Lesson 11: Magnetism and Induction (HL)

Magnetic fields exert forces on moving charges and currents, and changing magnetic flux generates emf. The catalogue marks this lesson HL because Theme D.4 (induction) is HL-only, but the motor effect and magnetic force on a charge (D.2/D.3) are SL core — both are developed here so the SL foundation is solid and the HL extension in Lesson 23 has its groundwork. Take this lesson after electric fields: the two field theories rhyme throughout.

The motor effect and force on a charge

A current-carrying conductor in a magnetic field experiences F = BIL sin θ (θ between the field and the current). A single moving charge experiences F = qvB sin θ. Direction comes from Fleming's left-hand rule: First finger Field, seCond finger Current (or positive-charge velocity), THumb Thrust (force). A negative charge (electron) feels the opposite direction. Because the magnetic force is always perpendicular to the velocity, it does no work: it changes direction, not speed — charges spiral or circle in uniform fields, with radius r = mv/(qB), the principle of the cyclotron and the mass spectrometer.

Fields are visualised by field lines (from north to south outside a magnet) and quantified by flux density B in tesla. The field around a long straight wire is circular with B ∝ 1/r; inside a solenoid the field is nearly uniform, B = μ₀nI.

Magnetic flux and Faraday's law

Magnetic flux through an area A is Φ = BA cos θ (θ between the field and the normal to the area); for a coil of N turns the flux linkage is NΦ. Faraday's law: the induced emf equals the rate of change of flux linkage, ε = −N ΔΦ/Δt. Lenz's law (the minus sign) says the induced effects oppose the change producing them — a direct consequence of energy conservation: the mechanical work you do pushing a magnet into a coil becomes the electrical energy produced.

An emf is induced whenever flux linkage changes: moving a magnet, switching a current, rotating a coil (the generator, Lesson 23), or changing the area of a circuit in a field (a sliding rod gives motional emf ε = BLv).

Worked problem 1 — motor effect and charge deflection

(a) A wire of length 0.35 m carries 4.2 A perpendicular to a field of 0.28 T. (b) An electron crosses the same field at 3.0 × 10⁶ m s⁻¹ perpendicular to it. Calculate both forces and the electron's radius of curvature.

(a) F = BIL = 0.28 × 4.2 × 0.35 = 0.41 N
(b) F = qvB = 1.60 × 10⁻¹⁹ × 3.0 × 10⁶ × 0.28 = 1.3 × 10⁻¹³ N
    r = mv/(qB) = (9.11 × 10⁻³¹ × 3.0 × 10⁶)/(1.60 × 10⁻¹⁹ × 0.28) = 2.73 × 10⁻²⁴ / 4.48 × 10⁻²⁰
    r = 6.1 × 10⁻⁵ m

The forces differ by twelve orders of magnitude, yet the same law governs both — scale awareness is an examinable skill.

Worked problem 2 — induced emf in a coil

A coil of 240 turns and cross-sectional area 3.6 × 10⁻³ m² sits in a magnetic field that falls uniformly from 0.85 T to 0.12 T in 0.050 s, field perpendicular to the coil's plane. Calculate the induced emf.

ΔΦ per turn = A ΔB = 3.6 × 10⁻³ × (0.85 − 0.12) = 2.63 × 10⁻³ Wb
ε = N ΔΦ/Δt = 240 × 2.63 × 10⁻³ / 0.050 = 12.6 V ≈ 13 V

If the coil's circuit resistance is 48 Ω, the induced current is 0.26 A; by Lenz's law it flows so as to create a field in the original direction, opposing the decrease.

Practical worked example — measuring a solenoid's field with a search coil

A search coil (N = 500 turns, area 2.2 × 10⁻⁴ m²) inside a solenoid is switched off; a datalogger records a peak emf of 0.84 V with the flux falling to zero in 12 ± 1 ms. Estimate B.

B = ε Δt/(N A) = 0.84 × 0.012/(500 × 2.2 × 10⁻⁴) = 0.01008/0.11 = 0.092 T

The timing uncertainty (1/12 ≈ 8%) dominates, so B = 0.092 T ± about 8%. Repeating with the current reversed and averaging both traces cancels any zero offset in the datalogger — a neat systematic-error control worth quoting in the IA.

Nature of Science

Faraday, largely self-taught, developed field lines from laboratory intuition; Maxwell later expressed the same ideas in four equations that predicted electromagnetic waves. The interplay of experimental insight and mathematical formalisation — each driving the other — is the pattern of much of modern physics.

Real-world application

Contactless payment and RFID work by induction: the reader's alternating field induces an emf in the card's coil, powering its chip, and the chip modulates the flux linkage to send data back. No battery, no contact — Faraday's law in your wallet. The same principle charges electric toothbrushes and some phones through plastic casings.

Common mistakes (examiner style)

• Calculating flux with θ measured from the plane of the coil instead of the normal (a cos/sin swap).
• Forgetting the N in flux linkage for multi-turn coils.
• Applying Lenz's law as "opposes the field" rather than "opposes the change": if flux is decreasing, the induced field reinforces it.
• Believing a magnetic force can speed up a charge (it cannot do work).
• In r = mv/(qB), using the particle's energy instead of its momentum, or forgetting that electrons curve the other way.

IB Exam Experience

Paper 1-style questions. 1. A bar magnet is dropped through a horizontal coil. As it approaches, the induced current opposes the approach. The emf is largest when the magnet: A. is held stationary above the coil B. moves slowest C. moves fastest near the coil D. is far away. Key: C — ε ∝ rate of change of flux linkage, greatest at greatest speed and strongest field gradient. 2. A proton enters a uniform magnetic field at right angles. Its path is: A. a straight line B. a parabola C. a circle D. a helix along the field. Key: C — F ⊥ v always, so uniform circular motion (a helix needs a velocity component along B). 3. Doubling both the number of turns and the rate of change of flux in a coil multiplies the induced emf by: A. 2 B. 4 C. 8 D. 1. Key: B — ε = N ΔΦ/Δt, both factors double.

Paper 2-style structured question (7 marks). A coil of 150 turns and area 2.0 × 10⁻³ m² rotates in a uniform field of 0.30 T. The flux linkage varies from maximum to zero in 0.010 s. (a) Calculate the maximum flux linkage. [2] (b) Calculate the average induced emf during this quarter-turn. [2] (c) The coil rotates at 25 revolutions per second. Explain why the emf is alternating and state its frequency. [2] (d) The coil is connected to a 12 Ω resistor. Calculate the average current during the quarter-turn. [1] *Marking notes:* (a) NΦ = NAB = 150 × 2.0 × 10⁻³ × 0.30 = 0.090 Wb-turns. (b) ε = 0.090/0.010 = 9.0 V. (c) Each half-turn reverses the sense of the flux change, so the emf changes sign twice per revolution; frequency = rotation frequency = 25 Hz. (d) I = 9.0/12 = 0.75 A. Candidates who use the peak emf formula ε₀ = NABω without comment lose the thread of the question — the question gives times precisely so that Faraday's law applies directly.

IA idea: Investigate how the induced emf in a falling-magnet-through-coil experiment depends on the magnet's drop height (speed), and whether ε ∝ v holds once flux geometry is accounted for.

Motional emf derived from flux

A conducting rod of length L slides at speed v on rails separated by L in a field B. In time Δt it sweeps area LvΔt, so the flux change is B × LvΔt and ε = ΔΦ/Δt = BLv — the formula of Lesson 11 from first principles, no memorisation needed. The same result comes from forces: the field separates charges in the moving rod (F = qvB on each carrier) until the electric field built up balances at E = vB, giving ε = EL = BLv. Two independent derivations agreeing is how physicists trust a result; being able to produce either is HL fluency.

Eddy currents: induction's double edge

In bulk conductors, changing flux drives swirling eddy currents whose fields oppose the change (Lenz). They are a nuisance in transformer cores (heating; fought by laminating the core into insulated sheets that break the loops) and a tool elsewhere: eddy-current braking stops trains and rollercoasters smoothly with no contact and no wear; induction hobs heat the pan directly; metal detectors sense the secondary field of eddies induced in buried metal. One phenomenon, three valuations — context decides whether an effect is a loss or a product.

The cyclotron: fields combined

A cyclotron uses a magnetic field to bend charged particles in circles (r = mv/qB) and an alternating electric field to accelerate them across a gap each half-turn. The elegance: the revolution frequency f = qB/(2πm) is independent of speed (non-relativistically), so a fixed-frequency oscillator stays in step as the spiral grows. At relativistic energies the frequency falls (γ grows), which is why modern machines either modulate the frequency (synchrocyclotrons) or vary B along the radius — a limitation Einstein's theory imposed on Newton's machine, visible in one line of algebra.

Self-induction: the coil resists change in itself

A coil's own changing current changes its own flux linkage, inducing a back emf ε = −L ΔI/Δt, where L is the inductance (henry). This is why switching off an inductive circuit draws a spark (the collapsing flux drives the emf high enough to ionise air) and why fluorescent starters and car ignition coils work at all — the ignition coil is a deliberate exploitation of a huge ΔI/Δt. Inductance's opposition to *change* in current is the exact analogue of inertia's opposition to change in velocity, and of capacitance's opposition to change in voltage: three components, three "reluctances", one pattern.