Define Self Inductance and Expression for Coefficient of It.

Define Self Inductance :

Self-inductance is the property of a coil (or any conductor) to oppose a change in the current flowing through it by inducing an electromotive force (EMF) in itself.

This phenomenon arises due to Faraday's law of electromagnetic induction. When the current through the coil changes, the magnetic flux linked with the coil also changes, inducing an EMF that opposes this change in current. This induced EMF is also known as back EMF.

Self inductance is also knowns as inertia of electricity.

Coefficient of Self Induction or Self Inductance :

The magnetic field at any point due to a current carrying coil is directly proportional to the current. Therefore, the magnetic flux ($\phi_B = BA$) through the area bounded by the current carrying coil is directly proportional to the current flowing in the coil, i.e.,

$ \phi_B \propto I $

$ \phi_B = LI $

where $L$ is the constant of proportionality and is known as \textbf{co-efficient of self induction or simply self inductance}.

If $I = 1$, then from eqn. (1),

$$ L = \phi_B \quad \dots (2) $$

Thus, \textbf{co-efficient of self induction or self inductance of a coil is defined as the magnetic flux linked with the coil when unit current flows through it.

Also, according to Faraday's law of electromagnetic induction, induced e.m.f. in the coil is given by

$$ \varepsilon = - \frac{d\phi_B}{dt} $$

Using eqn. (1), we get

$$ \varepsilon = - \frac{d}{dt} (LI) = -L \frac{dI}{dt} \quad \dots (3) $$

If $\frac{dI}{dt} = 1$ i.e. rate of decrease of current is unity, then from eqn. (3), we get, $L = -\varepsilon$.

Thus, \textbf{co-efficient of self induction or self inductance of a coil is defined as the induced e.m.f. produced in the coil through which the rate of decrease of current is unity.}

Units of Self Inductance :

SI Unit of self inductance is henry (H).

Since

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

$\therefore$ , $ 1 \text{ henry (H)}= \frac{1 \text{ volt}}{1 \text{ ampere/second}}$

$ 1 \text{ henry (H)}= 1 VA^{-1}s$

Also,

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

$\therefore$, $ 1 \text{ henry (H)} = \frac{1 \text{ weber}}{1 \text{ ampere}}$

$ 1 \text{ henry (H)} = 1WbA^{-1} $

Thus,

$ 1 \text{H} = 1 VA^{-1}s$

$ 1 \text{H} = 1 WbA^{-1} $

Hence, self inductance of a coil is said to be 1 henry (H) if 1 volt induced e.m.f. is produced in the coil due to $1 As^{-1}$ decrease in current in the coil.

Smaller units of Self inductance are : $1 \text{mH} = 10^{-3} \text{ H}$ and $1 \mu \text{H} = 10^{-6} \text{ H}$.

Dimensional formula for Self Inductance (L) :

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

$ L = \frac{\varepsilon}{dI/dt}$

$L= \frac{W/q}{I/t}$

$L= \frac{W}{q} \cdot \frac{t}{I}$

$L= \frac{\text{Work done}}{\text{Charge}} \cdot \frac{\text{Time}}{\text{Current}} $

$ [L] = \frac{[\text{ML}^2\text{T}^{-2}]}{[\text{AT}]} \cdot \frac{[\text{T}]}{[\text{A}]}$

$[L]= [\text{ML}^2\text{T}^{-2}\text{A}^{-2}] $

Inductor : An element of an electric circuit like a tightly wound coil of insulated wire which opposes the change in current flowing through it is called an inductor.

The symbol of an inductor in an electric circuit is shown as follows :