Electromagnetic Induction

23. Electromagnetic Induction

Content

23.1 Laws of electromagnetic induction

Learning Outcomes

(a) define magnetic flux and the weber.

(b) recall and solve problems using Φ = BA.

(c) define magnetic flux linkage.

* (d) infer from appropriate experiments on electromagnetic induction:

(i) that a changing magnetic flux can induce an e.m.f. in a circuit,

(ii) that the direction of the induced e.m.f. opposes the change producing it,

(iii) the factors affecting the magnitude of the induced e.m.f.

(e) recall and solve problems using Faraday's law of electromagnetic induction and

Lenz's law.

(f) explain simple applications of electromagnetic induction.

Magnetic flux

• The pattern of all magnetic lines should be continuous

• Early experimenters thought that there was a flow of something along these lines. This gave rise to the idea of magnetic ‘flux’ which means ‘flow’

• Magnetic flux density can be considered as the number of lines of magnetic force per unit area of an area at right angles to the lines

• Magnetic flux Φ (phi) can be thought of as the total number of lines

• So, magnetic flux is the product of the magnetic flux density and the area normal to the lines of flux

• For a uniform magnetic field of flux density B which makes an angle θ with an area A, the magnetic flux is given by the expression Φ = BA sin θ

• The unit of magnetic flux is weber (Wb). 1 Weber is equal to 1 tesla m2

• The concept of magnetic flux is used frequently when studying electromagnetic induction

Electromagnetic induction

• The link between electric current and magnetic field was discovered by Oersted in 1820. In 1831 Henry in US and Faraday in England demonstrated that an e.m.f. could be induced by a magnetic field. This effect is called electromagnetic induction

• Apparatus to demonstrate electromagnetic induction – pg 336 fig 12.35 Physics by Chris Mee

• Following observations were made:

– An e.m.f is induced when

• The wire is moved through the magnetic field across the face of the pole-pieces

• The magnet is moved so that the wire passes across the face of the pole-pieces

– An e.m.f. is not induced when

• The wire is held stationary between the pole pieces

• The magnet is moved so that the pole pieces move along the length of the wire

• The wire moves lengthways so that it does not change its position between the poles of the magnet

Conclusions to be drawn

• That an e.m.f. is induced whenever lines of magnetic flux are cut. The cutting may be caused by a movement of either the wire or the magnet

• The magnitude of the e.m.f. has been found to increase

– As the speed at which the wire is moved increases

– As the speed at which the magnet is moved increases

– If the wire is made into a loop with several turns

– As the number of turns on the loop increases

• Therefore the magnitude of the induced e.m.f. depends on the rate at which the magnetic flux lines are cut

• The term magnetic flux linkage is introduced to take into account 2 factors whereby change in magnetic flux linkage Δ(NΦ) = N ΔΦ

• All the above observations are summarised in Faraday’s Law of Electromagnetic Induction as, the e.m.f. induced is proportional to the rate of change of magnetic flux linkage

Now what about the direction of the e.m.f ?

• It has been noticed that the direction of the induced e.m.f. changes and that the direction is dependent on the direction in which the magnetic flux lines are being cut

• This can be determined using Fleming’s RH rule

– First finger is the direction of the magnetic field

– Thumb is in the direction of motion

– Second finger gives the direction of the induced e.m.f. or current

• Try pg 338 fig 12.38 Physics by Chris Mee

• Conclusion from the above fig considering the law of conservation of energy, is that

– an electric current is a form of energy and this energy must have been converted from some other form.

– movement of the wire against the electromagnetic force means that work has been done in overcoming this force and it is this force that is seen as electrical energy

Lenz’s Law

• Lenz’s law states that the direction of the induced e.m.f. is such as to cause effects to oppose the change producing it

• Faraday’s Law of Electromagnetic Induction and Lenz’s Law may be summarised using the equation,

E = - d(NΦ)/dt

where E is the e.m.f. induced by a rate of change of flux linkage of d(NΦ)/dt. The minus sign indicates that the induced e.m.f. causes effects to oppose the change producing it.

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Demonstation of convertion from mechanical energy to electrical energy

• Pg 339 fig 12.39 Physics by Chris Mee of spinning disc in a permanent magnetic field

• As the disc spins it cuts through the flux lines of the magnet causing e.m.f. to be induced in the disc

• But, because the rate of cutting varies from one part of the disc to another as the radius varies, the e.m.f. will have different magnitudes in different regions

• This e.m.f. induces currents of varying magnitude and direction in the disc called eddy currents which cause heating in the disc and dissipation of energy of rotation of the disc referred to as eddy current damping

• If the permanent magnet were to be replaced by an electromagnet, the spinning disc will slow down whenever there is a current in the electromagnet. This is the principle behind electromagnetic braking used in high speed trains

Example

• The uniform flux density between the poles of a magnet is 0.080 T. A small coil of area of cross-section 6.5 cm2 has 250 turns and is placed with its plane normal to the magnetic field. The coil is withdrawn from the field in a time of 0.26 s.

Determine:

a) the magnetic flux through the coil when it is between the poles of the magnet

b) the change in magnetic flux linkage when the coil is removed from the field

c) the average e.m.f. induced in the coil whilst it is being withdrawn

Solution

a) magnetic flux Φ = BA sin θ = 5.2 x 10-5 Wb

b) change in flux linkage = (NΦ)final – (NΦ)initial

= 0 – (250 x 5.2 x 10-5) = -1.3 x 10-2 Wb

c) induced e.m.f. = change in flux linkage/time taken = 0.050 V

The principle of the transformer

• In a transformer, an alternating current in one coil induces an alternating e.m.f. in the second coil.

• The ratio of the output (secondary) e.m.f. to the applied (primary) e.m.f. equals the ratio of the number of turns on the secondary coil to the number of turns on the primary coil

Vs/Vp = Ns/Np

• gtb

Transformer considerations

• For a transformer which is 100% efficient (i.e. ideal transformer),

input power = output power

VpIp = VsIs hence Ns/Np = Vs/Vp = Ip/Is

• In practice a transformer will have losses, some sources of losses being

– Loss of magnetic flux between the primary and secondary coils

– Resistive heating in the primary and secondary coils

– Heating of the core due to eddy currents

– Heating of the core due to repeated magnetisation and demagnetisation

• Although the use of soft iron reduces heating due to magnetisation and demagnetisation, due to its use to obtain high magnetic flux linkage, eddy currents cannot be prevented

• This is reduced by laminating the core i.e. building the core from thin strips of soft iron which are electrically insulated from one another

Transmission of electrical energy

• Generating plants are often situated a long way from the cities they serve

• This means that whatever the means of generation, it is likely to be transmitted over long distances

• Thus there is power loss due to the I2R effects

• These effects can be reduced if the power is transmitted at a high voltage using transformers as step-up and step-down transformers

• Transformers used in transmission networks have very high efficiencies often over 98%

Example

• A town several kilometres from an electricty generating station, requires 120 kW of power on average. The total resistance of the transmission lines is 0.40 ohms. Calculate the power loss if the transmission is made at a voltage of (a) 240 V (b) 24 kV

Solution

Current using 240 V is calculated to be 500 A, and using 24 kV is 5.0 A

Using power loss P = I2R,

for (a) we obtain power loss is 100 kW, i.e. more than 80% lost

(b) we obtain power loss is 10 W, i.e. less than one-hundredth of 1% of the power requirement

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Summary of laws of electromagnetic induction

Faraday's Law states that the magnitude of the emf induced is directly proportional to the rate at which the conductor cuts the magnetic field lines i.e. rate of change of magnetic flux

Lenz's Law states that the induced current always flows in a direction so that it opposes the change which is causing it

when the N pole of a magnet moves towards one end of a solenoid, that end will become a magnetic N pole to oppose the motion of the magnet

when the N pole is pulled away from the end of a solenoid, that end will become a magnetic S pole so as to oppose the motion of the magnet

The production of the electrical energy in the form of an induced current is actually the work done or a transformation from the mechanical energy required to go against the motion (conservation of energy).

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Application of electromagnetic induction

Electromagnetic induction is used to generate electricity in DC & AC generators

A small generator used to produce light in a bicycle lamp is called a dynamo

The huge generator that supplies electricity to our homes is called an alternator

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DC Generator

A generator is a device used for the generation of current through the rotation of rectangular conducting coils (armature) in a magnetic field.

Together with a commutator and carbon brushes to convert AC to DC in one direction only i.e. rectification

Reverse of a DC motor without the battery

When the coil is horizontal, the e.m.f. is a maximum. When the coil is vertical, the e.m.f. is zero.

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AC generator - Alternators

Alternators are generators of alternating current.

Similar to a DC generator but with the commutator replaced by 2 copper slip rings

The magnetic field is provided by an electromagnet which has a core of cast steel and is fed with direct current from a separate d.c. generator.

The armature is wound on an iron core which is shaped so that it can turn within the pole-pieces of the field magnet.

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A moving-coil microphone

Actually the reverse of a moving coil loudspeaker

Sound waves travel towards the microphone setting the diaphragm vibrating

These vibrations cause a small coil to move forward and backwards through the magnetic field which induces a small AC in the coil

The AC is amplified and sent into the loudspeaker

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A metal detector

Consists of 2 coils with one coil carrying current and the other coil in the magnetic field

When the detector is near a ferromagnetic material, the strength of the magnetic field through the second coil is increased

This increase induces current in the second coil which is detected

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Speedometers

Car speedometers work through the action of electromagnetic induction.

A permanent magnet, which rotates with the car's wheels, is suspended inside an aluminium drum. The drum can also rotate, though it is held back by a spring so that greater rotations require greater torque on the drum.

The needle of the speedometer is attached to the drum.

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Magnetostatics

Magnetostatics is the branch of electromagnetics dealing with the effects of electric charges in steady motion (i.e steady current or DC).

The fundamental law of magnetostatics is Ampere’s law of force. Ampere’s law of force is analogous to Coulomb’s law in electrostatics. Ampere’s law of force gives the magnetic force between two current carrying

circuits in an otherwise empty universe. Ampere’s law of force involves complete circuits since current must flow in

closed loops.

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Force Between Two Conductors

Parallel conductors carrying currents in the same direction attract each other

Parallel conductors carrying currents in the opposite directions repel each other

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Ampere’s Law of Force

Experimental facts: The magnitude of the force is inversely proportional to the

distance squared. The magnitude of the force is proportional to the product of the

currents carried by the two wires. If the 2 currents are l,m then F= klm/r2 where k is a constant

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Magnetic Effects of Electrons - Orbits

An individual atom acts like a magnet because of the motion of the electrons about the nucleus

each electron circles the atom once in about every 10-16 seconds

this would produce a current of 1.6 mA and a magnetic field of about 20 T at the center of the circular path

However, the magnetic field produced by one electron in an atom is often cancelled by an oppositely revolving electron in the same atom

The net result is that the magnetic effect produced by electrons orbiting the nucleus is either zero or very small for most materials

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Magnetic Effects of Electrons - Spins

Electrons also have spin The classical model is to consider

the electrons to spin like a top It is actually a quantum effect

The field due to the spinning is generally stronger than the field due to the orbital motion

Electrons usually pair up with their spins opposite each other, so their fields cancel each other

That is why most materials are not naturally magnetic

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Magnetic Effects of Electrons - Domains

Large groups of atoms in which the spins are aligned are called domains

When an external field is applied, the domains that are aligned with the field tend to grow at the expense of the others

This causes the material to become magnetized

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What now & next?

Cyclotron - developed in 1931 by E. O. Lawrence and M. S. Livingston at UC Berkeley, uses electric fields to accelerate and magnetic fields to guide particles at very high speeds

LHC – Large Hadron Collider

MRI

MagLev trains