Tutorial 8 - Electromagnetic Induction

Learning Objectives

To understand how a voltage is induced when flux changes.

To study the laws that explain the process.

To link electromagnetic induction with reluctance.

Key Questions

How do we demonstrate electromagnetic induction?

How does a generator work?

How is induction linked to inductance?

Demonstrating Electromagnetic Induction

If we pass a current in a wire in a magnetic field, we know that the wire will move.  It is therefore reasonable to suppose that if we move the wire in a magnetic field, and the wire is connected to an outside circuit, a voltage and current are induced.  If the wire is not connected, a voltage only is induced.  Consider this demonstration:

If we move the magnet parallel to the wire, the galvanometer hardly responds.  However, if we move the magnet across the wire, then we see a definite reading on the galvanometer.  The current (and voltage) induced on a single wire is rather small, but is increased by having more turns of wire.  For any voltage to be induced, we must move the magnet.  We call this voltage the induced electromotive force.  The e.m.f. is not a force at all; it’s a voltage.  It is often given the code E, a fancy letter ‘E’.

The direction of the current in the complete circuit can be worked out using Fleming’s Right Hand Rule.

Faraday’s Law and Lenz’s Law are two important rules that govern this effect.  Faraday’s Law is a formal definition of the effect:

The induced e.m.f. across a conductor is equal to the rate at which flux is cut.

Lenz’s Law says:

The direction of any induced current is such as to oppose the flux change that caused it.

The induced e.m.f. sets up a current that would oppose the force that is pulling the wire.  If the force were to assist the motion, we would get acceleration, and an increase in kinetic energy.  This would break the Law of Conservation of Energy.  In other words, we cannot get something for nothing.

Lenz’s Law is important in motors and generators.  As a motor speeds up, it acts like a generator to produce a back e.m.f. to oppose the current flowing in the motor.  Therefore the current through a fast running motor is quite small.  When it is running slowly, a big current flows.

The effect that we have seen is summed up in the relationship:

[N - number of turns; E – e.m.f (V); dF/dt - rate of change in flux (Wb s-1 )]

 Worked Example A single turn of wire of cross-sectional area 5.0 cm2 is at 90o to a magnetic field of 0.02 T, which is reduced to 0 in 10 s at a steady rate.  What is the e.m.f. induced? Answer Two formulae to use:  F = BA           and      E = - NDF                                                                               Dt               We need to work out the flux:                           F = BA = 0.02 T × 5 × 10-4 m2 = 1 ×  10-5 Wb               Now we can work out the e.m.f:                           E = - NDF = 1× 1 ×10-5 Wb = 1 ×10-5  V                          Dt           10 s

 A search coil has 2500 turns and an area of 1.5 ´ 10-4 m2. It is placed between the poles of a large horseshoe magnet.  It is rapidly pulled out of the field in a time of 0.30 s.  A data-logger records an average value for the e.m.f of 0.75 V.  What is the flux density between the poles of the magnet?

### EMF and speed

Consider a wire on two rails, w metres apart, travelling a distance l metres at a velocity of v metres per second in a time of t seconds.

Since F = BA, we can write:

Since A = lw and l = vt, we can change the equation:

And then:

Since the time interval is the same, all the time terms cancel out:

The minus sign is there to satisfy Lenz’s Law.

Questions on this involve the rather fatuous example of aeroplanes flying through the Earth’s magnetic field.  An EMF is induced due to the vertical component of the Earth’s magnetic field.  It’s no damned good to the aeroplane, which would have to fly along fixed rails to generate anything useful – a sky-train?  Here is one such question for you to try.

 An aeroplane is flying at a speed of 200 m s-1 above a geographical area where the vertical component of the Earth's magnetic field is 4.5 × 10-5 T.  Its wingspan is 30 m.  Calculate the EMF.

Dropping Magnets through Solenoids

We can move a wire through a magnet to get an e.m.f, OR we can move a magnet through a coil of wire.  It doesn’t matter, as long as there is relative movement.

When the magnet is moved towards the solenoid, we get a voltage induced according to Faraday’s Law.  However, Lenz’s Law tells us that the direction of the current in the solenoid will make a magnet that will oppose the movement of the bar magnet.  In other words, a North pole is induced, and it will try to repel the magnet.

The current goes anticlockwise.  If you put arrows on the ends of the N (for North), you will see it going anticlockwise.

Now let’s think about the magnet in the middle, as shown in the next diagram:

In this instance we see three different things:

• At the top, we see a North pole.  This is because the South pole of the magnet is moving downwards.  Lenz’s Law tells us that the direction of the current opposes the movement, so the current is trying to attract the magnet back to it.

• In the middle, the field lines are parallel to the direction of movement.  There is no change in flux, so no e.m.f. is induced.

• At the bottom, a North pole is induced to repel the North pole of the magnet.

You can see that the induced voltages are going in the opposite direction, so there is zero overall e.m.f. at this point.

Now the magnet is coming out at the bottom:

In this case, the South pole of the magnet is inducing a North pole at the bottom of the coil, which is trying to attract it back.  At the other end of the coil, there is a South pole induced.

The graph produced by the data-logger looks like this:

Notice that the second peak has a higher (negative) value.  This is because the magnet is accelerating, so its downwards velocity is changing all the time.   If the coil is connected to a voltmeter, the acceleration of the magnet will be very close to 9.8 m s-2, because the current will be very small, and the opposition to the movement will be tiny.  However, if there is a low resistance in the external circuit, there will be a noticeable effect on the acceleration.

The area under the graph is the change in flux.

Revision of Some Definitions

Flux is defined as product between flux density and area.

It is measured in Weber (Wb)

F = BA

Flux density is the same as the concentration of field lines.

It is measured in Tesla (T)

Flux linkage is the product of the flux and number of turns (NF).  The unit is Weber turns.

 How much flux is on the face of a face magnet if it measures 2 cm by 5 cm and the flux density is 20 mT?

The old-fashioned turntable and Vinyl LP records are making a comeback after many years.

The picture above shows a moving magnet pick-up cartridge on Hi-Fi record deck.  A tiny magnet is attached to a stylus made from an industrial diamond that tracks the grooves in a vinyl record.  The voltages generated in tiny coils mounted around the magnet.  These are amplified in an amplifier.  This type of cartridge is called a moving magnet cartridge.  It is possible to have a cartridge that employs a coil that is connected to the stylus.  It is vibrated in a strong magnetic field.  Audiophiles often say that the moving coil cartridge gives a better sound.

### The Generator Effect

The voltage of a simple generator is given by the formula:

• N - the number of turns

• v - the linear speed ( ms-1)

• w - width (m)

• B - the magnetic field strength (T)

• E - emf (V)

### Derivation

The diagram above is a simple alternating current generator.  It consists of a coil of N turns, radius r and length l spinning in a magnetic field of flux density B.  Its angular velocity is w radians per second.  The motion is, of course, circular.

We can use the fact that circular motion and simple harmonic motion are closely linked.  Simple harmonic motion describes the movement of an oscillating object, which means an object that moves to-and-fro in a period movement.  Examples include a swinging pendulum or an object bouncing up and down on a spring.   So we can use the equation for displacement:

x = A cos wt

Here the x term refers to the displacement from a fixed point.  We need to be aware of the term w.  It is the rate of rotation or the angular velocity.  In other words it is the angle turned per second.  It is measured in radians per second (rad/s).  “Radians” is a dimensionless unit, and some texts miss it out altogether.  In these notes I will always use the short-hand “rad”.

The symbol used for the Physics code, w, is omega, a Greek lower case long ‘o’ (ō).

We can make the point at which the coil is vertical the “rest position”. The maximum amplitude is r as in the diagram.

So our equation becomes x = r cos wt

Since the coil is rotating at a constant angular velocity, w, the linear speed of the edge of the coil is given as v = wr.  From SHM we can say:

v = -rw sin wt

EMF (zero current terminal voltage) and linear speed are linked by this equation:

• N - the number of turns

• v - the linear speed ( ms-1)

• w - width (m)

• B - the magnetic field strength (T)

• E - emf (V)

We can combine the two equations above to give:

The minus signs disappear and we can also say that A = rl.

Therefore:

This explains why the output of an AC generator is sinusoidal. The output of an AC generator is a sine wave.

The maximum value of the e.m.f. is when sin wt = 1.  Therefore:

E0 = BANw

 A simple generator consists of 50 turns of wire mounted in a rectangular coil of radius 1.5 cm and length 4 cm.  It is rotating at 20 revolutions every second.  Calculate the maximum emf.

The AC generator here is inefficient, but could be made more efficient by changing the shape of the magnet, and wrapping the coil around soft iron.  Practical AC generators have a rotating magnet (rotor) which passes between stationary coils (stator).  The alternating e.m.f is induced in these coils.  The machine is called an alternator.

Power station generators are massive.  They have a rotor that is connected to its own generator, called an exciter.  The stator coils are placed at 120 degrees to each other to allow 3-phase AC to be generated.  The voltage is 25 000 V, while currents of 15 000 A are common.  The whole machine is cooled by hydrogen gas, which has a particularly high specific heat capacity.  The picture above shows a power station alternator.

The generator is actually in the rectangular box on the right.  To the left is the low-pressure turbine.

Turbines and generators are so big that when the machine is off, the shaft has to be rotated slowly, otherwise it would sag and go out of shape (which is not a good idea).  It is driven by a barring motor.

Inductance

So far in this tutorial we have looked at how an EMF can be induced when there is relative movement between a wire and magnetic field.  We actually don't need to have movement, as long as flux changes.  If flux doesn't change, then there is no induced EMF.

We have seen how coils make a magnetic field when a current flows through them.  They have inductance which occurs whenever a an EMF is induced by a change in flux linkage.  If the EMF is induced in the same circuit in which the current is changing, then that inductance is called self inductance.  For example the coils of an electric motor will have a self inductance.  The self-inductance is given the physics code L and the units are Henry (H).

If the emf is induced by a change in flux in an adjacent circuit, then we describe the property as mutual inductance and give it the physics code M.  The units remain as Henry (H).

The formal definition of Inductance is:

A circuit has an inductance of one Henry when an EMF of one volt is induced by a current changing at a rate of one ampere per second.

From Faraday's and Lenz's Laws we know:

From the definition above, we can write:

 An ignition coil is required to give out an EMF of 5000 V. The current it takes is 5.0 A and the current decays to zero in a time of 10.0 ms. Calculate the inductance.

In Tutorial 6, we saw how when a magnetic field is generated, there is an EMF that opposes the supply voltage.  It is dependent on the rate of change of the current.  Since currents cause magnetic fields in the first place, it is reasonable to say that the EMF also results from the change in magnetic flux.  We can write:

But that is not the whole story, for the number of turns of wire in the coil is also important.  Lenz's Law applies, so there is a minus sign in the equation to show that the induced EMF is in the opposite direction to the applied voltage.  The equation becomes:

The term NF is called the flux linkage, and has the units Weber turns.  Turns is dimensionless, so the unit in unit analysis is Weber (Wb).  Therefore the EMF is the rate of change of flux linkage.

We know that:

F = BA

So we can rewrite the equation as:

The area is constant.  Only the flux density changes.  We can work out flux density for a solenoid with a core of relative permeability mr using:

The terms m0, mr, and n (= N/l) are all constant for a particular solenoid, so only the current changes.  So we can write:

And that tidies up to:

It gives us an expression for inductance that takes into account the properties of the solenoid:

where:

L = inductance (H);

A = area (m2);

m0 = 4p × 10-7 H m-1;

mr = relative permeability;

N = number of turns;

l = length (m).

The circuit symbols for inductors are shown below:

Inductors are useful for filtering alternating currents.  We will look at this in Tutorial 9.

Sometimes inductance can be a nuisance.  A resistor made by a coil of wire can have an inductance.  This can be reduced by bending the wire back on itself around an insulating material.

Any magnetising effect from the current going one way is neutralised by the current going the other way.

Energy in an Inductor

In tutorial 6 we also saw how the energy is related to inductance by:

 The energy stored in the magnetic field of an inductor is 80 J when the current is flowing is 2.0 A. Calculate the inductance.

From Tutorial 7, we saw that reluctance was the ratio of the magnetomotive force and the flux:

The magnetomotive force is the product of the current and the turns:

Fm = NI

We know that:

and:

So we can equate the two:

The minus signs and the dt terms obligingly cancel out for us:

Which rearranges to:

Since the dt terms have cancelled out, there is no time-dependent change.  Since F is dependent on I, there is no change in their relationship.  Therefore we can write the equation as:

From magnetic circuits (Tutorial 7) we know that the magnetomotive force is linked with flux and reluctance using:

Fm = NI = FS

So we can rearrange to:

So we can substitute:

The current terms cancel out to give us:

 A coil has 500 turns and is known to have an inductance of 0.35 H.  (a)  Calculate the reluctance. (b) Calculate the flux if the current is 100 mA.

We will look at mutual inductance in Tutorial 10.