text{ }\mu\text{H}=10^{-6}\text{ H}$

If $\varepsilon=1\text{ V}$ and $\frac{dI}{dt}=1\text{ A s}^{-1}$, then $L=1\text{ H}$. Thus, the self inductance of a coil is said to be one henry if the back emf of one volt is set up when the current in the coil changes at the rate of one ampere per second.

## Self Inductance of a Solenoid

Consider a long solenoid of length $l$, area of cross section $A$ and number of turns per unit length $n$. The magnetic field within the solenoid is given by

$B=\mu_0nI$

If the solenoid has $N$ number of turns, then $N=nl$. The magnetic flux $(\phi)$ linked with the length $l$ of the solenoid is given by

$\phi=NBA=(nl)(\mu_0nI)A$

Now, the self inductance of the solenoid is given by

$L=\frac{\phi}{I}=\frac{(nl)(\mu_0nI)A}{I}$

$\therefore L=\mu_0n^2lA\text{ __(1)}$

Since $n=\frac{N}{l}$, so equation $\text{(1)}$ becomes

$L=\mu_0\left(\frac{N^2}{l^2}\right)lA$

$\therefore L=\frac{\mu_0N^2A}{l}$

This gives the self inductance of the solenoid. From this relation, we can see that the self inductance of a solenoid depends on the total number of turns $(N)$, length $(l)$, area of cross section $(A)$ of the coil and the nature of material $(\mu_0)$ inside the coil.