Introduction

Electromagnetic induction is defined as the production of an electromotive force (i.e., voltage) across an electrical conductor due to its dynamic interaction with a magnetic field.

Faraday’s Experiment

Faraday’s observations are as follows:

1. When a magnet is moved inwards or away from the coil, the galvanometer shows sudden deflection.
2. Since, there is relative motion between the coil and the magnet, the deflection lasts for long.
3. The deflection becomes faster and slower if the magnet moves faster and slower into the coil.
4. The direction of deflection in the galvanometer is reversed when polarity of the magnet is reversed.

The deflection in the galvanometer is due to a current developed in the circuit. This current is called induced current and the emf developed in the coil is called induced e.m.f.

Magnetic flux

Magnetic flux through any area is defined as the number of magnetic lines of force passing normally through the surface.

If [latex]\phi[/latex] be the magnetic flux passing through the surface area ‘A’ placed in uniform magnetic field of flux density B forming an angle  with normal, then,

[latex]\phi = \vec{B}{A} = BAcos\theta[/latex]

If the coil of the area A consists of a large no. of turns N, then, it is called flux linkage and it is given by:

[latex]\phi = NAB cos\theta[/latex]

Its SI unit is Weber (Wb) and CGS unit is Maxwell.

1 Maxwell = [latex]10^{-8}[/latex] Weber

Electromagnetic Induction:

When a coil kept in a uniform magnetic field is rotated and emf. is produced in it. This phenomenon is called electromagnetic induction. When the two ends of the coil are connected through a conducting wire, then, a current flow through it and it is called inducing current.

Faraday’s law of electromagnetic induction:

The magnitude of induced emf across the coil is directly proportional to the rate of change of magnetic flux passing through it.

E ∝ [latex]\frac{d\phi}{dt}[/latex]

E = k[latex]\frac{d\phi}{dt}[/latex]

Where, k = proportionality constant. In SI system of unit, k = – 1, then,

E = -[latex]\frac{d\phi}{dt}[/latex]

Where, -ve sign indicates that emf produced in the coil always opposes a motion causing it.

The direction of induced emf is given by the following laws:

a) Lenz’s Law

The direction of induced current is such that it always opposes the change of magnetic flux which causes to produce it. Consider the magnet which is brought towards the coil due to which flux linked with the coil changes. This causes the induced emf in the coil and induced current flows in a circuit as shown in the figure below.

Hence, when we pull or push the magnet its motion will always be automatically opposed. The external agent that causes the magnet to move either towards the coil or away from it will always experience resisting force due to repulsion or attraction respectively and thus will be required to do work. This work done is converted into electrical energy in the form of induced emf in the coil. Hence, Lenz’s law is in accordance with the law of conservation of energy.

b) Fleming’s Right-hand rule

The direction of induced emf or current can be determined by using Flaming’s right-hand rule which states that, when a conductor such as wire attached to a circuit moves through a magnetic field, an electric current is induced in the wire due to Faraday’s law of induction. The current in the wire can have two possible directions. Fleming’s right-hand rule gives which direction the current flows.

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.

The thumb is pointed in the direction of motion of the conductor.

The first finger is pointed in the direction of the magnetic field. (North to South).

Then the second finger represents the direction of the induced or generated current (the direction of the induced current will be the direction of conventional current, from positive to negative).

Emf induced in a moving straight conductor

Consider, a straight conductor AC of length l is moving at right angles to the uniform magnetic field with velocity v. Suppose the conductor moves through a small distance ‘dx’ in time dt, then, the area swept out by the conductor is:

dA = ldx

According to Faraday’s law of electromagnetic induction:

E = [latex]\frac{d\phi}{dt}[/latex]

Or, E = [latex]\frac{d}{dt}(AB)[/latex]  [∵ [latex]\phi = ABCos\theta[/latex] = ABCos0^o = AB]

Or, E = B[latex]\frac{dA}{dt}[/latex]

Or, E = Bl[latex]\frac{dx}{dt}[/latex]

Or, E = Blv

If the conductor moves making an angle q with the direction of magnetic field, then, the emf induced is given by:

E = [latex]Blvsin\theta[/latex].

Emf induced in a rotating coil

Let us consider a rectangular coil having area A is placed in a uniform magnetic field B in such a way that it is normal to the plane of the coil making an angle δ with the magnetic field B as shown in figure. The component of field B at the right angle to the plane of the coil is BCosδ. The flux through the coil is ABCosδ.

If N be the no. of turns, then, the flux linkage f is given by NABCosδ ………….. (i)

If the coil turns, about an axis perpendicular to the file direction with the constant angular velocity ω or [latex]\frac{d\delta}{dt}[/latex], then, the induced emf in the coil is:

E = -[latex]\frac{d\phi}{dt} = -\frac{d(NABcos\delta)}{dt}[/latex]  = NBASinδ . [latex]\frac{d\delta}{dt}[/latex] ……………. (ii)

We also know that, ω = [latex]\frac{d\delta}{dt}[/latex] ………….. (iii)

On integrating,

ωt = δ ………………. (iv)

and, ω = 2πf ………….. (v)

now, from (ii), (iii), (iv), (v), we have,

E = 2πf NBASin2πft ……………. (vi)

If Sin2πft = 1, then, emf becomes maximum, so, the form of equation (vi) takes the form,

E_o = 2πfNBA ……….. (vii)

From equation (vi) and (vii), we have,

E = E_oSin2πft = E_oSinωt.

Thus a coil rotating with a constant angular velocity in a uniform magnetic field produces a sinosoidally alternating emf.

AC generator

AC generator is a device which converts mechanical energy into electrical energy in the form of alternating current. It is based on the principle of electromagnetic induction i.e., whenever the amount of magnetic flux linking a coil change, then, EMF is induced in it.

Construction / Structure

An AC generator consists of following parts:

1. Rectangular coils

A rectangular coil having N terms is kept in a uniform magnetic field which rotates due to an external source of energy.

2. Field magnets

There are two strong field magnets which produce uniform radial magnetic field in which the coil rotates.

3. Slip rings

There are two slip rings R1 and R2 with move along with the rectangular coil and provide output current.

4. Brushes

There are 2 carbon brushes B₁ and B₂ which are in touch with the slip ring and help to carry out the alternating current to the load.

Working

When the coil rotates the magnetic flux linking the coil continuously changes and hence, an emf is induced in the coil according to Faraday’s law of electromagnetic induction. When the coil is horizontal, the rate of change of flux linking the coil is maximum and thus the induced emf is maximum. When the coil is vertical, the rate of flux changing is instantaneously zero. So, the induced emf is also zero at that instant but during the vertical and horizontal position, flux changes smoothly and hence, a sinusoidal emf is produced.

Consider a rectangular coil having ‘N’ turns is rotating in uniform magnetic field ‘B’ then, the flux linkage through the coil is given by:

[latex]\phi = NABcos\theta[/latex]

Where, A = cross sectional area of the coil.

According to Faraday’s law of electromagnetic induction, the emf produced is given by:

E = [latex]\frac{d\phi}{dt}[/latex]

Or, E = -[latex]\frac{d(NABcos\theta)}{dt} = -NAB \frac{d(cos\theta)}{dt}[/latex]

= [latex]NABsin\theta. \frac{d\theta}{dt}[/latex]

                = NABωSin[latex]\theta[/latex]

                = E_o Sinωt.

Where, E_o = NABω

                & ω = [latex]\frac{\theta}{t}[/latex]

[latex]\therefore \theta[/latex] = ωt.

Self – inductance

When the current flowing in a coil change then EMF is induced in it. This EMF is called back EMF and the phenomenon is called self – induction. Due to self – induction, it opposes the growth or decay of the current flowing through the coil.

Consider a coil having n terms. When the current changes through the coil, then, EMF is induced in it.

According to Faraday’s Law of electromagnetic induction, the emf induced is given by:

E ∝ -[latex]\frac{d\phi}{dt}[/latex]

But the flux change is directly proportional to the current change in the coil, i.e.

[latex]\frac{d\phi}{dt}\propto \frac{dI}{dt}[/latex]

[latex]\therefore E \propto \frac{dI}{dt}[/latex]

Or, E = -L[latex]\frac{dI}{dt}[/latex]  ………… (i)

Where, L = coefficient of self – induction or self-inductance.

The unit of coefficient of self – inductance is Henry (H).

From the relation above,

When [latex]\frac{dI}{dt}[/latex] = 1 A/s and E = 1V, then, L = 1H.

Hence, the coefficient of self-induction of a coil is said to be 1 Henry if the current changes at the rate of 1 Ampere per second and produces an emf of 1 Volt.

From eq^n. (i), we have,

L = [latex]\frac{E}{\frac{dI}{dt}} = \frac{\frac{d\phi}{dt}}{\frac{dI}{dt}} = \frac{d\phi}{dI}[/latex]

Or, L = [latex]\frac{\phi}{I} = \frac{Flux\ change}{Current\ change}[/latex]

Self-Inductance of a coil (Solenoid)

Consider, a coil having ‘N’ turns and length ‘[latex]l[/latex]’. Let, ‘A’ be the cross-sectional area of the coil and ‘I’ be the current flowing in the coil. The magnetic field inside the coil is given by:

B = μ_o nI = μ_o[latex]\frac{N}{l}I[/latex]

The magnetic flux linking the coil is given by:

[latex]\phi = NAB[/latex]

Or, [latex]\phi = \frac{NA\mu_oNI}{LI}[/latex]

Or, L = [latex]\frac{\phi}{I} = \frac{NA\mu_oNI}{lI}[/latex]

Or, L = [latex]\frac{\mu_oAN^2}{l}[/latex]

Mutual Induction

When two coils are close to each other and when current changes in 1 coil, the flux linkage through the second coil changes. As a result, an emf is induced in the second coil. This phenomenon in which an emf is induced in a coil due to rate of change of current is in the other coil is called mutual induction.

Consider, two points having turns N₁ and N₂ are kept close to each other. Let, I₁ be the current flowing in the first coil. Let, [latex]\phi_2[/latex] be the flux linkage in the second coil. If the current changes in the first coil, [latex]\phi_2[/latex] also changes. As a result, an emf is induced in the second coil which is given by:

E₂ ∝ [latex]\frac{d\phi_2}{dt}[/latex]

But, the flux change in the second coil is directly proportional to current change in first coil.

[latex]\frac{d\phi_2}{dt} \propto \frac{dI_1}{dt}[/latex]

E₂ ∝ [latex]\frac{dI_1}{dt}[/latex]

[latex]\therefore E_2 = -M\frac{dI_1}{dt}[/latex]  ………….. (1)

Where, M is called co-efficient of mutual induction or mutual inductance. The unit of mutual inductance is 1 Henry (H). From eq^n. (i), if [latex]\frac{dI_1}{dt} = 1A/s[/latex] and E₂ = 1 Volt.

Then,

M = 1 H.

Hence, the coefficient of mutual inductance is said to be 1 Henry if the current changes at a rate of 1A/s in one coil and produced an emf of 1V in another coil.

Again, from eq^n. (i),

M = [latex]\frac{E_2}{\frac{dI_1}{dt}} = \frac{\frac{d\phi_2}{dt}}{\frac{dI_1}{dt}} = \frac{d\phi_2}{dI_1}[/latex]

[latex]\therefore M = \frac{\phi_2}{I_1} = \frac{Flux\ change\ in\ the\ second\ coil}{current\ change\ in\ the\ first\ coil}[/latex]

Mutual Induction of two long co-axial solenoid

Consider, two solenoids S₁ and S₂ such that the solenoid S₂ completely surrounds the solenoid S₁ and two solenoids are so closely arranged that they have same area of cross section ‘A’. Let, N₁ and N₂ to be the number of turns in the first and second coil respectively. If I₁ be the current flowing through the first coil. Then, magnetic field produced by it is given by

B₁ = μ_on₁I₁

Now, the magnetic flux linked with the second coil is given by:

[latex]\phi_2[/latex] = N₂AB₁

                = N₂Aμ_o[latex]\frac{N_1}{l}I_1[/latex]

M = [latex]\frac{\phi_2}{I_1} = \frac{N_2A\mu_oN_1I_1}{I_1l} = \frac{N_1N_2\mu_oA}{l}[/latex]

[latex]\therefore M = \frac{N_1N_2\mu_oA}{l}[/latex]

Transformer

It is a device which converts high AC voltage (low current) into low AC voltage (high current) and vice versa. There are two types of transformers i.e.:

i. Step Up transformer

ii. Step down transformer

i. Step of transformer:

The Transformer with converts low current or low AC voltage to high AC voltage or high current is called Step Up transformer.

ii. Step down Transformer:

The Transformer which converts high current or high AC voltage to low AC voltage or low current is called step down transformer.

Structure and working

When a magnetic flux is linked with a coil changes emf is induced in the nearby coil. It consists of two coils, primary and secondary of insulated copper wire on the laminated soft iron core. The soft iron core consists of thin strips of iron coated with vanish to insulate from each other. The primary coil is connected with the source of AC and the secondary is connected with the load.

When the current is passed through the primary coil, magnetic flux linked with the coil changes and emf is induced in the coil.

If θ is the flux linked with the primary coil and E_p is self – induction then the primary voltage E_p is given by:

E_p = −N_P [latex]\frac{d\theta}{dt}[/latex] …(i)

Where, N_p is the number of turns in the primary coil. If there is no loss of flux then the rate of flux linked with the secondary coil will be same.

Then the induced emf in secondary coil,

Or, E_S = N_s . [latex]\frac{d\theta}{dt}[/latex] …. (ii)

Where, N_S is the number of turns in secondary coil. Dividing equation (i) and (ii), we get,

Or, [latex]\frac{E_p}{E_s} = \frac{N_p}{N_s}[/latex]

Where, Np/Ns is called transformer ratio.

If N_P/N_S > 1 then E_P > E_S and the transformer is step down.

If N_P/N_S then E_S> E_P and the transformers in step up.

If L_p and L_S is the co – efficient of self – induction of primary and secondary coil respectively.

Then, [latex]\frac{L_p}{L_s} = \frac{N_p^2}{N_s^2}[/latex]      [L ∝ N²]

So, [latex]\frac{N_p}{N_s} = \sqrt{\frac{L_p}{L_s}}[/latex]

So, [latex]\frac{E_p}{E_s} = \sqrt{\frac{C_p}{L_s}}[/latex]

If C_P is the input current and I_s is the output current then, input = I_pE_p.

Similarly,

Output power = I_IE_S.

If there is no loss of energy then,

Input power = Output power.

Then,

I_pE_p = I_SE_S

Or, [latex]\frac{E_p}{E_s} = \frac{I_s}{I_p}[/latex]

Hence, the output voltage is increased, the output current will decrease and vice – versa. Therefore, for a long-distance transmission line, high voltage is used. If the secondary voltage is high then current will be below and hence the energy loss (I²Kt) due to resistance of wire is low.

Theory and working of an ac generator

 A.C. generator is an electronic mechanic which converts mechanical energy in to electrical energy. It works on the principle of electromagnetic induction i.e., when the coil is rotated in the uniform magnetic field then the emf is developed on the rotating coil.

Construction of A.C. generator

Armature: the rectangular coil (PQRS) consists large no. of turn which wounded the soft iron core is called armature.

Strong magnetic field: the rectangular coil PQRS is rotated in the strong magnetic field produce due to magnet N and S.

Slip ring: the rectangular coil PQRS are connected with two ring R₁ and R₂seperatly. The ring R₁and R₂rotated with the rectangular coil PQRS.

Brush: the two metallic brush B₁ and B₂ are fixed and connected with load through which output obtained.

Working

When the rectangular coil is rotated in the strong magnetic field then the magnetic line of force is cut and flux link changes then a/c to faraday’s law of electromagnetic emf is induced in the coil. The current flow out through the brush B₁ in one direction while in B₂ in other direction i.e., the direction of flow of current are in opposite to each half cycle. Similar process also occurring other half cycle. This phenomenon is repeated continuously.

Theory

Let the axis of the coil is perpendicular to the magnetic field and coil PQRS are rotated with constant angular velocity ‘ω’ as shown in figure.

Let [latex]\theta[/latex] be the angle normal to the coil and magnetic field B at any instant ‘t’ then we have

[latex]\theta[/latex] =ωt…………..(i)

The component of magnetic field to the plane of the coil is Bcosq then from equation (i) we have Bcosωt. Let, ‘A’ be the area of the coil then the magnetic flux

[latex]\phi[/latex] = BAcosωt

Since, the rectangular coil consisting large no. of turn say N then the magnetic flux due to N no. of coil is [latex]\phi[/latex] = BNAcosωt…………..(ii)

Differentiating the equation (ii) with respect time we get

E = [latex]\frac{d\phi}{dt}[/latex] = -BNAωsinωt………….(iii)

We also have from faraday’s law of electromagnetic induction

E = – [latex]\frac{d\phi}{dt}[/latex]………….(iv)

Now from eq. (iii) and (iv) we have

E = BNAωsinωt……..(v)

Let E_o = BNAω

Then from equation (v), we have,

E = E_osinωt………(vi)

Dividing the eq. (vi) by R we get I = I_osinωt.

Now the variation of induce emf of the coil with respect of the magnetic field are given below

[latex]\theta^o[/latex]	E=Eosinωt	Result
0	0	Plane of coil is B
90	Eo	Plane of coil is along B
180	0	Plane of coil is B
270	-Eo	Plane of coil is along B
360	0	Plane of coil is B

This table show that the E increase from zero to maximum and decrease from maximum to minimum.

Theory and working of a Dc generator

A DC generator converts the mechanical energy in to the electrical energy. It is based on the principle of the electromagnetic induction.

Construction

It consists of the split rings called commutator C₁ and C₂, the rectangular coil (PQRS) consists large no. of turn which wounded the soft iron core is called armature, field magnets and the brushes.

Working

When the armature rotates, commutator also rotates with it and alternatively come in contact with the brushes B₁ and B₂. When the coil is horizontal as in figure above, the commutator C₁ connects with the arm RS of the armature to the end A and the commutator C₂ connects with the arm PQ of the armature to the end B of the load. When the armature moves anti-clockwise, the arm PQ moves upward and the arm RS moves downward. An emf. is induced in the armature and current flows in the load (RL) from A to B in accordance with Fleming’s Right-Hand Rule.

When a half revolution is completed, the position of the armature is reversed. The side PQ of the coil is close to the N pole of the magnet and side RS is close e to the S-pole of the magnet. As the armature is rotating, the current in it flow in opposite direction. However, at this time the commutator C₁ and C₂ have also changed the position as they rotate with the armature. As a result, the commutator C₂ connects the arm PQ to the end A of the load while the commutator C₁ connects the arm SR to the end B of the load. Then, current passes through the load again from A to B.

Energy loss in transformer

The efficiency of a transformer is always less than 100% because of the following loss of energies.

i) Loss in copper coils

Some of the input power is lost in the primary and secondary coils when current passes through them in the form of heat. A thick copper wire of low resistance is used particularly for the coil carrying high current at low voltage so that the power loss can be minimised.

ii) Lost due to Eddy Currents

Due to changes in the flux through iron core, the current is produced and these currents are called Eddy Currents, which cause a loss of power in the form of heat. The iron core is laminated cutting across the part of any induced current i.e., thin sheets of laminated iron core are used. The high resistance of the sheets reduces the Eddy Currents.

iii) Hysteresis loss

During magnetization and demagnetization of the iron core, the magnetization lags behind the magnetizing field. This phenomenon is called magnetic hysteresis and causes the loss of input power. The core is made of very soft iron core which has small hysteresis loop so that there will be less energy loss.

iv) Lost due to flux linkage

Some magnetic flux may not link the coils and the core. Due to this, all input powers may not be received as the output power. This loss can be reduced by winding the secondary coil on top of the primary coil and making the iron core in the form of a closed loop.

Energy is stored in an inductor

When the current in the coil changes a back emf is developed across the coil. Some work has to be done to overcome the back emf and to keep the current steady in the coil. This work done against the back emf is stored in the coil in the form of magnetic energy.

The magnitude of back emf produced in the coil having inductance (L) is given by:

E = L[latex]\frac{dI}{dt}[/latex]

If I be the instantaneous current, the power used to over the back emf is:

P = EI

                = L[latex]\frac{dI}{dt}I[/latex]

                = LI[latex]\frac{dI}{dt}[/latex]

The total workdone when the current change from I = 0 to I = I_o is:

W = [latex]\int_{I = 0}^{I=I_o}Pdt[/latex]

                = [latex]\int_{I=0}^{I=I_o}LI\frac{dI}{dt}dt[/latex]

                = L[latex]\int_{I=0}^{I=I_o}IdI[/latex]

                = L[latex][\frac{I^2}{2}]_{I=0}^{I=I_0}[/latex]

                = [latex]\frac{L}{2}I_o^2[/latex]

This workdone is stored in the coil in the form of magnetic energy.

[latex]\therefore[/latex] Energy Stored (U_m) = [latex]\frac{1}{2}LI_o^2[/latex]