Inductance

Inductance Definition

In 1830, according to the results of experiments performed by Michael Faraday and Joseph Henry, electric currents were induced in a coil due to a change in magnetic fields surrounding it. In other words, an electric current is induced in coils due to a change in magnetic flux (number of magnetic field lines passing through an area) linked with the same coil.

Faraday provided two laws of electromagnetic induction based on his experimental results.

1. First law - An e.m.f. is induced whenever a magnetic field linked with the coil changes. This e.m.f. lasts as long as there is a changing magnetic field.

2. Second law - The magnitude of the e.m.f. induced in the coil is directly proportional to the rate of change of magnetic flux linked with the coil.

   \[E_i\propto\frac{\phi_2-\phi_1}{t}\]where \(\phi_1\) and \(\phi_2\) are the initial and final magnetic flux passing through the conductor.\[E_i=k\frac{\phi_2-\phi_1}{t}\] where \(k\) is the proportionality constant, and its value is 1 for all systems of units.So, the magnitude of an induced e.m.f. is \(E_i=\frac{|\phi_2-\phi_1|}{t}\).

The direction of this induced e.m.f was provided by Lenz's law. According to Lenz's law, the direction of the induced e.m.f. is such that it opposes the cause of its production, i.e., a change in the magnetic flux.

\[E_i=-\frac{\phi_2-\phi_1}{t}\]

or

\[E_i=-\frac{d\phi}{dt}\tag{1}\]

Here the negative sign indicates that an induced e.m.f. opposes the change in magnetic flux. The direction of the induced e.m.f will then determine the direction in which the induced current will flow.

Fig. 2 - The figure shows a coil placed perpendicular to the plane containing the magnet.

In the above diagram, the magnetic flux passing through the coil changes due to the movement of a magnet toward the coil, which induces an electric current.

The induced electric current is in the anticlockwise direction. Due to this electric current, the coil starts behaving as a magnet. According to the Right Hand Rule, the coil's north pole faces the approaching bar magnet's north pole, as shown in figure 2. This would result in a repulsive force that opposes the bar magnet's motion toward the coil.

This inductance feature of a coil is used in our houses through inductors to protect our electrical appliances from the sudden increase or decrease of an electric current. In the next part, we will learn about these inductors and how their inductance varies with electric current.

Inductance Formula

Imagine a coil is connected to an electric circuit. The change in an electric current through the circuit induces an e.m.f./electric current in the coil. This electric current is due to the change of magnetic flux passing through the coil. The induced current's direction opposes the change in electric current through the circuit.

An inductor usually consists of a coil of conducting material wrapped around a ferromagnetic material's core. An inductor's core provides a medium to concentrate and set the maximum limit of the magnetic flux passing through the coil.

This magnetic flux \(\left(\phi\right)\) is directly proportional to the electric current flowing through the coil.

\[\phi\propto I\] or \[\phi=LI\tag{2}\] where \(L\) is a constant of proportionality which is called the coefficient of self-inductance or self-inductance of a coil.

When \(I=1\,\mathrm{ampere}\), then the equation (2) becomes \(L=\phi\).

Inductance Unit

From equations (1) and (2),

\[\begin{align*}E_i&=-\frac{d}{dt}\left(LI\right)\\E_i&=-L\frac{dI}{dt}\\L&=-\frac{E_i}{\frac{dI}{dt}}\tag{3}\end{align*}\]

The SI unit of \(L\) is henry. From equation (3),

\[1\,\mathrm{henry}=\frac{1\,\mathrm{volt}}{1\,\mathrm{ampere\,per\,sec}}\]

The coefficient of self-inductance \(\left(L\right)\) depends upon the physical properties of the coil.

• The number of turns in the coil,
• The area of the cross-section of the coil,
• The material of the core on which the coil is wound.

In the next section, we will learn about the self-inductance of a long solenoid and how it varies with its physical properties.

Inductance Equation

When a coil is wound into a tightly packed helix, it behaves as a solenoid. In the case of a long solenoid, the length of the solenoid is large compared to the radius of its cross-section.

Imagine an electric current \(I\) is passing through the solenoid of length \(l\) and with a number of turns \(N\). The magnetic field \(B\) at any point inside the solenoid is

\[B=\frac{\mu_0NI}{l}\tag{4}\]

where \(\mu_0\) is the absolute magnetic permeability of free space/vacuum.

Let \(A\) be the area of the cross-section of the solenoid. Then the magnetic flux passing through each turn of the solenoid in terms of the magnetic field and area is \(\phi=B\cdot A\) or \(\phi=BA\cos(\theta)\) where \(\theta\) is the angle between magnetic field lines and the area vector of a solenoid.

Inductance measurement

Let's say the solenoid is placed perpendicular to the direction of magnetic field lines, such that the angle between the area vector and the magnetic field lines is \(0^\circ.\) Therefore

\[\phi=BA.\]

Using the value of equation (4), the above equation becomes

\[\phi=\frac{\mu_0NI}{l}A\]

Total magnetic flux through the solenoid with \(N\) number of turns is

\[\begin{align*}\phi&=\frac{\mu_0NIA}{l}\times N\\\phi&=\frac{\mu_0N^2IA}{l}\tag{5}\end{align*}\]

From equations (2) and (5),

\[\begin{align*}LI&=\frac{\mu_0N^2IA}{l}\\L&=\frac{\mu_0N^2A}{l}\end{align*}\]

This equation shows the dependence of the inductance of a solenoid on its physical properties. From the equation, it is clear that the inductance of a solenoid is

1. directly proportional to the square of the number of turns

2. directly proportional to the area of the cross-section

3. inversely proportional to the length of the solenoid

In these last two parts, we have discussed how an induced e.m.f. depends upon various physical parameters and how it opposes the flow of electric current through the circuit. This shows the resemblance of self-inductance with the inertia corresponding to the mass of an object in mechanics. The amount of work done by the electric charges flowing through the coil against the opposing induced e.m.f. to maintain the flow of electric current is stored in an inductor as magnetic potential energy. In the next part, we study about the magnetic potential energy and how it relates to the self-inductance of an inductor.

Magnetic Potential Energy and Inductance

When an alternating current flows through an inductor, the current increases from 0 to some maximum value, say \(I_0\). Let \(I\) be the value of current passing through an inductor at time \(t\). Due to the change in electric current, an e.m.f. is induced in the inductor. The magnitude of this induced e.m.f. in terms of the inductance \(L\) is,

\[E_i=L\frac{dI}{dt}\tag{6}\]

This e.m.f is also known as back e.m.f as it opposes the flow of an electric current through the circuit. During the flow of an alternating current through the inductor, the current amount increases or decreases at a constant rate. Then, the work done by the electric charge while flowing through the inductor against this induced e.m.f. is \(W=E_iq\).

By differentiating the above equation while keeping induced e.m.f. constant (due to constant value of \(L\) and \(\frac{dI}{dt}\))

\[\frac{dW}{dt}=E_i\frac{dq}{dt}\]

where \(\frac{dq}{dt}\) is an electric current flowing through the inductor.

\[\therefore\quad \frac{dW}{dt}=E_iI\tag{7}\]

From equation (6) and (7)

\[\begin{align*}\frac{dW}{dt}&=L\frac{dI}{dt}I\\\end{align*}\] If we ignore the resistive losses, then the total amount of the work done by the electric charge to maintain an electric current \(I\) through the inductor is

\[\begin{align*}\int dW&=\int_0^I LIdI\\W&=L\int_0^I IdI\\W&=L\frac{I^2}{2}\\W&=\frac{1}{2}LI^2\tag{8}\end{align*}\]This work done by the electric charge is stored in the inductor in the form of magnetic potential energy i.e. \(U_{L}=\frac{1}{2}LI^2\).

In mechanics, the kinetic energy of particle of mass \(\left(m\right)\) moving with velocity \(\left(v\right)\) is \(K=\frac{1}{2}mv^2\tag{9}\).

By comparing equations (8) and (9), we find that as mass is the measure of inertia in mechanics, inductance is analogously the measure of 'inertia' in electronics.