Magnetic Fields

This guide covers Topic 20, Magnetic fields, in full — subtopics 20.1 Concept of a magnetic field, 20.2 Force on a current-carrying conductor, 20.3 Force on a moving charge, 20.4 Magnetic fields due to currents and 20.5 Electromagnetic induction — from Cambridge International AS & A Level Physics 9702, 2025–2027 series. This is A Level content and the most extensive single topic in the electricity and magnetism strand.

Before studying this

This resource assumes electric current from Electricity: Current, Potential Difference and Resistance, and circular motion from Motion in a Circle.

Syllabus coverage

CAMBRIDGE INTERNATIONAL AS & A LEVEL PHYSICS 9702 — A Level, Topic 20

20.1 Concept of a magnetic field — understanding that a magnetic field is an example of a field of force produced either by moving charges or by permanent magnets; representing a magnetic field by field lines.

20.2 Force on a current-carrying conductor — recalling and using F = BIL sin θ for the force on a current-carrying conductor in a magnetic field; understanding how the force on a current-carrying conductor can be used to define magnetic flux density, and defining the tesla; explaining the forces between current-carrying conductors and predicting the direction of these forces.

20.3 Force on a moving charge — recalling and using F = BQv sin θ for the force on a charge moving in a magnetic field; understanding how the magnetic force can provide the centripetal force needed for the circular motion of charged particles in a uniform magnetic field; deriving and using the Hall voltage V_H = BI/(ntq), and explaining how the Hall effect is used in a Hall probe to measure magnetic flux density; describing the use of crossed (perpendicular) electric and magnetic fields in velocity selection.

20.4 Magnetic fields due to currents — sketching magnetic field patterns due to currents in a long straight wire, a flat circular coil and a long solenoid; understanding that the magnetic field due to a current increases with current and decreases with distance from a current-carrying conductor; understanding the effect of a ferrous core on the strength of the magnetic field produced by a solenoid.

20.5 Electromagnetic induction — defining magnetic flux and magnetic flux linkage; understanding and explaining experiments that demonstrate induction, the direction of the induced e.m.f. and the factors affecting its size; recalling and using Faraday’s law of electromagnetic induction; recalling and using Lenz’s law to determine the direction of an induced e.m.f.

The magnetic field concept

A magnetic field is a field of force produced by moving charges (electric currents) or permanent magnets, represented using field lines that form closed loops, running from north to south poles outside a magnet.

Force on a current-carrying conductor

A current-carrying conductor placed in a magnetic field experiences a force:

F = BIL sin θ

where B is the magnetic flux density, I is the current, L is the length of conductor in the field, and θ is the angle between the current direction and the field. This defines magnetic flux density: the tesla is the flux density that produces a force of 1 N per metre of conductor carrying 1 A, perpendicular to the field.

Forces between current-carrying conductors

Each current-carrying conductor produces its own magnetic field, so two parallel conductors each experience a force due to the other’s field. Two wires carrying current in the same direction attract; two wires carrying current in opposite directions repel. The size of each force follows from F = BIL, with B taken as the flux density produced by one wire at the location of the other — so the force per unit length increases with both currents and decreases with the separation between the wires.

Force on a moving charge

A charge moving through a magnetic field also experiences a force:

F = BQv sin θ

Because this force acts perpendicular to velocity, it does no work and changes only the direction of motion, not its speed. When a charged particle enters a uniform field perpendicular to its velocity, this magnetic force provides exactly the centripetal force needed for uniform circular motion — connecting directly to F = mv²/r from circular motion.

Worked example. An electron (charge 1.6 × 10⁻¹⁹ C) moves at 2.0 × 10⁶ m s⁻¹ perpendicular to a magnetic field of flux density 0.50 T. The force on it:

F = BQv = 0.50 × 1.6 × 10⁻¹⁹ × 2.0 × 10⁶ = 1.6 × 10⁻¹³ N

The Hall effect and the Hall probe

When a current-carrying slice of material (typically a semiconductor) is placed in a magnetic field perpendicular to the current, the magnetic force on the moving charge carriers pushes them sideways, building up a charge imbalance across the slice until the resulting electric field exerts an equal and opposite force on the carriers. This sideways potential difference is the Hall voltage:

V_H = BI / (ntq)

where B is the magnetic flux density, I is the current, n is the number density of charge carriers, t is the thickness of the slice in the direction of B, and q is the charge on each carrier. Because V_H is proportional to B for a given slice, current and carrier density, this effect is used in a Hall probe: a calibrated Hall slice whose voltage output gives a direct reading of magnetic flux density.

Deriving the Hall voltage. Let the slice have width d (the dimension across which V_H appears) and thickness t (in the direction of B), so its cross-sectional area is A = dt. Each carrier, of charge q and drift speed v, feels a magnetic force Bqv. The Hall voltage builds up until the electric force from the Hall field E = V_H/d balances it:

qE = Bqv   ⟹   V_H / d = Bv   ⟹   V_H = Bvd

The current is I = nAvq = n(dt)vq, so v = I/(ndtq). Substituting:

V_H = B × I/(ndtq) × d = BI/(ntq)

The width d cancels, which is why V_H depends on the thickness t but not on the width of the slice. A small n (as in a semiconductor) gives a larger, more easily measured Hall voltage.

Velocity selection

A velocity selector uses crossed (perpendicular) electric and magnetic fields to allow only charged particles of one particular speed through undeflected. The electric force (qE) and the magnetic force (qBv) act in opposite directions on the particle; only when these balance,

qE = qBv₀   ⟹   v₀ = E/B

does the particle pass through in a straight line — particles with any other speed experience a net sideways force and are deflected out of the beam.

Magnetic fields due to currents

A long straight current-carrying wire produces a field of concentric circles around it, whose flux density increases with current and decreases with distance from the wire. A flat circular coil produces a field resembling that of a short bar magnet at its centre. A long solenoid produces a strong, uniform field inside it, similar to a bar magnet’s field but concentrated and controllable via the current.

Placing a ferrous (iron) core inside a solenoid greatly increases the flux density produced for the same current, because the core itself becomes magnetised and adds its own much stronger field to the solenoid’s — this is the basis of an electromagnet, and is why relays and electric bells use an iron-cored coil rather than an air-cored one.

Electromagnetic induction

Magnetic flux Φ through an area is the product of magnetic flux density and the area perpendicular to the field. Magnetic flux linkage is the product of flux and the number of turns of a coil, NΦ.

Faraday’s law states that the magnitude of an induced e.m.f. is proportional to the rate of change of magnetic flux linkage. Lenz’s law states that an induced e.m.f. always acts in a direction to oppose the change producing it — this determines the direction (sign) of the induced e.m.f. and is a direct consequence of conservation of energy.

Experiments that demonstrate electromagnetic induction

The syllabus (20.5) asks you to understand and explain experiments showing three things: that a changing magnetic flux can induce an e.m.f. in a circuit; that the induced e.m.f. is in a direction that opposes the change producing it; and which factors affect the size of the induced e.m.f. The standard demonstrations use a coil of many turns (or a single wire) connected to a sensitive centre-zero ammeter (galvanometer), which shows both the size and the direction of a small current.

1. A changing flux induces an e.m.f.

• Magnet and coil. Push a bar magnet into the coil: the meter deflects briefly. Hold the magnet still inside the coil: the reading returns to zero, even though flux still links the coil. Pull the magnet out: the meter deflects the other way. Moving the coil instead of the magnet gives the same results. An e.m.f. is induced only while the flux linkage is changing.
• Wire and magnet. Move a straight wire, connected to the meter, down through the field between the poles of a U-shaped magnet: the meter deflects while the wire cuts across the field lines. Moving it up gives a deflection the other way. Holding it still, or moving it parallel to the field lines, gives no deflection, because the flux through the circuit does not change.
• Two coils, no motion. Wind two coils on the same iron core, one connected to a cell and switch, the other to the meter. The meter deflects briefly when the switch is closed and, the other way, when it is opened, but not while a steady current flows. A changing flux induces an e.m.f. even when nothing moves.

The meter detects the current that the induced e.m.f. drives round the complete circuit. The e.m.f. is induced whether or not the circuit is complete; it can be shown with a high-resistance voltmeter or an oscilloscope instead.

2. Factors that affect the size of the induced e.m.f.

With the magnet-and-coil arrangement, the maximum deflection is larger when:

• the magnet is moved faster (the flux linkage changes in less time);
• a stronger magnet is used (a larger change in flux density B);
• the coil has more turns N (a larger change in flux linkage NΦ);
• the coil is wound on an iron core and the magnet is moved towards its end, so that the iron concentrates more of the magnet’s flux through the turns.

A larger coil area does not help here: a wider loop around a bar magnet also encloses more of the magnet’s return field, which points the opposite way, so the net flux linked is not increased. Area matters for a coil in a uniform field (for example between the poles of a U-shaped magnet), where a larger area gives a larger flux linkage NBA.

Each of these increases the rate of change of flux linkage, which is exactly what Faraday’s law says the induced e.m.f. is proportional to. Pushing the magnet in and pulling it out at the same speed gives deflections of the same size in opposite directions.

3. The direction of the induced e.m.f. (Lenz’s law)

First find which way the meter deflects for a known current direction (for example, briefly connect a cell through a large resistor). Then:

• as the north pole approaches the coil, the induced current makes the end of the coil nearest the magnet a north pole, which repels the magnet and opposes its approach;
• as the north pole is withdrawn, the induced current reverses and that end becomes a south pole, which attracts the magnet and opposes its withdrawal.

The pole produced at the coil’s end can be checked against the direction of the induced current using the right-hand grip rule for a coil. Because the induced effect always opposes the motion, work must be done to move the magnet, and that work is the source of the electrical energy — Lenz’s law is conservation of energy. In the same way, a magnet dropped through a vertical copper tube falls more slowly than through a plastic one: currents induced in the copper oppose its fall.

Common mistakes

• Forgetting the sin θ term in F = BIL sin θ or F = BQv sin θ — the force is maximum when the current/velocity is perpendicular to the field (θ = 90°) and zero when parallel to it (θ = 0°).
• Assuming the magnetic force on a moving charge does work on it — since the force is always perpendicular to velocity, it changes direction only, never speed or kinetic energy.
• Confusing magnetic flux (Φ, through one turn) with flux linkage (NΦ, for a coil of N turns) — Faraday’s law is stated in terms of flux linkage.
• Getting the direction from Lenz’s law wrong — the induced effect always opposes the change that caused it, never reinforces it.
• Mixing up the directions in the parallel-conductor force rule — same directions attract, opposite directions repel; check with F = BIL and the right-hand/left-hand rule rather than trying to memorise it directly.
• Forgetting that a velocity selector only passes one speed — any charge moving faster or slower than v₀ = E/B feels a net force and is deflected.

Quick revision checklist

• F = BIL sin θ, and defining the tesla from it
• Forces between current-carrying conductors: same direction attracts, opposite repels
• F = BQv sin θ, and how it provides centripetal force for charged particles in circular motion
• V_H = BI/(ntq), the Hall voltage: its origin, its derivation from qE = Bqv and I = nAvq, and how a Hall probe measures B
• Velocity selection: v₀ = E/B when the electric and magnetic forces balance
• Field patterns for a straight wire, flat coil and solenoid, and the effect of a ferrous core
• Faraday’s law (rate of change of flux linkage) and Lenz’s law (opposing direction)
• The induction experiments: magnet and coil with a galvanometer, a wire moved through a magnet’s field, the factors that increase the induced e.m.f., and finding its direction with Lenz’s law

Related resources

• Capacitance — the previous A Level topic
• Alternating Currents — the next A Level topic, built on electromagnetic induction
• Cambridge AS & A Level Physics hub

Written against Cambridge International AS & A Level Physics 9702, 2025–2027 series. Always check the current syllabus for your examination year.