Magnetic fields

All five subtopics here are printed under "A Level subject content" in the 9702 syllabus, and every objective carries the tier "A Level". None of it is AS. It is examined on Paper 4. Paper 1 is the AS multiple-choice paper, and the whole 9702 bank on this site comes from Paper 1, so no practice is tagged to this topic.

This is the largest topic on the A Level physics paper, with nineteen objectives across five subtopics, so allow it more time than its neighbours.

Syllabus points

20.1 Concept of a magnetic field

• Understand that a magnetic field is an example of a field of force produced either by moving charges or by permanent magnets.
• Represent a magnetic field by field lines.

20.2 Force on a current-carrying conductor

• Understand that a force might act on a current-carrying conductor placed in a magnetic field.
• Recall and use the equation F = BIL sin θ, with directions as interpreted by Fleming's left-hand rule.
• Define magnetic flux density as the force acting per unit current per unit length on a wire placed at right angles to the magnetic field.

20.3 Force on a moving charge

• Determine the direction of the force on a charge moving in a magnetic field.
• Recall and use F = BQv sin θ.
• Understand the origin of the Hall voltage and derive and use the expression V_H = BI/(ntq), where t = thickness.
• Understand the use of a Hall probe to measure magnetic flux density.
• Describe the motion of a charged particle moving in a uniform magnetic field perpendicular to the direction of motion of the particle.
• Explain how electric and magnetic fields can be used in velocity selection.

20.4 Magnetic fields due to currents

• Sketch magnetic field patterns due to the currents in a long straight wire, a flat circular coil and a long solenoid.
• Understand that the magnetic field due to the current in a solenoid is increased by a ferrous core.
• Explain the origin of the forces between current-carrying conductors and determine the direction of the forces.

20.5 Electromagnetic induction

• Define magnetic flux as the product of the magnetic flux density and the cross-sectional area perpendicular to the direction of the magnetic flux density.
• Recall and use Φ = BA.
• Understand and use the concept of magnetic flux linkage.
• Understand and explain experiments that demonstrate: that a changing magnetic flux can induce an e.m.f. in a circuit; that the induced e.m.f. is in such a direction as to oppose the change producing it; the factors affecting the magnitude of the induced e.m.f.
• Recall and use Faraday's and Lenz's laws of electromagnetic induction.

What produces a magnetic field

A magnetic field is a region in which a moving charge or a magnetic material experiences a force. It is produced either by moving charges, meaning an electric current, or by a permanent magnet, where the field comes from the motion of electrons within the atoms.

Field lines run from north to south outside a magnet, they never cross, and their spacing shows the strength of the field. Conventional symbols: a dot for a field out of the page, a cross for a field into the page, from the point and the flights of an arrow.

Force on a current-carrying conductor

Put a current-carrying wire in a magnetic field and, unless the current is parallel to the field, a force acts on it. This is the motor effect.

F = BIL sin θ

where θ is the angle between the wire and the field. Two special cases follow at once:

• The wire perpendicular to the field, θ = 90°: the force is a maximum, F = BIL.
• The wire parallel to the field, θ = 0: sin θ is zero, so there is no force at all.

Magnetic flux density B is defined from that maximum case: it is the force acting per unit current per unit length on a wire placed at right angles to the field. Its unit is the tesla, where 1 T is 1 N A⁻¹ m⁻¹. Getting the words "at right angles" into the definition is required.

Fleming's left-hand rule gives the direction. First finger for the field, second finger for the current, thumb for the motion, that is the force, and all three mutually perpendicular. Use the left hand for the motor effect and the right hand for induction, and remember that the current is conventional current, so for electrons the second finger points opposite to their motion.

Worked example. A wire of length 8.0 cm carries a current of 3.5 A at 30° to a field of flux density 0.25 T.

F = 0.25 × 3.5 × 0.080 × 0.5 = 0.035

so the force is 3.5 × 10⁻² N. At right angles it would have been twice that, since sin 90° is 1.

Force on a moving charge

A current is charge in motion, so the same effect applies to a single charge:

F = BQv sin θ

with the direction again from Fleming's left-hand rule, taking the direction of conventional current as the direction of motion of a positive charge. A negative charge moving to the right is a conventional current to the left, so its force is in the opposite direction.

Circular motion in a magnetic field

If a charged particle moves perpendicular to a uniform magnetic field, the force is always perpendicular to the velocity. From topic 12, that is exactly the condition for circular motion at constant speed.

The force does no work, because it is perpendicular to the motion, so the speed and kinetic energy are unchanged. Only the direction changes.

Equating the magnetic force to the centripetal force:

BQv = mv² / r, so r = mv / (BQ)

This is the basis of the mass spectrometer, the cyclotron and the bending magnets of a particle accelerator. Reading the equation: a faster or more massive particle follows a larger circle, while a stronger field or a larger charge tightens it.

Worked example. An electron travelling at 2.0 × 10⁷ m s⁻¹ enters a field of 4.0 mT at right angles.

r = 9.11 × 10⁻³¹ × 2.0 × 10⁷ / (4.0 × 10⁻³ × 1.60 × 10⁻¹⁹) = 0.02847

so the radius is about 2.8 cm.

If the particle enters at some other angle, the component of velocity along the field is unaffected and the perpendicular component goes in a circle, so the path is a helix. That is beyond the required detail, but it explains why the perpendicular condition is always stated.

Velocity selection

Cross an electric field and a magnetic field so that the two forces on a moving charge act in opposite directions. The electric force is qE and does not depend on speed; the magnetic force is Bqv and does. So there is exactly one speed at which they balance:

qE = Bqv, so v = E / B

Charges at that speed pass straight through. Faster ones are deflected by the larger magnetic force, slower ones by the electric force. Note that the selected speed does not depend on the charge or the mass, so a velocity selector passes particles of one speed regardless of what they are, which is precisely why it is used to feed a mass spectrometer.

Worked example. A velocity selector uses plates 5.0 cm apart with 2.0 kV across them and a magnetic flux density of 0.080 T.

E = 2000 / 0.050 = 40000

v = 40000 / 0.080 = 500000

so it selects particles travelling at 5.0 × 10⁵ m s⁻¹.

The Hall effect

Pass a current through a thin slice of conductor with a magnetic field perpendicular to it. The charge carriers moving along the slice experience a force sideways, so they build up on one edge, leaving the opposite edge with the opposite charge. That separation creates an electric field across the slice, and therefore a potential difference: the Hall voltage.

Charge stops building up when the electric force balances the magnetic force. That equilibrium gives the expression.

The electric field across the slice of width d is V_H/d, so the electric force on a carrier is qV_H/d, and setting it equal to the magnetic force Bqv:

qV_H / d = Bqv, so V_H = Bvd

The current through a slice of thickness t and width d is I = nqvA with A = td, so

v = I / (nqtd)

Substituting:

V_H = B × I / (nqtd) × d, which gives V_H = BI / (ntq)

Read the result: the Hall voltage is proportional to B for a fixed current, which is what makes it a way of measuring a field, and it is larger for a thin slice and for a material with a small n. That last point is why Hall probes use a semiconductor rather than a metal: n in a semiconductor is many orders of magnitude smaller, so the Hall voltage is large enough to measure easily.

A Hall probe is that slice with a constant current through it and a voltmeter across it, calibrated so the reading gives B directly. It must be held with its face perpendicular to the field, since it responds only to the perpendicular component, and rotating it until the reading is a maximum is how you find the field direction.

Magnetic fields due to currents

Three patterns must be sketched, and the right-hand grip rule gives every one of them: point the thumb along the conventional current and the fingers curl the way the field goes.

• Long straight wire. Concentric circles around the wire, spaced further apart with distance, in a plane perpendicular to the wire.
• Flat circular coil. Circles close to the wire on each side, becoming a roughly straight field through the centre of the coil, with the field strongest at the centre.
• Long solenoid. Field lines run straight and evenly spaced inside the solenoid, so the field there is strong and uniform, and spread out outside, where the pattern looks like that of a bar magnet. The end from which the lines emerge is the north pole.

Adding a ferrous core, such as soft iron, to a solenoid greatly increases the flux density, because the core becomes magnetised and its own field adds to that of the current. That is what makes an electromagnet useful, and soft iron is chosen because it loses its magnetism when the current is switched off.

Forces between current-carrying conductors

Two parallel wires each sit in the other's magnetic field, so each experiences a force from the motor effect. Applying the right-hand grip rule to find one field and Fleming's left-hand rule to find the force on the other wire gives the result:

• Currents in the same direction: the wires attract.
• Currents in opposite directions: they repel.

That is the opposite way round from charges, where like repels, and it is worth noting the contrast deliberately so the two do not get merged.

The explanation the mark scheme wants is the two-step one: wire 1 produces a magnetic field at wire 2; wire 2 carries a current in that field and so experiences a force. Then the same for the other wire, and by Newton's third law the two forces are equal and opposite.

Magnetic flux and flux linkage

Magnetic flux Φ is the product of the magnetic flux density and the cross-sectional area perpendicular to the direction of the flux density:

Φ = BA

measured in webers, where 1 Wb is 1 T m². If the field is at an angle, only the perpendicular component counts, so Φ = BA cos θ where θ is the angle between the field and the normal to the area.

Magnetic flux linkage is the flux multiplied by the number of turns in a coil:

flux linkage = NΦ = NBA

measured in weber-turns. The distinction matters because the induced e.m.f. depends on the flux linkage, so a coil of 500 turns produces 500 times the e.m.f. of a single loop in the same changing field.

Electromagnetic induction

Three experiments demonstrate the effect and are worth being able to describe:

• Moving a magnet into a coil connected to a sensitive ammeter. A current is registered while the magnet moves, and stops the instant it stops, showing that it is the change of flux that matters, not the presence of flux.
• Reversing the direction of motion, or reversing the magnet, reverses the current, and the direction is always such as to oppose the change.
• Varying the speed, the number of turns and the strength of the magnet, each of which increases the reading, showing what the magnitude depends on.

Faraday's law

The magnitude of the induced e.m.f. is proportional to the rate of change of magnetic flux linkage.

E = -d(NΦ)/dt

Worked example. A coil of 250 turns and area 1.5 × 10⁻³ m² sits in a field of 0.40 T perpendicular to its plane. The field falls to zero in 0.020 s.

Initial flux linkage:

250 × 0.40 × 1.5 × 10⁻³ = 0.15

E = 0.15 / 0.020 = 7.5

so an e.m.f. of 7.5 V is induced.

Lenz's law

The direction of the induced e.m.f. is such that it opposes the change producing it. That is what the minus sign in Faraday's law records.

Lenz's law is a statement of conservation of energy, and saying so is worth a mark. If the induced current helped the change instead of opposing it, the motion would accelerate on its own and energy would appear from nowhere. Because it opposes, work must be done against the induced force, and that work is exactly the electrical energy generated.

The practical consequences follow: pushing a magnet north pole first into a coil induces a current that makes the near end of the coil a north pole, repelling it, so you have to push. Pulling it out induces a current that makes the near end a south pole, attracting it, so you have to pull. Dropping a magnet down a copper tube is slow for the same reason.

Common mistakes

• Using the right hand for the motor effect. Left hand for the force on a current, right hand for induction.
• Forgetting that a wire parallel to the field feels no force, or using sin θ where the angle given is to the normal rather than to the field.
• Saying the magnetic force on a moving charge changes its speed. It is always perpendicular to the velocity, so it does no work.
• Defining magnetic flux density without the condition that the wire is at right angles to the field.
• Saying the Hall voltage is bigger in a metal. It is bigger in a semiconductor, because n is smaller.
• Saying like currents repel. Parallel currents in the same direction attract.
• Confusing flux with flux linkage, and leaving N out of an induced e.m.f. calculation.
• Saying an e.m.f. is induced because there is a flux through the coil. It is induced by a change of flux.
• Quoting Faraday's law as "proportional to the flux linkage" rather than to its rate of change.
• Stating Lenz's law without saying that it follows from conservation of energy.
• Forgetting to convert an area given in cm² to m², which costs a factor of 10⁴.