Electricity and Magnetism

MAGNETS AND POLES

A compass contains a small magnet,  pivoted so that it can turn horizontally. When the compass is held horizontal, the magnet experiences a force which makes it turn till one end points approximately to geographic north. This end is called the north-seeking pole or just the north pole (N) of the magnet. The other end is the south pole (S).

There are only a few materials attracted by a magnet, referred to as ‘ferromagnetic’ materials. These include iron, nickel and cobalt. Steel is mostly made out of iron, so steel is also ferromagnetic.

Forces between poles

The forces increase as poles get closer together, and like poles repel and unlike poles attract:

Inducing magnetism

We can form a small chain of nails and paperclips as shown below:

The bar magnet ‘induces’ magnetism in each nail and paperclip – i.e. each becomes a small magnet. The poles induced are as shown above – and the nails and clips are attracted to the magnet and each other because unlike poles attract.

If the top nail is held and the magnet removed, the nails fall apart, because the magnetism of the iron is only temporary. Iron is said to be magnetically ‘soft’ – it is easy to magnetise but also loses its magnetism easily.

If the same is done with the paperclips, they hold together, because the magnetism of the steel is permanent. Steel is said to be magnetically ‘hard’.

MAGNETIC FIELD LINES

We imagine a magnet to be surrounded by an invisible ‘magnetic field’, which we represent by field lines or lines of force. These are lines which ‘flow’ from north to south. When two magnets are close together, it is their fields which are believed to exert forces on each other.

A plotting compass, which is simply a small compass about one cm in diameter, can be used to plot out the shape of a magnetic field.

Note that, as indicated in the following diagrams:

• the direction of the magnetic field at a point is taken as the direction that a north pole would tend to move if it were at that point
Bar magnet
The spacing of field line represents the strength or intensity of the field. Near the poles they are closer together than those further away, because the magnet is strongest near the poles.

Nearby magnets

When magnets are close together, with their fields overlapping, the fields combine to produce a resultant field which acts in one direction at any given point.

The point X is called a neutral point. The forces due to both magnets cancel each other, i.e. there is no net force, at X.

The Earth's magnetic field

The Earth’s magnetic field is produced as if it had a short bar magnet inside it, with its south pole in the northern hemisphere.

FORCE ON A CONDUCTOR IN A MAGNETIC FIELD

The motor effect

• When the current is in the direction shown the wire is forced upwards
• Reversing the current or reversing the poles reverses the direction of the force
Note that the current is only turned on for a short while, since the wire could get hot.

Fleming’s left-hand rule ('the motor rule')

This is a way to remember the relative directions of the magnetic field, the current and the direction of motion.

The thumb and first two fingers of the left hand are held at right angles to each other:

• The first finger is pointed in the direction of the field (North to South)
• The second finger is pointed in the direction of the current (conventional direction, from + to -)
• The thumb then indicates the direction of motion of the wire
Resultant field pattern

When a current flows through a wire, it produces a magnetic field around the wire in the form of concentric circles (see later). This field then interacts with that of the magnet, producing a force on the wire.

The following represents the magnet and an end-on view of the wire, with the field each produces represented separately:

The following represents the fields combined, i.e. the resultant field:
If we think of magnetic field lines as being like ‘stretched elastic bands trying to contract’, then we can see that the wire is going to be forced downwards in the diagram - as the left-hand rule predicts.

Moving-coil loudspeaker

In this, a radial magnetic field is produced between the pole-pieces of a magnet.

The turns of the coil are at right angles to the field lines, so when a current passes through the coil it experiences a force to the right or left (by Fleming’s left-hand rule). If the current is alternating, i.e. repeatedly changing direction, the coil is forced to vibrate to the right and left, as does the stiff paper cone attached to it. The vibrating cone makes the air vibrate, producing sound at the same frequency as that of the alternating current.

MAGNETIC FIELD STRENGTH, B

To define the strength of a magnetic field we make use of the motor effect.

We denote magnetic field strength by B, and define its magnitude by:

For a small magnet, B is typically about 0.03T, and for the Earth B is about 5*10-5 T

Expression for F

This is for a conductor at right angles to the magnetic field, as in the previous diagram.

MAGNETIC FIELDS DUE TO CURRENTS

If a small compass (the needle itself is a small magnet) is held close to a wire, the needle flicks momentarily when a current is turned on and off.

This implies that an electric current produces a magnetic field, and the field affects the compass needle. It is believed that all magnetic fields are produced by the motion of electric charges. The cause of the Earth's magnetic field is not well understood, but it is assumed that it is due to electric currents circulating inside the Earth.

The field produced by a conductor depends upon its shape:

A long straight wire

(i) Field lines

The ‘right-hand grip rule’ can be used to give the direction of the magnetic field lines produced by a current in a long straight wire:
• The wire is gripped with the right-hand, with the thumb pointing in the direction of the current, and the fingers then curl in the direction of the field lines
Note: The direction of the current means the direction of conventional current, from + to -

Note: You may have heard of the 'right-hand screw rule', which can also be used to give the direction of the field lines.

(ii) Flux density

The ‘permeability’ of a medium is a measure of the effect that the medium has on the flux density. Some substances, such as iron have a large value of permeability.

A plane circular coil

The direction of the field lines can be found by applying the right-hand grip rule to a small section of wire.

The magnitude of B at the centre is given by:

A solenoid (a wire wound into a long cylindrical shape)

(i) Field lines

The field outside the above solenoid is very similar to that of a bar magnet.

The right-hand grip rule, as stated above, can be applied to a small section of the conductor to determine the direction of the field lines.

Alternately, there is a different version of the right-hand grip rule which applies to the solenoid:

• The solenoid is gripped in the right-hand, with the fingers turning in the direction of the current, and the thumb then points to the N pole of the solenoid
(ii) Flux density
The value of B on the axis and well inside a solenoid is given by:

The field is fairly uniform over the cross-section of the solenoid, i.e. B is almost the same on and off the axis. The expressions for B are only strictly true for an infinitely long solenoid. In practice they are reasonably accurate if the length l is at least 10 times the diameter.

FORCES BETWEEN CURRENTS

Two parallel wires carrying currents will either attract or repel each other.

Consider diagram (a):
• Apply the right-hand grip rule to the left-hand conductor - this indicates that the magnetic field at the right-hand conductor due to the current in the left-hand conductor is into the paper
• Now apply Fleming’s left hand rule to the right-hand conductor - this indicates that the field produces a force on the right-hand conductor to the left, as shown
The directions of all the forces can be determined in a similar way.

Again, consider diagram (a). The flux density B1 produced by the left-hand conductor at the right-hand conductor is given by:

The above expression gives the magnitude of the force on either wire, since by Newton’s third law the forces on the pairs of wires are equal in magnitude, but opposite in direction.

Example

Definition of the ampere

• The ampere is that current which, if flowing in two straight, parallel wires of infinite length, placed one metre apart in vacuum, will produce on each of the wires a force of 2 * 10 -7newtons per metre
Notice that this definition makes no reference to other electrical quantities such as volts. Thus, the phenomenon of forces between current carrying wires provides a non-electrical way of defining the ampere. The definition is the basis for a device for measuring current called a 'current balance'.

MOTION OF CHARGES IN MAGNETIC FIELDS

Force on a single charge

Consider a section of a conductor carrying a steady current I.

The electrons which are able to take part in conduction are called ‘free electrons’, as opposed to those that are fixed to atoms, and unable to move about inside the conductor. The free electron move under the influence of an applied p.d. They collide repeatedly with atoms, and so do not have a constant velocity. However, they do have an average velocity, called their ‘drift velocity’.

If the conductor is at right angles to a magnetic field of strength B, then the force F/ on it (see earlier notes) is given by:

Though the expression is derived for electrons in a conductor, it applies equally to electrons or other charges moving, say, through a vacuum, in which case v would be their actual velocity rather than their drift (average) velocity.

Note - in applying the left-hand rule to determine the direction of the force on an individual charge, the second finger points:

• in the direction of motion of the charge if it is positive, since it is moving in the conventional current direction
• in the opposite direction to the motion if it negative (as is an electron)
Path of a charge moving at 900 to a magnetic field

The charge Q is treated as being positive for the purposes of the diagram, so it is equivalent to a current to the right. Applying the left hand rule implies that it will experience an upward force when it enters the field:

Now, the force F is at 900 to the direction of motion, and so does not change the value of v, since it has no component in the direction of v. Thus, the size of the force remains constant: F = BQv. Also, the force is initially at 900 to the motion and remains at 900 at every point on the curved path.

A force with a constant magnitude and which is always at 900 to the direction of motion is exactly the same type of force as keeps a cork on a circular path when spun at a constant speed on the end of a thread. Hence the charge Q will move along a circular path. This type of force is called a 'centripetal force'.

If the radius of the circle is r and the mass of the charged particle is m, then:

If the charge is an electron, then replace Q by e.

Example

Electron beams are sometimes called ‘cathode rays’, and the following is one version of a ‘cathode ray tube’, designed to make the rays visible:

Electrons ‘boil off’ the cathode and accelerate towards the positive anode, which has a hole in it to allow the electrons to pass through. The tube is in a uniform magnetic field, which is at right angle to the paper, and so the electron are forced to follow a circular path. The glass envelope contains low-pressure hydrogen, which glows as the electrons pass through it (due to becoming ionised) and so the path of the electrons is revealed.

a) An electron is emitted from the cathode and is accelerated by a p.d. of 4000V. Calculate the speed gained by the electron. (specific charge of electron, i.e. charge to mass ratio, e/m = 1.8 * 1011 C kg-1)

The work done on a charge = charge * p.d. = its gain in kinetic energy, so:

b) The electron enters at right angles a uniform magnetic field of flux density 0.0025 T. Determine the radius of the path followed.

A RECTANGULAR COIL IN A MAGNETIC FIELD

Suppose that a rectangular coil PQRS of N turns is pivoted so that it can rotate about a vertical axis (the dotted line):

When the current flows, each side of the coil experiences a force.

The forces on PQ and SR lie on the same line and are equal in size but opposite in direction and so cancel each other

The forces on PS and QR have opposite directions and both have the magnitude given by F = BIlN (since there are N lengths l on each side of the loop, and each length experiences a force BI l ). These forces tend to produce rotation and constitute a couple (a couple being a pair of equal but opposite forces whose lines of action do not coincide).

A couple has a property called moment or torque, T, which is a measure of the size of its turning effect, and is given by:

Example

A coil of 50 turns with area 5cm2 carries a current of 10A and is placed with its plane parallel to a field of strength 0.20T. What torque is needed to hold it stationary in the field?

The applied torque must balance the torque produced by the field, and so is found from:

MOVING COIL GALVANOMETER

A galvanometer detects small currents (it can be calibrated to act as an ammeter or voltmeter). It consists of a coil of thin, insulated copper wire which is suspended in a magnetic field and is able to rotate, carrying a pointer with it.

The curved pole pieces and the soft-iron cylinder produce a radial field in the air gap, so the plane of the coil is always parallel to the field. Thus, a constant torque is produced when a constant current flows through the coil.

The rotation of the coil is limited by the opposing torque produced by hair springs. These are wound opposite ways, so that the rotation is opposed, whichever way it is.

DC MOTOR

In the set up above, apply Fleming’s left hand rule to side AB.
• the First finger (Field) points N to S
• the seCond finger (Current) points in the direction of I
• the thuMb (Motion) then points upwards
Thus, force on AB is upwards. The force on CD is opposite to that on AB, since the current is opposite to that in AB. These forces tend to rotate the coil.

The commutator attached to the ends of the coil rotates with the coil and its ‘split-ring’ design ensures that the current in the coil is always in such a direction that the forces always produce rotation in the same direction. Thus, the coil can rotate in the same direction continuously.

Practical motors consist of several coils of many turns of wire wound in slots in a soft iron cylinder (the 'armature'), connected to a commutator with an appropriate number of segments. Also, curved pole pieces are used to produce a radial field. This arrangement produces greater and more uniform torque. However, the iron core is a conductor, and whenever a conductor moves through a magnetic field, currents are induced in it called eddy currents, and these generate wasted heat. Eddy currents are very much reduced by using a laminated cylinder - the iron is in the form of thin slices which are insulated from each other by paper or some other insulator.

As the coil of a motor rotates, it cuts the flux of the magnetic field, and this creates an emf in the coil which opposes the applied voltage V, and is called a back emf E.

If the current in the coil is I and the resistance of the coil is r, then:

Each term represents power: VI is the power supplied to the motor, I2r is the power dissipated by the coil ('copper losses'), so VI- I2r = EI, is the mechanical power output of the motor.

When the motor is first turned on, the armature is stationary, so the back emf is zero. At that moment the current would be V/r and, as r is small (typically less than 1 ohm), the current could be big enough to burn out the coil. To prevent this, a ‘starting’ resistance is placed in series with the motor, and its resistance decreases as the motor speeds up. As it speeds up, the back emf increases until, at operating speed, the back emf is just a little less than the applied voltage, and the current is quite small.

In large motors, electromagnets provide the magnetic field. In a ‘shunt-wound’ motor, the field coils are in parallel with the armature. We can represent the circuit by:

An arrangement such as below can be used to investigate the above relationships:
With the support bar fixed at its lowest position, say, the applied voltage is set to the normal working voltage of the motor. The voltage V, current I and readings F1 and F2 recorded. Then the following can be calculated:

A stroboscope, which provides a flashing light at a known frequency, can be used to determine the frequency of rotation of the pulley. Starting with a high frequency, this is gradually reduced until the reference mark appears to be at rest. The strobe frequency then equals the frequency f of the pulley. Now the following can be calculated:

The results can be represented graphically:

A Level Physics - Copyright © A C Haynes 1999 & 2004