Electromagnetism
1. What This Chapter Covers
Electric motors, generators, calling bells and cranes are all around us. How do they work? Is there any relation between electricity and magnetism, and can we produce magnetism from electricity?
The answer came from a single lecture demonstration. Hans Christian Oersted (1777-1851) was one of the leading scientists of the nineteenth century, popular as a public lecturer. During one such lecture in April 1820 he carried out an experiment never performed before: he placed a compass needle underneath a wire and turned on the current. The needle deflected.
Oersted recognised at once what he had done. Until then electricity and magnetism were believed to be two different, unconnected sciences. He had shown they were related phenomena.
The consequences the book lists are worth reading as a chain: scientists influenced by the experiment went on to build the dynamo and the electric motor, and out of that technology came radio, television and fibre optics. The unit of magnetic field strength is named the oersted in his honour, and he was made a foreign member of the Royal Swedish Academy of Sciences in 1822.
Activity 1 reproduces the experiment. Fix two thin wooden sticks with slits into a thermocole sheet, run a 24 gauge copper wire through the slits, and connect it in series with a 3 V or 9 V battery and a key. Place a magnetic compass below the wire.
- First bring a bar magnet close to the compass. The needle deflects — nothing surprising there.
- Now take the bar magnet far away and switch on the circuit. The needle deflects again, with no magnet anywhere near it.
So the current-carrying wire exerts a force on the needle, without contact. That is a field force, and the field responsible is the magnetic field.
The chapter is allotted 14 periods in November — the largest allotment in the whole book — and runs from textbook page 209 to page 236.
2. The Magnetic Field
We use the term field when a force is applied on an object by another object without any physical contact between them.
Activity 2 maps one. Place a bar magnet in the middle of a sheet of paper, put a compass near it, and mark dots on either side of the needle. Join the dots and draw an arrow from the south pole of the needle to its north pole. Repeat with the compass at many positions.
Three observations come out of it:
- The needle settles in different directions at different positions — so the field has a direction that varies from point to point.
- Far from the magnet the needle points north-south, as it would with no magnet at all — so the strength of the field falls off with distance.
- Hold the compass above the magnet and it still responds — so the field exists in all directions around the magnet and is three dimensional.
A magnetic field exists in the region surrounding a bar magnet, and is characterised by strength and direction.
3. Magnetic Field Lines
Activity 3 draws them. Mark the earth's north-south line with a compass, lay the bar magnet along it with its north pole to geographic north, then place the compass at the magnet's north pole and put a dot at the compass needle's north pole. Move the compass to that dot, mark the next one, and continue until you reach the magnet's south pole.
Joining the dots gives a curved line, and starting from different points near the north pole gives a family of them.
These curves are magnetic field lines. They are imaginary lines drawn to help us understand the field. Two rules follow:
- The tangent drawn to a field line at a point gives the direction of the field there, because a compass placed on the line comes to rest along the tangent.
- Where the lines are crowded the field is strong; where they are spread apart it is weak. Near the poles they crowd; far from the magnet they spread.
The book is careful on one point. The lines appear to be closed loops, but you cannot conclude that from the picture, because we do not know how the lines are aligned inside the bar magnet. The chapter settles that question later, with the solenoid.
A field is non-uniform if either its strength or its direction changes from point to point. It is uniform only if both are constant throughout.
The field of a bar magnet is therefore non-uniform on both counts.
4. Magnetic Flux and Magnetic Flux Density
Imagine a plane of area A placed perpendicular to a uniform field. A certain number of field lines pass through it, and that number estimates the strength of the field there.
Magnetic flux Φ is the number of lines passing through a plane of area A taken perpendicular to the field. Its SI unit is the weber.
Flux depends on how the plane is oriented; here we take the perpendicular case. Now define the strength itself:
Magnetic flux density B is the magnetic flux passing through unit area taken perpendicular to the field. It is also called magnetic field induction.
B = Φ / A, so Φ = B A
The unit of flux density is weber per square metre, also called the tesla.
The general case
If the plane is tilted, let θ be the angle between the field B and the normal to the plane. The effective area perpendicular to the field is then A cos θ, so
B = Φ / (A cos θ), giving Φ = B A cos θ
The book leaves you a question worth answering: what is the flux through a plane taken parallel to the field? With the plane parallel to B, the normal is perpendicular to B, θ is 90°, cos θ is zero, and the flux is zero.
5. Magnetic Fields Produced by Currents
Activity 1 already told us that a current-carrying wire produces a magnetic field. Three arrangements matter.
Straight wire — Activity 4
Pass a 24 gauge copper wire vertically through a hole in a wooden plank and a retort stand, connect it to a 3 V or 9 V battery through a switch, and arrange 6 to 10 compass needles in a circle around the hole. Switch on.
The needles all point along tangents to the circle. So the field lines around a straight wire are circles, and this can be confirmed by sprinkling iron filings around the wire. That settles the earlier question: magnetic field lines are closed lines.
The direction depends on the current direction:
| Current | Field lines |
|---|---|
| Vertically upwards, out of the page | Anticlockwise |
| Downwards, into the page | Clockwise |
Right hand thumb rule: grab the wire with your right hand so that the thumb points along the current; the curled fingers then show the direction of the magnetic field.
Circular coil — Activity 5
Wind insulated copper wire 4 to 5 times through two holes in a plank to make a coil, and trace the field lines with a compass as before.
Placing the compass at the centre of the coil shows that the field is perpendicular to the plane of the coil. Placing it in front of one face and noting which pole faces the coil identifies that face's polarity — the needle's south pole turns towards the coil's north pole.
| Current in the coil | Field direction |
|---|---|
| Anticlockwise as you look at it | Points towards you |
| Clockwise as you look at it | Points away from you |
Right hand rule: curl your right hand fingers in the direction of the current, and the thumb gives the direction of the magnetic field.
Solenoid — Activity 6
A solenoid is a long wire wound in a close-packed helix. Wind one through equidistant holes in a plank, switch on the current, sprinkle iron filings and jerk the plank gently. An orderly pattern appears.
The pattern is the decisive one in this chapter, because the field lines of a solenoid resemble those of a bar magnet — one end behaves as a north pole and the other as a south pole. More importantly:
- Outside the solenoid the field lines run from north to south.
- Inside they run from south to north.
- The lines outside are continuous with those inside.
So the magnetic field lines are closed loops — and, as the book points out, this is true for the bar magnet too, which answers the question left open in section 3.
The conclusion drawn from all three: electric charges in motion produce magnetic fields.
6. Magnetic Force on a Moving Charge
Activity 7 is the one everyone remembers. Bring a bar magnet near an old CRT television screen and the picture is distorted; move it away and the picture clears. The electrons travelling to the screen are being deflected, so a magnetic field exerts a force on moving charges. That force is the magnetic force.
For a charge q moving with velocity v perpendicular to a field B, experiment gives
F = q v B
and for a general angle θ between v and B,
F = q v B sin θ
The special case matters more than the formula. When the charge moves parallel to the field — along it or against it — θ is 0° and sin θ is zero, so the charge experiences no force at all.
Finding the direction
Two rules are given, and they are not interchangeable:
- The curling rule, for any angle. Point your right hand fingers along the velocity, curl them towards the field, and the thumb gives the direction of the force.
- The right hand rule, generally used when v and B are perpendicular. Stretch the fore-finger, middle finger and thumb mutually perpendicular. Fore-finger along the velocity or current, middle finger along the field, thumb gives the force.
Both are stated for a positive charge. For a negative charge, find the force on a positive charge first and then reverse it.
In every case, the magnetic force is perpendicular to both the velocity and the field.
Example 1: the circular path
A charge q moves with speed v perpendicular to a field B. Because the force F = qvB is always perpendicular to the velocity, it acts exactly like a centripetal force, and the particle moves in a circle.
q v B = m v² / r, so r = m v / (q B)
and the time period T = 2πr / v becomes
T = 2π m / (q B)
Notice what has dropped out: the period does not depend on the speed.
7. Force on a Current-Carrying Wire
A current is charges in motion, so a current-carrying wire in a magnetic field experiences a force. The same special case applies first: a wire kept along the field feels no force, because each charge is moving parallel to the field.
Deriving F = ILB
Take a straight wire carrying current, perpendicular to a uniform field B directed into the page, with only a length L of it inside the field. The charges drift with velocity v.
The force on a single charge is F₀ = q v B. If Q is the total charge inside the field,
F = Q v B ... (1)
The time for the charge to cross the field is t = L/v, so v = L/t ... (2). Substituting,
F = Q L B / t = (Q/t) L B ... (3)
But Q/t is precisely the current I, so
F = I L B ... (4)
and for a general angle θ between the current and the field,
F = I L B sin θ ... (5)
Activity 8, and why the wire moves
Pass a copper wire through slits in two wooden sticks, connect it to a 3 V battery through a switch, and bring a horseshoe magnet near it. The wire deflects. Reverse the magnet's polarity and it deflects the other way; reverse the current and it does so again.
The right hand rule predicts which way it deflects. But the book is explicit that the rule does not explain why it deflects, and then supplies the reason:
- The horseshoe magnet has its own field between the poles.
- The current in the wire produces its own circular field.
- Where the two point the same way — the upper part, in the book's figure — the fields add and the net field is strong. Where they oppose — the lower part — the net field is weak.
- The result is a non-uniform field around the wire, and the wire moves towards the weaker field region.
8. The Electric Motor
Take a rectangular coil ABCD in a uniform magnetic field, carrying a current.
Sides AB and CD are always at right angles to the field, so they always feel a force. Applying the right hand rule, the force on AB acts inward and the force on CD acts outward.
Sides BC and DA are different. Their angle to the field changes as the coil turns. When the currents in them are parallel to the field no magnetic force acts on them; when they are perpendicular, the force pulls the coil up at BC and down at DA.
Net force zero, yet it rotates
The force on AB is equal and opposite to the force on CD, because they carry equal currents in opposite directions. Their sum is zero, and so is the sum of the forces on BC and DA. The net force on the coil is zero — and yet the coil turns.
The book's analogy is exact: opening the cap of a bottle. Two equal and opposite forces applied on either side of the cap produce no net push, but they do produce rotation. The same pair of equal and opposite forces on the two sides of the coil makes it rotate.
Why it would stall, and what fixes it
If the current direction never changed, the coil would rotate to the vertical, carry past it by inertia, and then find the forces on its sides acting the other way round — trying to rotate it anticlockwise. It would halt and turn back, and keep oscillating.
The fix is to reverse the current in the coil every half rotation. That is done with brushes B₁ and B₂ connected to the battery, and slip rings C₁ and C₂ fixed to the ends of the coil and rotating with it. Initially C₁ touches B₁ and C₂ touches B₂; after half a rotation each ring meets the other brush, so the current through the coil reverses. This happens every half rotation, and the coil keeps turning the same way.
In an electric motor, electrical energy is converted into mechanical energy.
The Do you know? box adds an alternative rule. Fleming's left hand rule: stretch the left hand thumb, middle finger and forefinger mutually perpendicular; the forefinger gives the field, the middle finger the current and the thumb the force. The box notes it can be used to explain the working of the electric motor.
9. Electromagnetic Induction
The reverse question: what happens when a coil without current is moved in a magnetic field?
Activity 9 is one of the experiments in the long series carried out by Faraday and Henry. Connect a coil to a sensitive ammeter or galvanometer. With nothing moving there is no emf, and no deflection.
| What you do | What the galvanometer does |
|---|---|
| Push a bar magnet towards the coil, north pole facing it | Deflects — a current has been set up |
| Hold the magnet at rest near the coil | No deflection |
| Move the magnet away from the coil | Deflects in the opposite direction |
| Repeat the whole thing with the south pole facing the coil | Same behaviour, deflections exactly reversed |
The crucial finding is that it is the relative motion that matters: it makes no difference whether the magnet is moved towards the coil or the coil towards the magnet.
Faraday's law
Whenever there is a continuous change of magnetic flux linked with a closed coil, a current is generated in the coil.
That current is the induced current, set up by an induced emf, and the phenomenon is electromagnetic induction.
Faraday further observed that more rapid changes of flux generate a greater induced emf, which gives the quantitative form:
The induced emf generated in a closed loop is equal to the rate of change of magnetic flux passing through it.
ε = ΔΦ / Δt ... (6)
If a single turn links flux Φ₀ and the coil has N turns, the flux linked with the coil is
Φ = N Φ₀ ... (7)
10. Lenz's Law
Faraday's law gives the size of the induced emf but not its direction. Lenz's law supplies that, and the book derives it from conservation of energy rather than asserting it.
Push a bar magnet towards a coil, north pole facing it. Suppose the induced current were clockwise as seen from the magnet. Then the coil would behave like a magnet with its south pole facing the magnet's north pole, so the bar magnet would be attracted — it would gain kinetic energy for free. That contradicts conservation of energy, so the assumption must be wrong.
The induced current must therefore be anticlockwise, making the coil's north pole face the magnet's north pole. The two now repel, so you must do work to push the magnet in — and that work is what is converted into electrical energy in the coil.
Pulling the magnet away reverses everything: the coil now opposes the magnet's withdrawal, which requires its south pole to face the magnet's north pole.
Lenz's law: the induced current appears in such a direction that it opposes the change in the flux in the coil.
Put simply: when flux increases the coil opposes the increase, and when flux decreases the coil opposes the decrease. The law was discovered by the Russian physicist Heinrich Lenz.
11. Deriving Faraday's Law from Conservation of Energy
Two bare parallel conductors sit l metres apart in a uniform field B, with a galvanometer closing one end and a cross wire laid across them. Slide the cross wire to the left and the needle deflects one way; slide it right and it deflects the other way.
Let the cross wire move a distance s in time Δt, and let the emf be ε. Conservation of energy says the electrical energy must come from the work we did in moving the cross wire. Ignoring friction, that work is F s.
The force on the cross wire, from equation (4), is
F = B I l ... (8)
This force opposes the applied force, so we are doing positive work:
W = F s = B I l s ... (9)
As the cross wire moves left, the area of the loop decreases, and so does the flux through it. With B perpendicular to the area,
ΔΦ = B l s ... (10)
Comparing (9) and (10), W = (ΔΦ) I. Dividing both sides by Δt:
W/Δt = I (ΔΦ/Δt) ... (11)
The left side is electric power, and power is also the product of emf and current, P = ε I ... (12). Comparing, ΔΦ/Δt is the induced emf — which is Faraday's law, obtained from energy conservation alone.
Motional emf
Dividing (9) by Δt a different way:
W/Δt = F s/Δt = B I l s/Δt ... (13)
Here s/Δt is the speed v of the cross wire, so P = F v = B I l v ... (14). Setting (12) equal to (14), ε I = B I l v, so
ε = B l v
This is the motional emf. The book adds an important caveat: this is not Faraday's law of induction, because it is not related to a loop. It applies when a conductor moves in a uniform magnetic field.
The book's worked examples
Example 1. A coil of 400 turns; the flux through each single turn rises by 0.001 Wb in 0.1 s, then stays constant from t = 0.1 s to 0.3 s.
ε = N ΔΦ/Δt = 400 × 0.001 / 0.1 = 4 V
From 0.1 s to 0.3 s there is no change in flux, so no emf is generated — the number of turns and the field strength are irrelevant if nothing is changing.
Example 2. A conductor moves at 10 m/s perpendicular to a field of 0.8 T and induces 8 V. Find its length.
ε = B l v, so 8 = 0.8 × l × 10, giving l = 1 m
12. Applications of Faraday's Law
The book lists four, and they are worth knowing as examples rather than as a list to recite:
| Application | How the flux changes |
|---|---|
| Security check walk-through | A large upright coil produces a weak AC magnetic field. Carrying significant quantities of iron changes the flux linked with the coil, and the induced current triggers an alarm |
| Tape recorder | Plastic tape coated with iron oxide is magnetised more in some parts than others. As it moves past the small coil in the head, the field from the tape changes and current is generated in the coil |
| ATM card | The chapter sets this one as a discussion question — what happens when the magnetic strip is swiped through a scanner |
| Induction stove | A metal coil beneath the cooking surface carries AC, producing an alternating magnetic field. That field crosses the bottom of a metal pan, inducing an emf and hence a current; the pan's finite resistance turns that current into heat, which is conducted to the water |
13. Magnetic Levitation
Fix a soft iron cylinder upright on a wooden base, wind copper wire around it, and drop a metal ring — slightly wider than the cylinder — over it. Connect the coil to an AC source and switch on.
The ring levitates. Switch off and it jumps into the air dramatically. Now replace the AC with DC and the ring lifts up and falls down immediately.
Why AC levitates it
By Newton's second law, the net force on a hovering ring is zero, so an upward force F must balance its weight w.
AC changes both its magnitude and its direction at regular intervals, so the coil's ends keep swapping polarity. For the ring to be pushed up continuously, the ring must behave like a magnet that also swaps polarity at the same intervals, but always in the sense that puts like poles together.
Working it through: if the current in the solenoid is clockwise viewed from the top, its upper end is a south pole. An upward force needs the ring's lower face to be a south pole too, which requires an anticlockwise current in the ring. When the solenoid reverses, the ring must reverse with it — and it does, because the changing field changes the flux linked with the ring.
Note what is and is not changing: the ring's area is constant, but the field through it changes, so the flux through it changes, and that is what drives the induced current.
Why DC does not
With no current there is no flux linked with the ring. Switching the DC on links flux suddenly — that is a change, so the ring rises. After that instant the flux is steady, there is no further change, and the ring falls. Switching off is also a change, so the ring lifts again and falls once more.
14. The Electric Generator
Hold a rectangular coil between the poles of a curved permanent magnet and rotate it. The flux through the coil changes continuously, so by the law of induction a current is induced.
Follow one rotation:
| Position of the coil | Induced current |
|---|---|
| Vertical, side A at top, at rest | Zero |
| First quarter, rotating clockwise | Rises from zero to a maximum, flowing A to B, with the peak when the coil is horizontal |
| Second quarter | Falls back to zero as the coil returns to vertical, now with B at top |
| Second half of the rotation | Same pattern again, but with the direction of current reversed |
AC generator
The ends of the coil are joined to two slip rings, and two carbon brushes press against them to draw the current out to external devices. Because the current reverses direction in every half cycle, it is alternating current, and so has a frequency. This arrangement is the AC generator.
DC generator
Replace the two slip rings with two half rings — a commutator. During the first half rotation the current rises from zero to a maximum and falls to zero. As the coil moves past that position, the ends of the coil change over to the other half rings, so the reversal that happens inside the coil is undone at the output.
The current delivered during the second half rotation is therefore identical in direction to the first, and the output is direct current.
In generators, mechanical energy is converted into electrical energy — the exact reverse of the motor.
Key words from the chapter
Magnetic flux, magnetic flux density, electric motor, slip rings, induced current, induced emf, electric generator, DC and AC currents, rms values.
15. Summary
Oersted, in April 1820, placed a compass needle under a current-carrying wire and saw it deflect, proving that electricity and magnetism — until then thought unconnected — are related.
A magnetic field is the region around a magnet or a current where a force acts without contact, characterised by strength and direction, and it is three dimensional. Field lines are imaginary: the tangent gives the field direction, crowded lines mean a strong field, and a field is uniform only if both strength and direction are constant.
Magnetic flux Φ is the number of lines through a perpendicular area, measured in webers; flux density B is the flux per unit perpendicular area, measured in tesla or weber per square metre. In general Φ = B A cos θ, so the flux through a plane parallel to the field is zero.
Currents produce fields. Around a straight wire the lines are circles, given by the right hand thumb rule. A circular coil produces a field perpendicular to its plane, given by the right hand rule. A solenoid produces a field like a bar magnet's, running north to south outside and south to north inside — which proves that field lines are closed loops, for the bar magnet as well.
A charge moving in a field feels F = q v B sin θ, and no force at all when it moves parallel to the field. Because the force stays perpendicular to the velocity, a charge moving across a field travels in a circle of radius r = mv/qB with period T = 2πm/qB, independent of its speed.
A current-carrying wire feels F = I L B sin θ, derived by substituting I = Q/t into F = QvB. A wire deflects because its own circular field adds to the external field on one side and opposes it on the other, so the wire moves towards the weaker region.
In a motor, a rectangular coil in a uniform field feels equal and opposite forces on opposite sides. The net force is zero but the couple rotates it, like two fingers opening a bottle cap. Without intervention it would stall and oscillate, so slip rings and brushes reverse the current every half rotation, and electrical energy becomes mechanical energy.
Faraday's law states that a continuously changing flux through a closed coil generates a current, and that the induced emf equals the rate of change of flux, ε = ΔΦ/Δt, with Φ = NΦ₀ for N turns. What matters is relative motion — a stationary magnet near a coil induces nothing.
Lenz's law follows from energy conservation: the induced current must oppose the change in flux, because the alternative would let the magnet accelerate for free. The cross-wire derivation recovers Faraday's law from the work done against F = BIl, and gives the motional emf ε = Blv — which the book is careful to say is not Faraday's law, since it concerns a conductor, not a loop.
Induction runs the security walk-through, tape recorder, ATM card reader and induction stove, and levitates a metal ring on an AC solenoid — while DC only lifts it once, because after the switching instant the flux stops changing.
Finally, the generator reverses the motor. A coil rotating in a field gives an emf that is zero when the coil is vertical and maximum when horizontal, reversing every half turn. Taken off two slip rings that gives alternating current; taken off two half rings it gives direct current, and mechanical energy becomes electrical energy.
