In this article, we shall study the rotation of coil in uniform magnetic field, the concept of alternating current.

The principle of Electrical Generator:

Consider a circular coil of ‘n’ turns of each area A placed with its plane perpendicular to a uniform magnetic field of induction B. At time t = 0, the plane of the coil is normal to field. Therefore, the magnetic flux passing through it is given by

ø = nAB.

Where n = number of turns

A = Area of the coil

B = Uniform magnetic induction

The coil is now rotated with constant angular speed ‘ω’ about a diameter. In a time ‘t’, the plane of the coil makes an angle θ with the initial position. Therefore, the angle traced by the coil in time ‘t’ is given by

θ = ωt

To find the new flux, the magnetic induction is resolved into two mutually perpendicular components. The component B sin θ is along the plane of the coil and hence it does not contribute flux. However, component B cos θ is perpendicular to the plane of the coil. Therefore, the flux through the coil at time t is given by

ø =  nAB cos θ= nAB cos ωt

Therefore the induced e.m.f. in the coil is given by

Also ω= 2 π f, where f is the frequency of rotation.

Where e = instantaneous induced e.m.f.

e₀  =  peak e.m.f.

n  = number of turns of the coil

A  = Area of the coil

B = Uniform magnetic induction

Since sin ωt changes from +1 to -1, the e.m.f. induced changes periodically. The e.m.f. given by the above equation i.e. e = e₀ sinωt is known as alternating e.m.f.

When        sin ωt = ±1

e = e₀ (±1) = ± e₀ = Peak e.m.f.

The peak value of induced e.m.f. (e) is defined as the maximum value of alternating e.m.f.

Peak e.m.f.  = e₀ =  2πfnAB

Graphically an alternating e.m.f. e = e₀ sin ωt is as shown. The graph shows that the induced e.m.f. changes its magnitude and direction periodically.

Simple A.C. Circuit With Resistance Only (Alternating Current):

Consider an A.C. Circuit consisting of a resistance R connected to a source of e.m.f.,

e = e₀ sin ωt.

By Ohm’s law, the current through the circuit is given by,

Peak current (I0) is defined as the maximum value of alternating current.

Comparing the equations of A.C. e.m.f. and A.C. current it is obvious that they are in phase.

Terminology of AC Circuits:

Peak value of alternating e.m.f.:

The magnitude of the maximum value of alternating e.m.f. is called the peak value of e.m.f. It is denoted by e₀.

Peak Value of Alternating Current:

The magnitude of the maximum value of alternating current is called the peak value of the current. It is denoted by I₀.

Average value of Alternating e.m.f.:

The constant value shown by a D.C. voltmeter during one period of alternating e.m.f. (T) is called the average value of alternating e.m.f.  It is denoted by e_av. The average alternating e.m.f. is zero.

Average Value of Alternating Current:

The constant value shown by a D.C. ammeter during one period of alternating current (T) is called the average value of alternating current. It is denoted by I_av.  As the average alternating e.m.f. is zero the average alternating current in the circuit is also zero.

R.M.S. Value of e.m.f.:

The root mean square (r.m.s.) value of e.m.f. is that steady e.m.f., which is to be applied across the given resistor to produce the same amount of heat in a given time as it is produced by given alternating e.m.f. It is denoted by e_r.m.s.

R.M.S. Value of Current:

The root mean square (r.m.s.) value of current is that steady current which is passed through given resistor to produce the same amount of heat in a given time as it is produced by given alternating current. It is denoted by I_r.m.s..

Expression for Power in AC Circuit:

An alternating current of peak value ‘I’ is equivalent to a constant direct current of value I₀/√2. This is known as the effective value, virtual value or r.m.s. value of current.

I_r.m.s. = I₀/√2

The power generated with the D.C. circuit containing resistance ‘R’ is given by

P  =  I²R

Therefore, the power generated in an A.C. circuit through which current I_r.m.s. is flowing through resistance ‘R’ is given by

But according to Ohms Law,

This is an expression for the power dissipated in A.C. circuit.

Science > Physics > Electromagnetic Induction > Rotation of a Coil in Uniform Magnetic Field

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In this article, we shall study the concept of self and mutual induction and also construction and working of the transformer.

Concept of Self Induction:

When a current flowing through a coil changes the magnetic flux linked with the coil itself changes. Due to which an induced e.m.f. is generated in the same coil. This property of coil of producing induced e.m.f. in a coil due to change in the current through the same coil is called self-induction.

Let ø be the magnetic flux linked with the coil. At any instant, this flux is directly proportional to the current in the coil.

Where L = Coefficient of self-induction

Differentiating both sides w.r.t. time t, we get

By Faraday’s law of electromagnetic induction we have

Thus induced e.m.f. in the coil is given by

Considering magnitude only we have

Thus, the self-inductance of a coil can be defined as the induced e.m.f. in the coil itself due to a unit rate of change of current in the same coil. The S.I. unit of self-inductance is henry (H).

Thus the self-inductance of a coil is one henry, if an e.m.f. of 1 volt is induced in the coil when the current passing through the same coil changes at the rate of 1 ampere per second.

Concept of Mutual Induction:

When two coils are placed very close to each other if the current is passed through one coil (primary) a magnetic field is set around this coil. Now the second coil (secondary)  is kept in the magnetic field created by the primary coil. Thus magnetic flux is linked with the secondary.

When a current flowing through primary changes the magnetic flux linked with the secondary changes. Due to which an induced e.m.f. is generated in the secondary coil. This property of producing induced e.m.f. in secondary due to change in the current through the primary is called mutual induction.

Let ø_s be the magnetic flux linked with secondary and i_p be the current through the primary. At any instant, the magnetic flux linked with secondary is directly proportional to the current in the primary.

Where M = Coefficient of mutual induction

Differentiating both sides w.r.t. time t, we get

By Faraday’s law of electromagnetic induction we have

Thus induced e.m.f. in the secondary coil is given by

Thus, the mutual inductance of coil can be defined as the induced e.m.f. in the secondary due to unit rate of change of current in the primary coil. The S.I. unit of mutual inductance is henry (H).

Thus the mutual inductance of a coil is one henry, if an e.m.f. of 1 volt is induced in the secondary coil when the current passing through the primary coil changes at the rate of 1 ampere per second.

Transformer:

The transformer is a device which converts the alternating voltage from one value to another. It works on the principle of mutual induction.

Construction:

A transformer consists of two sets of a coil, primary coil and secondary coil, which are well insulated from each other. The primary coil is input coil and the secondary coil is output coil. The two coils are wound on a soft iron core either one above other or on the separate arm.

Assumptions:

• The primary current and resistance of the primary current are small.
• Same magnetic flux links both the primary and secondary.

Working:

When an alternating source of voltage say V_P is applied across the primary. it creates a changing magnetic flux which is linked to the secondary coil. The value of the flux linked with the coils depends on the number of turns of both the coils. Let ø be the magnetic flux per unit turn linked with the coil at a time ‘t’.  Let the rate of change of flux liked with the coil be dø/dt.

The e.m.f. induced in the secondary at the instant is given by

Where Ns = number of turns of the secondary coil.

The back e.m.f. induced in the secondary at the instant is given by

Where, N_P = Number of turns of the primary coil.

For open-circuit E_S =V_S and E_P = V_P

This equation is known as the equation of the transformer.

For ideal transformer,

Power Input = Power Output

From equations (1) and (3) we get

Types of Transformers:

Depending upon the ratio of the output voltage to the input voltage, transformers are classified into two types.

Step-up transformer:

When the output voltage is greater than the input voltage (VS > VP or NS > NP), the transformer is called a step-up transformer.

Step-down transformer:

When the output voltage is less than the input voltage (V_S < V_P or N_S < N_P), the transformer is called a step-down transformer.

Energy Losses in Transformer:

Flux leakage:

If the windings are. not proper one over the other or there are air gaps, then the magnetic flux due to primary never gets fully linked with the secondary. This loss can be minimized by winding the two windings one over the other carefully.

Copper Losses:

Copper wires used for windings, both the windings possess some resistance. Hence,  some energy will be lost in the form of heat given by I²R. This loss can be minimized by reducing resistance by using thick wires.

Eddy Current Losses:

The core of the transformer is made up of soft iron. The continuously changing magnetic flux produces eddy currents in it. Which tend to heat the core. Thus some energy is lost due to eddy currents in the core. These losses can be minimized by making core laminar.

Hysteresis Losses:

Iron has a tendency to retain residual magnetism even if the magnetizing field is removed. Thus for each cycle, some energy is lost in destroying the residual magnetism of the previous cycle. This loss cannot be minimized and it is irrecoverable

Science > Physics > Electromagnetic Induction > Concept of Self and Mutual Induction

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If a block of metal is moved in a magnetic field or kept in a changing magnetic field, the free electrons in the conductor experience force and begin to circulate. This gives rise to induced currents in closed circular paths and it is known as eddy current or Foucault currents. These induced eddy currents flow in the form of eddies in such a direction so as to oppose the motion of the conductor in the field.

Eddy currents produce a large amount of heat in the soft iron core of transformers, induction coil, electric motors, etc. This results in a decrease in the efficiency of such machines. However, eddy currents can be usefully employed in many electrical devices.

Uses of Eddy Currents:

Deadbeat Galvanometer:

The coil of a moving coil galvanometer which is used for measuring current is wound on a copper or aluminium frame. When a current flow, the coil gets deflected and as soon as the current stops the coil tends to oscillate about its equilibrium position. However, the eddy currents generated in the metal frame opposes the oscillatory motion. As a result, it to oscillate. This makes the moving coil galvanometer deadbeat.

Induction Motor:

The induction motor consists of a long metallic axle mounted lightly in a uniform magnetic field with its axis at a right angle to the field. If the magnetic field is rotated with respect to the stationary axle, eddy currents are developed in it. The eddy currents thus produced try to reduce the relative motion by rotating the axle in the same direction of rotation of the magnetic field. This is the principle of an induction motor.

Induction Furnace:

The heating effect of the eddy current is used for melting metal in an induction furnace. Eddy currents of large magnitude are produced field. The changes in the magnetic field are so rapid that very large eddy currents are generated and heat produced is sufficient to melt quickly. An induction furnace is used for producing alloys of different metals.

Electric brakes:

It is an efficient system of brakes employed in electric trains. The axle of a train is surrounded by a coaxial cylindrical drum. When the train is to be stopped, a strong magnetic field is applied to the rotating drum. This results in the generation of large eddy currents which oppose the relative motion between the stationary field and the axle. Thus the train slows down and comes to rest quickly and smoothly.

Speedometer:

In a speedometer, the magnet rotates inside a pivoted metallic drum. The speed of rotation depends on the speed of the vehicle. Eddy currents in the drum cause the magnet to rotate with the drum. It rotates through an angle which is proportional to the speed of the vehicle. A pointer attached to the magnet gives a reading of the speed of the vehicle on a calibrated scale.

Disadvantages of Eddy Currents:

• There is a major heat loss during cycling eddy currents due to friction in the magnetic circuit, especially where the core is saturated. Thus there is the loss of useful electrical energy in the form of heat.
• There is magnetic flux leakage.
• To avoid losses due to eddy current the core of induction coils and transformer is not constructed as a block but og number of thin strips.

Science > Physics > Electromagnetic Induction > Eddy Currents

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In this article, we shall study the concept of electromagnetic induction i.e. conversion of magnetic energy into electrical energy.

The phenomenon of producing an induced e.m.f. in a conductor or conducting coil due to changing magnetic flux or field is called electromagnetic induction. This phenomenon was discovered independently by Michael Faraday and Joseph Henry.

Magnetic Flux:

The total number of magnetic field line passing through a given area is called the magnetic flux through that area. It is denoted by symbol ‘ø’. Its S.I. unit is weber (Wb)

Magnetic Induction:

Magnetic induction is defined as magnetic flux per unit area. It is denoted by letter ‘B’. Mathematically, B = ø/A. its S.I. unit is weber per square metre Wb/m² or tesla (T).

Faradays Coil Magnet Experiment:

A coil of insulated copper wire is connected to a centre zero galvanometer. A bar magnet is held near the coil. Initially, the galvanometer shows zero deflection. When the magnet is moved towards the coil, a momentary deflection is observed towards one side. When the magnet comes to rest, the deflection also becomes zero. If the speed of motion of the magnet is increased, the deflection also increased.

When the magnet is moved away from the coil, a momentary deflection is observed in the opposite direction. This direction is also observed as long as the magnet is in motion.

Faradays Coil and Coil Experiment:

Two coils of insulated copper wire a primary coil P and secondary coil S kept very near to each other. The coil P is connected to a cell and tap key K. The other coil S is connected to a centre zero galvanometer G.

By pressing the tap key, when a current is passed through P. the galvanometer shows a momentary deflection towards one side. When the current in P is switched off, a momentary deflection is observed in the galvanometer in the opposite direction.

Faraday’s Laws of Electromagnetism:

Faraday’s First Law of Electromagnetic Induction:

Whenever there is a change in magnetic flux linked with a coil, an e.m.f is induced in the coil.

Faraday’s Second Law of Electromagnetic Induction:

The magnitude of e.m.f. induced in a coil is directly proportional to the rate of change of magnetic flux linked with it.

Explanation:

If dø is the change in magnetic flux in time dt, then the induced e.m.f.

In S.I. system, e is measured in volts, dø in weber and dt in seconds

This formula gives the magnitude of induced e.m.f.  The direction is given by Lenz’s law.

Lenz’s Law of Electromagnetic Induction:

Lenz’s law states that the direction of e.m.f. induced in a coil is son as to oppose the change in magnetic flux that causes it.

Applying Lenz’s Law, the Faraday’s Law can be expressed as

Negative sign shows that the induced e.m.f. opposes the rate of change of flux.

Explanation:

Lenz’s Law is a direct consequence of the principle of conservation of energy. When N-pole of a magnet approaches a coil, the induced current flows in such a direction that its face attains N-polarity. Hence the magnet is repelled. Therefore, work has to be done in moving the magnet against repulsion, in order to maintain induced current. In other words, there is a conversion of mechanical energy into electrical energy.

Proof of Faraday’s Laws of Electromagnetic Induction:

Consider a rectangular loop of conducting wire PQRS of width ‘l’ be arranged with its plane in the plane of the page.  A uniform magnetic field is applied perpendicular to the loop i.e. perpendicular and towards the page.  The magnetic field arranged in such a way that a part loop is in the field.  If ‘x’ is the width of the loop in the field, then the area of the loop in the field is

A = l . x

Also, the magnetic flux passing through this area due to the field is

ø  = BA

∴  ø  = B . l . x

The loop is now pulled out of the field towards the right with a uniform velocity v.  The flux linked with the loop i.e. B l x undergoes a change.  If dx is the displacement of the loop in time dt, then

But, dx/dt = v = the velocity of the loop

This shows that the flux linked with the loop changes at a constant rate.  Therefore by Faraday’s Law an induced e.m.f. is produced in the loop.  The induced e.m.f. sets up an induced current which flows through the loop in a clockwise direction. A conductor which carries a current in a magnetic field experiences a force given by

The direction of the force is given by Fleming’s left-hand rule. Therefore the side PS of length ‘l’ experiences a force acting towards left is given by

F = I l B sin 90° = F = I l B

This force is unbalanced.

However, the force acting on side PQ and SR are of the same magnitude but opposite in direction. Therefore they balance each other.

When the loop is moved towards the right through distance ‘dx’ in time ‘dt’ in the ‘dt’, the work done against force is given by

dW =  -Fdx

Negative sign shows that displacement is opposite to force.   Substituting for F

dW =  – I l B dx

Therefore, the rate of doing work is given by

This is the mechanical power supplied to the loop the mechanical power generates an equal amount of electrical power by maintaining an induced current. If ‘I’ is the induced current due to an induced e.m.f. ‘e’, then

Electrical power = e i

According to the principle of conservation of energy

Electrical power = Mechanical power

e i  =  -Bi l v

e   = -B l v   …. (2)

Substituting from equation (1) in equation (2), we get

Thus Faraday’s law of electromagnetic induction is proved

Fleming’s Right-Hand Rule:

Stretch the forefinger, middle finger and thumb of right hand mutually perpendicular to each other. If the forefinger points the direction of the magnetic field, and the thumb shows the direction of motion of the conductor then the middle finger shows the direction of induced e.m.f.

Science > Physics > Electromagnetic Induction > Introduction to Electromagnetic Induction

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