Electricity and Magnetism




LENZ'S LAW

We have seen that an electric current can create a magnetic field. The opposite effect was discovered in 1831, i.e. that a magnetic field can induce an electric current. This is called electromagnetic induction.

This can be done (1) by relative motion and (2) by changing a magnetic field:

1. Electromagnetic induction by relative motion

i) The dynamo effect

The wire is made to move up and down between the poles of the magnet. When the wire is moved up, the galvanometer needle flicks one way, and when it is moved down the needle flick the other way. When the wire is stationary, there is no current, no mater how strong the magnetic field. If the wire is moved directly from the N to the S, or the other way, there is no current. Thus, the conductor has to ‘cut’ the magnetic field lines, for a current to be induced.

Now, an electric current is produced by a voltage. So though it is a current that the above meter detects, it is actually a voltage (an electromotive force, 'emf') that is induced, which then produces the current.

When pushing the wire though the magnetic field, a force can be felt resisting the motion. We find that the current is always induced in such a direction that the force on it due to the permanent magnet opposes the motion of the wire.

This is an example of Lenz’s law of electromagnetic induction, which says that:

ii)
When the coil is moved or when the magnet is moved the galvanometer indicates that a current flows. It is the relative motion that matters. If there is no motion there no current.

Also, as indicated in the diagrams, the current is always induced in such a direction that the magnetic field it produces opposes the direction the magnet moves.

This is another example of Lenz’s Law.

We can understand Lenz’s law as special case of the principle of conservation of energy. Consider the left-hand diagram in the last set-up. Suppose that as soon as the N pole were pushed towards the magnet, that the induced current in the coil produced a S pole at the top end instead of a N pole. This S pole would then attract the N pole. This could make the magnet accelerate towards the coil without any outside force applied. This would in turn produce a bigger current in the coil, increasing the coil’s field, and attracting the magnet even more strongly.

Thus, we could have increasing kinetic energy of the magnet, and increasing electric energy in the coil for no energy input, apart from the initial nudge of the magnet. Unfortunately (?) this does not occur. We cannot get energy for nothing. It contradicts the principle of conservation of energy, which says that energy cannot be created or destroyed but only changed from one form to another.

2. Electromagnetic induction by changing a magnetic field

The transformer effect

In this case the magnetic field produced by one coil (the ‘primary coil’) passes through a second coil (the ‘secondary coil’). If the field produced by the primary coil changes, an emf is induced in the secondary coil.

Observations: Explanation:

Initially the current in the primary coil is zero. When the switch is closed, the current in the primary coil rises quickly from zero towards a certain steady value. As we have seen, a current produces a magnetic field. The rising current produces a rising magnetic field. This produces a rising magnetic field in the soft-iron core, which passes through the secondary coil. A changing magnetic field induces a voltage (called an emf). Thus, the rising magnetic field induces an emf, and hence a current, in the secondary coil. Hence, the ammeter needle moves.

As soon as the current in the primary is steady, the magnetic field is no longer changing and so there is no more current induced in the secondary. Only a changing magnetic field induces a current. No matter how large the magnetic field is, if it stays constant there is no current induced. Hence, the ammeter needle just flicks indicating a momentary current.

When the switch is now opened, the ammeter needle flicks the other way. Everything is now reversed. The current in the primary was steady, and so the magnetic field it produced was steady. When the switch is opened, the current falls to zero, and so does the magnetic field it produced. This changing field again passes through the secondary, and induces an emf and so a current. But the field is falling instead of rising, as it was the first time, so the current is in the opposite direction to the first time and the needle flick the other way.

The above diagram represents the basic structure of a transformer, to which we return to later.


MAGNETIC FLUX

To be able to discuss induced emfs numerically, we introduce the ideas of magnetic flux and flux linkage. We already have a definition of flux density, B, in terms of the force on a current carrying conductor in a magnetic field. Now consider a single loop of wire, of area A, in a magnetic field:

Definition


Thus, the unit of magnetic flux density, T = Wb per m2 = Wb m-2. This is why we express the strength of a magnetic field as ‘flux density’. It is the ‘flux per square metre’.

Flux linkage


FARADAY'S LAW

While Lenz's law indicates the direction of an induced emf, its size is given by Faraday's law of electromagnetic induction:


EMF IN A MOVING ROD

The light-beam galvanometer, G, is very sensitive to electric current. The light spot moves left or right depending on the direction of the current through G. When the rod is made to move either left or right, G gives a deflection while the rod is moving. The emf is due to the rod ‘cutting the magnetic flux’.

Suppose that the rod is made to move a distance x in time t:

Fleming's right-hand rule (or 'dynamo rule')

This is a way of remembering the relative directions of the magnetic field, the motion and the induced current, when a conductor is moved through a magnetic field.

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

Note that Fleming's: Suppose that a conductor is initially at rest in a magnetic field. There is no induced current because there is no motion. Now suppose that it is forced to move to the left (in the direction of v). Using the right-hand rule, the first finger points down, the thumb points to the left and the second finger points in the direction of the induced current I, as in the diagram:
We now have a current I flowing in a magnetic field. Using the left-hand rule, the first finger points down, the second finger points in the direction of  I, and the thumb then points to the right, which is the direction of the force F on the conductor.

Notice that the direction of the force F on the induced current is such as to oppose the cause producing the current - as expected from Lenz’s law. Again, this can be viewed as a consequence of the principle of conservation of energy. If the opposite occurred, i.e. if the induced current and resulting force encouraged the motion producing it, we could get increasing kinetic energy and current produced from nothing (apart from the initial nudge given to the rod).


EDDY CURRENTS

A piece of metal moving in a magnetic field, or exposed to a changing one, has emfs induced in it which can cause currents, called eddy currents. According to Lenz’s law, such currents will move in such a way that the forces on them oppose the cause producing them. The forces on the induced currents can produce electromagnetic damping or braking:

In the left hand set up, if the solid copper cylinder is set spinning, it comes rapidly to rest - the electromagnetic braking effect.

In the right hand diagram, the solid copper is replaced by a similar cylinder, but made now of coins. The braking effect is found to be much less. The dirt between the coins, and the fact that they only touch at a relatively few points, very much reduces the size of the eddy currents, and therefore reduces the braking effect.

Electric motors, dynamos and transformers all contain iron, which improves their performance. However, the iron is either moving in a magnetic field or subject to a changing field, and so is subject to eddy currents. This can produce energy loses in the form of unwanted heat. The effect is considerably reduces by using laminated iron (similar to the coin arrangement above). This is in the form of thin sheets, separated by paper or another insulator, which significantly reduces the eddy currents, and therefore their heating effect.

The heating effect of eddy currents is not always unwanted. For example, the effect is exploited in ‘induction heating’, in which high frequency ac is used to induce currents and melt metals.

Damping a meter

When a current passes through the coil of a moving coil meter, the coil starts to rotate. It can be brought to rest more quickly by using electromagnetic damping. This can be achieved by winding the insulated coil on a metal frame. As this moves in the magnetic field, eddy currents are induced in it which, according to Lenz’s law, oppose the motion producing them, i.e. the motion of the coil. This is a practical use of electromagnetic damping

If the damping is critical, the pointer comes to rest quickly, and the movement is said to be ‘dead beat’.

A 'ballistic galvanometer' is one in which there is deliberately very little damping. It can be shown that if a current flows through such a galvanometer for a short time, then the first deflection is proportional to the total charge passing through the coil. Thus, a ballistic galvanometer can be calibrated to measure charge.


A COIL MADE TO ROTATE IN A MAGNETIC FIELD

The following represents a coil, shown as a single loop of wire, for simplicity, forced to rotate in a magnetic field:

End view of coil with N turns:


Hence, the emf alternates sinusoidally, and would produce an alternating current in a complete circuit.


T = period = time for one rotation.

Example

A coil of areas 10cm2 and with 500 turns rotates at 600 revolutions per minute ('rpm') in a magnetic field of strength 0.10T. Calculate the maximum emf induced.

We need to SI units, e.g. m2 for area:

The alternator (this is an alternating current (ac) generator)
 

The diagram represents a coil of wire forced to rotate in a magnetic field. The ends of the wire are attached to two slip rings that rotate with the coil. Carbon brushes rub against these to provide continuous contact, and an alternating emf is extracted.

As previously mentioned, when a wire is forced to move through a magnetic field, an emf is induced in it, and a force acts on the conductor opposing its motion. In the above set-up, the coil experiences a force opposing its motion, and so energy has to be supplied to keep it rotating. The energy used to maintain the rotation of the coil gets changed to electrical energy.

In practical generators, several coils are wound in slots in a soft iron cylinder (an arrangement called the 'armature'), the cylinder being laminated to reduce eddy currents.


SELF INDUCTION

We’ve seen that if the current in one coil changes, then the magnetic field it produces also changes, and if a second coil is exposed to this changing field, then an emf is induced in the second coil. This phenomenon is called mutual induction.

However, if the current in a coil changes, the coil itself is exposed to the changing field. Thus, an emf is induced in the coil. This emf opposes the change producing it (by Lenz’s law) and is called a ‘back emf’. This phenomenon is called self induction. A coil is said to have a property called self-inductance, or just inductance, and is called an inductor.

Observation

The resistor and the inductor have the same value of resistance. When the switch is closed, the same current will (eventually) flow through both the bulbs, and they will reach equal brightness.

However, we observe that while bulb (1) reaches maximum brightness almost instantly, bulb (2) takes a fraction of a second to do so. The latter is due to the back emf induced in the coil opposing the rising current.

The graph represents the rise in current through each device.

Self-induction also opposes the fall of a current. When a circuit is broken, the self-induced emf can be large enough to produce a spark at the switch.

Definition and units


The constant of proportionality is denoted by L and is called the inductance of the coil. Since the back emf opposes the change in current, we write the equation as:


Inductance of a solenoid

It can be shown that the inductance L is given by:


Relative permeability

Example

Find L for a solenoid of 500 turns of length 25cm, of area 50cm2 and wound on:




Energy stored in an inductor

The energy W stored in the magnetic field surrounding an inductor, of inductance L, carrying a current I is given by:


GROWTH OF CURRENT IN AN LR CIRCUIT
The slope of the graph at any point equals the slope of the tangent drawn at that point, and gives the growth rate of the current. So, at time t/:


The steepest part of the graph is at the very start, when t = 0. Thus, the greatest rate of growth of current is when the switch is first closed.


THE TRANSFORMER

The basic structure of a transformer was described earlier.

Energy losses in a transformer

A transformer is designed so that as little energy as possible is wasted, and most are extremely efficient in this regard. However, some energy losses do occur due to:

1. Resistance of the windings

The copper wire used for the windings is a good conductor, but does have some resistance, so heat losses, referred to as 'I2R losses' or 'copper losses', occur. (recall that P=IV and V=IR so power P = I (IR) = I2R)

2. Eddy currents

The alternating flux in the soft-iron core induces eddy currents and causes heating. This is considerably reduced by having a laminated core, i.e. in the form of thin slices of iron separated by an insulating material.

3. Flux leakage

The flux due to the primary may not all link to the secondary if the core is poorly designed.

Another arrangement is represented below, in which the insulated wires are wound on top of each other on the iron core. This ensures that all the magnetic field produced by the primary is inside the secondary.
4. Hysteresis loss

When an alternating emf is applied to the primary coil of a transformer, the iron core is repeatedly magnetised in one direction and then in the opposite direction. This process produces some wasted energy, which heats the core, and is called 'hysteresis loss'.

Transformer equations


In a well-designed transformer, energy losses are small, so:

Observation

In dia.1 a coil of wire with 800 turns is placed on the left hand arm of the iron core and 200 volts are applied to the coil.

A small bulb is attached to a length of insulated wire a couple of metres long. The wire is wound loop by loop onto the right hand arm of the iron core. After a few loops are on, the bulb stars to light faintly. It is seen to be nearly fully bright with about 10 loops of wire on the arm.


In the set-up in dia.1 the bulb may not be fully bright because of ‘flux leakage’.

However, in dia.2, when the top part of the iron core is added to complete the core, the bulb suddenly becomes significantly brighter, and may even ‘blow’. This is because of a more efficient transfer of flux through the core.

Observation

As the p.d. applied to the primary coil is increased to 240V, the nail is seen to glow white hot and then melt - and steel melts at well over 10000C!

Step-up and step-down transformers

The 'turns ratio' is the ratio of the number of turns of wire on the coils, and this determines if the voltage output at the secondary is greater or less than the voltage applied to the primary:


TRANSMISSION OF ELECTRICITY

Suppose that 100,000W of electrical power is to be send through power cables.

Power, P = V I (watts = volts*amps), so this might be done as, for example:

Now the power loss in the cables, i.e. heat loss per second, equals I2R, so the lower the current, the smaller the heat loss. Hence, the high voltage ('high tension'), low current option is preferred.

The following represents a power station voltage being stepped up for transmission, and then stepped down for use:

Any wasted heat represents extra fuel used at the power station - so reducing the heat loss saves the power station fuel, and therefore saves money.

The above explains why alternating voltages (rather than d.c.) are used for power transmission:

 

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