Magnetic Fields

Force On A Current Carrying Wire In A Magnetic Field

You will be familiar with the basic notion of a magnetic field, in which magnetic materials experience a magnetic force.  However it is worth revising some of the basic ideas that you will have come across in early secondary school.

Revision

• Magnetic fields can be shown by field lines, which go from North to South.

• The field lines in a strong magnetic field are more closely packed than in a weak field.

• Unmagnetised materials are attracted to either pole.

• Like poles repel; unlike poles attract.

• In the Earth’s magnetic field, the North pole will align itself to point to the North, if the magnet is allowed to swing freely.

• The Earth has a magnetic field like a bar magnet.  Notice that the S-pole is under the North geographic pole.  Be careful not to be confused by this.

• We never get single magnetic poles; if there is an N-pole, there must also be an S-pole to go with it.

• Magnets are thought to result from the action of tiny atomic magnets called domains.  This can be explained by the movement of electrons that represent a tiny electric current that results in magnetism.  In most materials, the currents cancel out.

• Only iron, cobalt, and nickel and their alloys are magnetic.

• When a magnetic material is unmagnetised, the domains are all jumbled up.  If some of the domains are lined up, then the material is partially magnetised.  If the domains are fully lined up, the magnet is saturated, and cannot be magnetised further.

• Some materials like soft iron lose their magnetism quickly.  These are used for temporary magnets.  Permanent or hard magnetic materials do not lose their magnetism.

• Electric currents always produce a magnetic field, even if the wire itself is not made of a magnetic material.  The magnetic field of a single current carrying wire is like this:

• The direction of the current is determined by the Screwdriver Rule.

• The magnetic field of a solenoid is like a bar magnet.

You will be familiar with the motor effect.  If we put a current carrying wire in a magnetic field, we see that there is a force.

As we pass a current through the wire, there is a force that acts on the wire at 90^o to the direction of the magnetic field.  This is given by Fleming’s Left Hand Rule with which you will be familiar.

We can work out the force that is exerted on the wire quite simply.  Experiment shows us that the force is proportional to:

• The current;

• The strength of the magnetic field ;

• The length of wire within the magnetic field.

This is summed up in a simple formula:

 F = BIl

[B – magnetic field strength; I – current in A; l – length in m]

The term ` is called the magnetic field strength, or the flux density, and is measured in Tesla, T.  The magnetic flux density can be thought of as the concentration of field lines.  We can increase the force by increasing any of the terms within the equation.  If we coil up the wire, we increase its length within the magnetic field.

Worked Example

This relationship holds true as long as the current is at 90^o to the magnetic field.  If the wire is at an angle to the field, the relationship takes this into account by changing to:

F = BIl sin q

Force on a Charge

We know that a magnetic field and an electric current interact to produce a force.  Since a current is a flow of charge, it is reasonable to suppose that a magnetic field exerts a force on individual charge carriers.

We find that in a magnetic field, the force acts on a stream of electrons always at 90^o to the direction of the movement.  Therefore the path is circular.

Consider a charge q moving through a magnetic field B at a constant velocity v.  The charge forms a current that moves a certain distance, l, in a time t.

We know:       

• Velocity = distance ¸ time

• Current = charge ¸ time

•  F = BIl

So we can substitute into this relationship to give us:

            F = B ´ (q/t) ´ vt = Bqv

So the formula now becomes:

F = Bqv

[ F- force in N; B – field strength in T; q – charge in C; v – speed in m/s]

The charge is usually the electronic charge, 1.6 ´ 10^-19 C.

Path of charged particles in a Magnetic Field

We have seen that the force always acts on the wire at 90^o, and that gives us the condition for circular motion.  We can combine the relationship a = v²/r with Newton II to give us:

Therefore:        

This rearranges to give us:

 Worked Example

 

Question 3

In a particle physics experiment, a detector is place in a magnetic field of 0.92 T.  A particle is found to produce a track of radius 0.5 m.  Other experiments have shown that the particle carries a charge of +1.6 ´ 10-19 C and that its speed was 3.0 ´ 107 m/s.  What is the mass of the particle?  How does it compare to the mass of an electron (9.11 ´ 10-31 kg)?

ANSWER

The cyclotron is a particle accelerator that relies on this idea.  The machine’s main components are two D-shaped electrodes ("Dees") in an evacuated chamber, placed between the poles of a large electromagnet

From the top it looks like this:

Notice that the beam of particles is not circular, but a spiral.  This is because the particles are being accelerated by the electric field between each D-shaped electrode (called a dee).  As their speed increases, so does the radius of the curved path.

If a particle of charge Q enters one of the dees with a speed v, it will move in a semi-circular path of radius r. 

Þ rearranging gives us

We can work out from t = s/v what time it takes for the charge to travel:

The voltage will change twice in each alternating current cycle, therefore in the period of one cycle, the time is 2t.  Since f = 1/T, we can write the expression as:

Magnetic Flux Density, Flux, and Flux Linkage

We have seen how B is called the magnetic field strength, or the flux density, and is measured in Tesla, T.  The magnetic flux density can be thought of as the concentration of field lines.  We can increase the force by increasing any of the terms within the equation.  If we coil up the wire, we increase its length within the magnetic field.

If we look at the magnetic field of a solenoid, we know that it is like a bar magnet:

We can see that the magnetic field strength is uniform within the solenoid.  However the flux density becomes less at the ends, as the field lines get spread out.

We need a term that tells us the number of field lines, and it is called the magnetic flux.  It is given the physics code F (‘Phi’, a Greek capital letter ‘Ph’), and has the units Weber (Wb).  The formal definition is:

The product between the magnetic flux density and the area when the field is at right angles to the area.

In code we write:

F = BA

Remember that flux density is the number of field line per unit area, not unit volume!

The flux linkage is the flux multiplied by the number of turns of wire.  If each turn cuts (or links) flux F, the total flux linkage for N turns must be NF.  We can also write this as NBA.  In other words:

Flux linkage = number of turns of wire ´ magnetic field strength ´ area

The flux linkage can be changed in two ways:

• We can alter the strength of the magnetic field;

• We can alter the area at 90^o to the magnetic field by moving the coil.  If we are turning the coil, the new flux linkage is given by NBA sin q where q is the angle the area makes to the magnetic field.  When we move a coil across a magnetic field, the area swept is the change in area (just like the change in distance is the distance moved).

We give the change in flux linkage the physics code DF.

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.  It is often given the code Î, a fancy letter ‘E’.

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.  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; E– e.m.f (V); DF/Dt – rate of change in flux (Wb/s)]                         

Worked Example

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

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

• Speed the magnet passes through a coil or vice versa.

• Number of turns in the coil.

• Strength of the magnet.

2.      Changing a magnetic field.  We don’t 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.

The transformer is a machine that is simplicity itself.  It consists of:

• A primary coil connected to the alternating power source.  This provides the changing magnetic field.

• A secondary coil connected to the load (in this case, the rectifier and control equipment in the locomotive).

• A laminated soft iron core.

The two coils are electrically completely different circuits.  Either of the coils can act as a primary.  The laminated core is made up of layers of soft iron separated by an insulating layer of varnish or glue.  This reduces losses from eddy currents.  Soft iron is NOT soft like putty; it is heavy and hard.  However "soft" means that it loses its magnetism immediately the current is turned off.  Therefore the magnetic field can change forwards to backwards as the current changes.

The ratio of the input voltage to the output voltage is the same as the ratio of the number of turns on the primary to the number of turns on the secondary.  We can write this as:

Number of turns on the primary    = Primary Voltage

Number of turns on the secondary    Secondary voltage

In Physics code:

N1 = V1

N2    V2

If N1 is greater than N2, we have a step-down transformer, because the voltage is reduced.  A step-up transformer increases the voltage.

If a transformer is 100 % efficient (and it nearly is) we can say that:

 power in = power out

V1I1 = V2I2

Therefore we can say that when the voltage is lower, the current is bigger.  We can rewrite the transformer equation in terms of current to give us:

N1 = I2

N2    I1

In practice, the transformer is about 97 % efficient.  When a large transformer is transferring a lot of energy, even 3 % losses produce a fair amount of heat.  Therefore the transformer is cooled with oil which is pumped to heat exchangers.

The picture above shows the huge transformers used at a power station.  You can see the massive cables coming out from the generator (they are like pipes).  In this power station, the generator voltage is 15 000 V, and this transformer steps the voltage up to 275 000 V.

Transmission of Electricity

To carry the kinds of currents you worked out in Question 7, you would need very thick wires.  Heavy currents make even thick wires get hot.  The power lost is worked out using:

P = I²R

Your answer would be about the whole output of a big power station.  That energy would be lost as heat, simply to warm up the countryside.  So electrical energy is distributed at very high voltages, and relatively low currents.

You can see that the lost power is much less.

In power stations, alternating current is generated typically at 25 000 V (with a current of 30 000 A) from each alternator (ac generator).  The alternators are connected by short and massive cables to a step up transformer immediately outside the building.  The voltage is stepped up to 275 kV.  Much thinner cables can carry the electricity to where it's needed, using a network of high voltage transmission lines called the National Grid.  The transmission lines are carried by transmission towers (pylons) to substations where the voltage is reduced by step-down transformers:

• 33000 V for local distribution;

• 25000 V for railways;

• 11000 V for heavy industry;

• 415 V for light industry;

• 230 V for our homes.

While such high voltages are potentially extremely dangerous, the distribution from large power stations is much more efficient that lots of small local power stations, so less fuel needs to be burned overall.