This term paper consists of the basic knowledge regarding the topic Self-induction and mutual induction. I have tried my level best to impart the maximum information of the topic as much as I can with the help of different books and Internet. It consists of the basic knowledge about Electromagnetic induction and the different laws and rules which are necessary for the study of electromagnetic induction where from the concept of Self-induction and mutual induction has evolved.

Faraday’s laws have been described briefly in this paper that is enough to understand them. Lenz’s law & Fleming’s right hand rule have been discussed to some extent to get Self-induction and mutual induction properly. Finally detailed study material plus some practical information regarding the main topic Self-induction and Mutual induction has been provided such that everything about the topic will be gained by students who will read out it.

At the end applications of Self-induction and Mutual induction have been given plus the future prospective. Numericals regarding the topic have been given at the end of the paper.

In order to study self-induction & mutual induction we need to study firstly the basic principle.The basic principle of self-induction and mutual induction is explained by Electromagnetic Induction which is as follows:

Electromagnetic induction was discovered by Michael Faraday and Joseph Henry in 1831; however, but it was Faraday who first published the results of his experiments. Faraday’s first experiment demonstration of electromagnetic induction was carried by him in August 1831, he wrapped two wires around opposite sides of an iron rod. Based on his experiments he discovered properties of electromagnets, he observed that when current started to flow in one wire, a sort of wave would travel through the ring and cause some electrical effect on the opposite side. He plugged one wire into a galvanometer, and watched it as he connected the other wire to a battery. He saw a transient current (which he called a “wave of electricity”) when he connected the wire to the battery, and another when he disconnected it. Faraday had found several other manifestations of electromagnetic induction. For example, he saw transient currents when he quickly slid a bar magnet in and out of a coil of wires, and he generated a steady (DC) current by rotating a copper disk near a bar magnet with a sliding electrical lead (“Faraday’s disk”).

On the bases of lines of force Faraday explained electromagnetic induction but among top scientists at the time was ready to accept his results & rejected widely his theoretical ideas, mainly because they were not formulated mathematically

Lenz’s law, formulated by Heinrich Lenz in 1834, describes “flux through the circuit”, and gives the direction of the induced electromotive force and current resulting from electromagnetic induction.

## Electromagnetic induction:-

Whenever a conductor is moving through a magnetic field, there is always production of voltage across the conductor this phenomenon is called as electromagnetic induction. Michael Faraday is generally credited with the discovery of the induction phenomenon in 1831. A decade earlier the discovery of electromagnetic induction in 1831 was preceded by Danish physicist Hans Christian Oersted (1777-1851). Oersted showed that an electric current produces a magnetic field. That is, if we place a simple magnetic compass near any of the electrical wires that are carrying a current, a magnetic field around the wires can be detected. If an electric current can produce a magnetic field, physicists reasoned, perhaps the reverse effect could be observed as well. So they set out to generate an electric current from a magnetic field. This effect was first observed in 1831 by English physicist Michael Faraday (1791-1867) and after him by American physicist Joseph Henry (1797-1878). The principle on which the Faraday-Henry discovery is based is shown in the figure below. A long piece of metal wire is wound around a metal bar. The two ends of the wire are connected to a galvanometer, an instrument used to measure electric current. The bar is then placed between the poles of a magnet.

## Fig A

## Technical details:-

Faraday on the bases of his experimentsfound that the electromotiveforce (EMF) produced around a closed path is proportional to the rate of change of the magnetic flux through any surface bounded by that path. The conclusion of his experiments is that the electric current will be induced in any closed circuit when the magnetic flux through a surface bounded by the conductor changes. This applies whether the field itself changes in strength or the conductor is moved through it.

Electromagnetic induction is responsible for the operation of generators, all electric motors, transformers, induction motors, synchronous motors, solenoids, and most other electrical machines.

## Faraday’s law of electromagnetic induction states that:

## First law:-

It tells us about the condition under which emf is induced in a conductor & can be stated as: When the magnetic flux linking a conductor or coil changes, an e.m.f is induced in it.

## Second law:-

Faraday’s second gives us the magnitude of the induced e.m.f. in a conductor or coil which is directly proportional to the rate of change of flux linkages. Mathematically

E = – dÎ¦B / dt

Where E is the electromotive force (emf) in volts, Î¦B is the magnetic flux in webers.

For the common but special case of a coil of wire, having N loops with the same area, Faraday’s law of electromagnetic induction states that

E = – N (dÎ¦B /dt)

Where E is the electromotive force (emf) in volts,N is the number of turns of wire; Î¦B is the magnetic flux in webers through a single loop.

The result of the minus sign in the above equation is because of the direction of induced e.m.f. explained by Lenz’s law.

## Lenz’s law

Emil Lenz put forward a simple rule to find the direction of induced current: The induced current will flow in such a direction so as to oppose the cause that produces it.

Let us apply Lenz’s law to figure given above. When the N-pole of the magnet is approached to a coil of several turns as the N-pole of the magnet is moved towards coil, the magnetic flux linking the coil increases. Therefore, an e.m.f and hence current is induced in the coil according to faraday’s laws of electromagnetic induction. Therefore according to Lenz’s law, the direction of the induced current will be such so as to oppose the cause that produces it. In the present case, the cause of the induced current is the increasing magnetic flux linking the coil. Therefore, the induced current will set upmagnetic flux that opposes the increase in flux through the coil. Therefore, the induced current will set up magnetic flux that opposes the increase in flux through the coil. This is possible only if the left hand face of the coil becomes N-pole. Once we will know the magnetic polarity of the coil face, the direction of the induced current can be easily determined by applying right hand rule for the coil.

The Lenz’s law can be summed up as under: If the magnetic flux Ð¤ linking a coil will flow in such a direction so as to oppose the increase in flux i.e. the induced current will produce flux as shown in figure given below,

If magnetic flux Ð¤ linking a coil is decreasing, the induced current i in the coil will flow in such a direction so as to oppose the decrease in the flux i.e. the induced current will produce flux Ð¤ to aid the flux Ð¤ as shown in figure given below

## .

## Fleming’s right hand rule:-

It is named after British engineer John Ambrose Fleming, who invented it.

Fleming’s right hand rule

Fleming’s right hand rule shows the direction of induced current when a conductor moves in a magnetic field. If the right hand is held with the thumb, first finger and second finger mutually perpendicular to each other (at right angles), as shown in the diagram above.Then the thumb points the direction of motion of conductor, the first fingure points the direction of magnetic field and the second fingure points the direction of induced current.

## Applications of electromagnetic induction:-

Electric generators

Electric motors

Transformer

Electric guitar

Electric bell

## Self-induction:-

The property of a coil that opposes any change in the amount of current flowing through it is called self-induction. The property of self-inductance is a particular form of electromagnetic induction. It can be also defined as is the induction of a voltage in a current-carrying wire when the current in the wire itself is changing. In the case of self-inductance, the magnetic field created by a changing current in the circuit itself induces a voltage in the same circuit. Therefore, the voltage is self-induced. In circuit diagram, a coil or wire is usually used to show an inductive component. If we have a closer look at a coil will help understand the reason that a voltage is induced in a wire carrying a changing current. The alternating current running through the coil creates a magnetic field in and around the coil that is increasing and decreasing as the current changes. The magnetic field forms concentric loops that surround the wire and join to form larger loops that surround the coil as shown in the figure below. When the current increases in one loop the expanding magnetic field will cut across some or all of the neighboring loops of wire, inducing a voltage in these loops. This causes a voltage to be induced in the coil when the current is changing.

By studying this image of a coil, it can be seen that the number of turns in the coil will have an effect on the amount of voltage that is induced into the circuit. Increasing the number of turns or the rate of change of magnetic flux increases the amount of induced voltage. Therefore, Faraday’s Law must be modified for a coil of wire and becomes the following.

VL = NdÎ¦/dt

Where VL = induced voltage in volts, N = number of turns in the coil, dø/dt = rate of change of magnetic flux in webers/second

The equation simply states that the amount of induced voltage (VL) is proportional to the number of turns in the coil and the rate of change of the magnetic flux (dø/dt). In other words, when the frequency of the flux is increased or the number of turns in the coil is increased, the amount of induced voltage will also increase.

In a circuit, it is much easier to measure current than it is to measure magnetic flux, so the following equation can be used to determine the induced voltage if the inductance and frequency of the current are known. This equation can also be reorganized to allow the inductance to be calculated when the amount of inducted voltage can be determined and the current frequency is known.

VL = L di/dt

Where VL = the induced voltage in volts, L = the value of inductance in henries, di/dt = the rate of change of current in amperes per second.

This L in above equation is known as self-inductance.

## Factors affecting inductance:-

The inductance of a conductor upon these factors:-

Shape & number of turns.

Relative permeability of material surrounding the conductor.

The speed with which magnetic field changes.

Anything that affects magnetic field affects inductance.

## PRACTICAL REPRESENTATION OF SELF INDUCTANCE:-

Make a circuit containing a bulb, a solenoid, a 12 V DC battery and a switch as shown in figure below. Observe the intensity of light by switching on the circuit.

Connect a 12V AC source to the circuit in the place of the 12V DC battery. On observing we will find that the intensity of light decreases while using an AC in the circuit. It is because there is a change in magnetic flux linked with the solenoid when AC flows through it. Due to this change in flux an induced emf develops in the solenoid. This emf is opposite to the emf applied in the circuit. Therefore theresultant emf in the circuit decreases. The brightness of the bulb also decreases. Self-induction is the phenomenon of inducing an emf in a coil caused by the variations of magnetic flux produced by a varying current in the same coil. In the above experiment, a soft iron rod is slowly introduced into the coil and then taken out from it. We will have many observations of these two cases.

Inductors are coils which can oppose the changes of current in a circuit. They are used for reducing current in AC circuits without any loss of electrical energy. For these purposes resistors can also be used. But while using resistors electrical energy is wasted in the form of heat.

## DIAGRAMATICAL VIEW OF SELF INDUCTION:-

## Mutual Induction:-

The property of two neighbouring coils to induce voltage in one coil due to change of current in the other is called mutual induction. Also can be defined as when an emf is produced in a coil because of the change in current in a coupled coil, the effect is called mutual inductance. The emf is described by Faradays law & its direction is always opposite to the change in magnetic field produced in it by the coupled coil (Lenz’s Law). The induced emf in coil 1 is due to self-inductance L. Figure below shows mutual inductance

The induced emf in coil 2 caused by the change in current I1can be expressed as

Emf2 = -N2AÎ”B/Î”t = -MÎ”I1/Î”t

The mutual inductance M can be defined as the proportionality between the emf generated in coil 2 to the change in current in coil 1 which produced it. The mutual inductance M of two coupled inductances L1 and L2 is equal to the mutually induced voltage in one inductance divided by the rate of change of current in the other inductance:

M = E2/ (di1/dt)

M = E1 / (di2/dt)

If the self-induced voltages of the inductances L1 and L2 are respectively E1s and E2s for the same rates of change of the current that produced the mutually inducedvoltages E1 and E2, then:

M = (E2m / E1s)L1

M = (E1m / E2s)L2

Combining above two equations:

M = (E1mE2m / E1sE2s)(L1L2)= kM(L1L2)

where kM is the mutual coupling coefficient of the two inductances L1 and L2.

If the coupling between the two inductances L1 and L2 is perfect, then the mutual inductance M is:

M = (L1L2)

The most common application of mutual inductance is the transformer.

## Factors affecting mutual inductance:-

Shape of circuits.

(Loopy ——– Large M)

(Straight ————– Small M)

Size of circuits.

Number of turns in each circuit.

Distance between circuits.

Orientation of circuits.

## Mutual Inductance: Transformer as an application

Transformer is one of the most well-known application known to almost every human being. When more current flows in the secondary of a transformer as it supplies more power, then more current must flow in the primary as well since it is supplying the energy. This coupling between the primary and secondary is most conveniently described in terms of mutual inductance. The mutual inductance appears in the circuit equations for both the primary and secondary circuits of the transformer.

## PRACTICAL REPRESENTATION OF MUTUAL INDUCTANCE:-

If we wound an insulated copper wire around one end of a soft iron core and the ends of the coil are connected to a battery through a switch and another insulated copper wire wound around the other end of the iron core. Connecting the ends of this coil to a galvanometer. Of these, the circuit which is connected to the battery is called primary circuit and that connected to the galvanometer is called secondary circuit.

An experiment to explain mutual induction.

Note down the deflections in the galvanometer when the primary circuit is switched on or off.Why does the galvanometer needle deflect?

That is because an emf is produced in the secondary. Then how is it produced.

## Is this the emf of the cell?

When there are two nearby coils the variation of current in one of them produces a change in the magnetic flux around it. The second coil is situated in this region of varying magnetic flux. Therefore by electromagnetic induction an emf is induced in the secondary coil. This phenomenon is called mutual induction.

## Applications & future prospective:-

Self-induction & mutual induction has got numerous applications in different fields of electrical engineering. We cannot imagine any electrical field in which induction is not present starting from a power plant up to our households.

It is used in Electric generators.

It is used in Electric motors.

It is used in Electric guitar.

It is used in Electric bell.

It is used in transformers.

It is used in Electric heating purposes.

Presently electrical engineers are trying to make a generator which will be brushless based on electromagnetic induction. Being a vast subject and having numerous applications it is expected that it may have many more applications.

## Example 1:- A coil has 1000 turns , a current of 5 A causes a flux of 6 mWb to link the coil. What is the coil inductance?

Sol:-Here N = 1000; Î¦ = 6* 10-3 Wb; I = 5 A

Therefore, Coil inductance, L = NÎ¦/I

= 1000 * 6*10-3 / 5

= 1.2 H Answer.

## Example 2:-Two coils A & B ofturns 600& 500 respectively. A current of 8 A flowing in coil A produces a flux of 0.05 Wb. If coefficient of coupling is 0.2. Find mutual inductance?

Sol :-Here NA = 600; NB = 500 Î¦A= 0.05 & IA = 8 A

Mutual inductance M = kÎ¦ANB /IA

= 0.2* 0.05 * 500 / 8

= 0.625 H Answer.