Magnetic Fields Tutorial 5 - Electromagnetic Induction

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 (emf).  It is often given the code ő, a fancy letter ĎEí.

 

The emf depends on:

The maximum emf occurs when the wire is moved at 90 degrees to the field.
 

 

The direction of the current is determined by Fleming's Right Hand Rule.

 

 

 

Faraday's and Lenz's Law

Faradayís Law and Lenzís Law are two important rules that govern this effect.

 

Faradayís 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.

 

Lenzís 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.

 

 

 

Lenzís 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.  Electric motors are therefore very suited to railway use, where big currents are needed to get trains moving, and there is no need for a gearbox that is needed for a diesel engine.

 

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

                       

[N - number of turns; EĖ e.m.f (V); DF/Dt Ė 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?

Two formulae to use: 

F = BA  

and       

We need to work out the flux:

                        F = BA = 0.02 T ◊ 5.0 ◊ 10-4 m2 = 1.0 ◊10-5 Wb

Now we can work out the e.m.f:

                       ő = - NDF = 1 1.0 ◊10-5 Wb = 1.0 ◊10-5  V

                                   Dt           10 s

 

Question 1

A search coil has 2500 turns and an area of 1.5 ī 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 emf of 0.75 V.  What is the flux density between the poles of the magnet? 

Answer

 

Cutting Flux

Consider 2 coils cutting field lines in a magnetic field of flux density B. Both are moving at constant speed v.

 

 

In diagram A we can see that a certain number of field lines are being cut every second. In diagram B we see that the area is doubled, so twice as many field lines are cut every second.

Since
F = BA, there is twice as much flux cut every second.
 

 

 

Linking EMF and the speed of a wire in a magnetic field

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.

 

 

Faradayís and Lenzís Laws:

The minus sign shows that Lenz's Law applies.

 

Since F = BA, we can write:

 

 

Also A = lw and l = vtSo we can write:

 

 

And then:

 

 

The t terms cancel out to give us:

 

The minus sign is there to satisfy Lenzís Law.

 

Questions on this often involve the rather fatuous example of aeroplanes flying through the Earthís magnetic field.  An e.m.f. is induced due to the vertical component of the Earthís magnetic field.  Itís no damned good to the aeroplane, which would have to fly along fixed rails to generate anything useful Ė a sky-train?

 

Question 2

 A coil of length 50 mm and width 80 mm with 30 turns is passed through a perpendicular magnetic field of value 0.245 T at a velocity of 1.20 m s-1.

(a) Which data item is irrelevant?

(b) Calculate the EMF

Answer

 

Magnetic Fields in Coils

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 doesnít matter, as long as there is relative movement.

 

 

When the magnet is moved towards the solenoid, we get a voltage induced according to Faradayís Law.  However, Lenzís 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.

 

 

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

 

Now letís think about the magnet in the middle, as shown in the next diagram:

 

In this instance we see three different things:

 

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 shown 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.

 

Question 3

A conducting liquid flowing through a pipe in a magnetic field cuts lines of magnetic flux and generated an emf across opposite sides of the liquid.  The emf can be used to determine the flow rate of the liquid. 

 

In a brewery beer flows through a 35 cm diameter pipe at a rate of 0.4 m3 s-1.  The pipe is in a magnetic field of 5.0 ◊ 10-3 T. 

 

What is the emf between the opposite sides of the liquid?

Answer

 

The Transformer Effect

We can use a magnetic field to induce a voltage in two ways:

 

1.      Relative movement.  The size of the voltage depends on:

2.      Changing a magnetic field.  We donít have to make the magnetic field move.  If we turn the current on or off, there is a change in the magnetic field, and that induces a voltage in a second unconnected coil.  This is called the transformer effect or mutual induction.

 

We can also make the magnetic field go forward and backwards by using an alternating current.  We will look at the transformer effect in a later tutorial.  However it is worth noting now that radio broadcasts use the transformer effect.  The changing magnetic field (made by the alternating current) in the transmitter induces a very tiny alternating voltage in the receiver.  This is amplified to enable us to hear the broadcast.

 

 

Investigation of the effect on magnetic flux linkage of varying the angle using a search coil and oscilloscope (Required Practical)

This experiment uses a circular coil which is connected to an AC supply and a search coil connected to a CRO.  The apparatus is set up like this:

 

 

 

The search coil is clamped so that it remains in the same place.  It should be placed at the centre point of the flat coil (sometimes called a Helmholtz coil).  The coil is then rotated to an angle like this and the display on the CRO will change. 

 

 

 

The flat coil will need to be held in a clamp to make sure that the angle doesn't change.  If your coils are like this, then they can be easily held with a boss on a clamp-stand.

 

 

 

You will need to do a preliminary experiment to determine the best current to use.  The flat coils shown can carry a maximum of 2 amps, which should not be exceeded, otherwise they will get hot.  They are expensive.  An ammeter can be inserted to ensure that the current is not excessive.  You will need an AC ammeter.  If you have a power signal generator, you might want to determine the best frequency.  You will also need to set up the CRO to give a reading that reduces the uncertainty to a minimum.  You also should consider ways of measuring angles with less uncertainty than using a protractor.  The diagram above might give you a few hints... 

 

Do NOT be tempted to use an AC voltmeter in this assessment.  Part of the assessment is how well you can use the CRO.

 

 The experiment will NOT work with DC. 

 

If you are not sure on how to use the CRO, refer to the Electricity Tutorial 10 using the link HERE.

 

You should include details of the preliminary experiment in your report, including your chosen values and ranges.

 

You will need harvest data from angles 0o to 90o.  You need to measure the voltage on the CRO.  You will need to take repeats as well. 

 

The theory tells us that:

 

The graph of EMF against cos q should look like this:

 

 

The gradient is the product BANf.  The number of turns will be written on the search coil.  Area is easily worked out by measuring the diameter of the search coil.  The frequency is that on the signal generator (which you can check on the CRO), or the mains frequency, which is kept very close to 50 Hz.  Therefore the magnitude of the magnetic flux density can be worked out by:

 

 

You could also compare the flux density worked out from the gradient with the flux density produced by the coil.  Hopefully they should be about the same...

 

Theory

We know from Faraday's and Lenz's Law that:

 

We know also from our previous tutorial that:

 

NF = BAN cos q

 

 

We can combine these to write:

 Getting rid of the D term, we can write:

 

 

Since f = 1/t, we can now write:

 

The minus sign shows that there is a phase change between the current in the coil and the emf induced in the search coil.

 

We can measure E on the CRO.  N is the number of turns which will be written on the search coil.  Area, A, is easily worked out by measuring the diameter of the search coil.  The frequency, f, can be checked on the CRO as well.  It should be about the same as the frequency displayed on the signal generator, depending on how well the signal generator is calibrated.  If you are using an AC supply, the frequency is 50 Hz.  The problem comes with measuring B, the flux density.

 

Consider a flat coil of N turns, and radius r metres. It is carrying a current of I amps. The current is going clockwise as we look at it.

The magnetic field looks like this:

 

The maximum magnetic field strength is at the very centre of the coil.  The flux density at the centre is directly proportional to the current, I and the number of turns, N. It is inversely proportional to the radius, r, of the coil.

The formula for the flux density at the centre is this:


 

The term m0 ("mu-nought") is a constant which has the value 4p ◊ 10-7 H m-1 (Henry per metre).  It is called the permeability of free space.

 

Do not mix up the permeability of free space m0 with the permittivity of free space e0. The two words sound similar.

 

 The number of turns for the flat coil will be different to the number of turns in the search coil.  The radii will be different as well.