17.16 Self Induction

17.16 Self Induction

Consider a circuit in which a coil is connected in series with a battery, a switch and a rheostat as shown in Fig. 17.18. If we move the rheostat quickly, the primary current through the coil will change. The magnetic flux through the coil will also change, which finally induces an emf in the coil itself.

The phenomenon, in which changing current induces an emf inside the coil itself, is called self-induction. Let $\phi$ represents flux passing through one loop of the coil. The flux passing through the coil of $N$ turns would be $N\phi$. As this flux $\phi$ is proportional to the magnetic field produced which is in turn proportional to the current $I$, therefore, $N\phi \propto I$

where $L$ is proportionality constant and is called self-inductance of the coil. The self-inductance $L$ depends upon the following factors:

1. Number of turns of the coil.
2. Area of cross-section of the coil.
3. Core material, by winding the coil around ferromagnetic iron core, the magnetic flux and hence inductance can be increased significantly relative to that for an air core.

(Fig. 17.18: A coil showing production of induced current and induced emf in itself.)

By Faraday's law, the emf induced in the coil is given by

$\epsilon_{i} = - \dfrac{\Delta(N\phi)}{\Delta t}$ Using Eq. (17.28); $\epsilon_{i} = - \dfrac{\Delta(LI)}{\Delta t}$

or

Here $L$ is self-inductance of the coil and minus sign indicates that the induced emf opposes the applied voltage. In the above expression, If we take $\epsilon_{i} = 1$ volt and $\Delta I/\Delta t = 1\,\mathrm{A\,s^{-1}}$, then $L = 1$ henry ($\mathrm{H}$), defined as:

If the emf induced in the coil is $1\,\mathrm{V}$ when the current flowing through it changes at the rate of $1\,\mathrm{A\,s^{-1}}$, the coil has self-induce in a coil of $1$ henry ($\mathrm{H}$). The self-inductance of the coil can also be defined as ratio of the emf to the rate of change of current in the same coil. The SI unit of self-inductance $L$ is $\mathrm{Vs\,A^{-1}}$ having dimensions $[\mathrm{ML^{2}T^{-2}A^{-2}}]$. By equating Eqs.17.30 and 17.31, we have $-N \dfrac{\Delta\phi}{\Delta t} = -L \dfrac{\Delta I}{\Delta t}$

which gives another form of equation for self-inductance of the coil.

Example 17.8

A coil has an inductance reactance of $160\,\Omega$ at a frequency of $50\,\mathrm{Hz}$. Calculate self-inductance of the coil.

Solution

Induced reaction $X_{L} = 160\,\Omega$, Frequency $f = 59\,\mathrm{Hz}$ We know that; $X_{L} = 2\pi f L$ or $L = X_{L} / 2\pi f$ $L = 160\Omega / (2 \times 3.14 \times 50\,\mathrm{Hz}) = 0.509\,\mathrm{H}$