D.4 Induction | IB DP Physics

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This is the closest thing to a one-to-one match with the old syllabus, but it is narrower than the retired Unit 11: that unit also carried power generation, transmission and capacitance, which are not part of D.4. Old notes and past papers on this topic will cover more than you now need.

🎯What you need to be able to do

Calculate magnetic flux and flux linkage.
Apply Faraday’s law to find an induced emf.
Apply Lenz’s law to find the direction of an induced current, and relate it to conservation of energy.
Use \( \varepsilon = BvL \) for a conductor moving through a field.

📚The physics

Magnetic flux \( \Phi = BA\cos\theta \) measures how much field passes through a loop, where \(\theta\) is the angle between the field and the normal to the area — not the plane of the loop. Get that wrong and every sine becomes a cosine. Flux is maximum when the field is perpendicular to the loop’s plane, and zero when the field lies in the plane. The unit is the weber.

Flux linkage is \( N\Phi \) for a coil of \(N\) turns, because each turn links the same flux.

Faraday’s law is the heart of the topic:

\[ \varepsilon = -N\frac{\Delta\Phi}{\Delta t} \]

The induced emf depends on the rate of change of flux linkage, not on the flux itself. A coil sitting in the strongest field imaginable has no emf induced in it as long as nothing changes. Something must vary — the field strength, the area, or the orientation.

Lenz’s law is that minus sign made physical: the induced current flows in the direction that opposes the change producing it. Push a magnet north-pole-first towards a coil and the coil’s near face becomes a north pole, repelling it. Pull it away and the near face becomes a south pole, attracting it. Either way the coil resists what you are doing.

Why it must be that way is worth being able to argue, because it is a standard extended-response question. Suppose the induced current helped the motion instead. The magnet would accelerate, inducing a larger current, accelerating it further — energy from nothing. Lenz’s law is conservation of energy expressed in the language of induction. The work you do pushing against that opposition is exactly the electrical energy that appears.

A conductor moving through a field is the simplest case: \( \varepsilon = BvL \), where \(L\) is the length in the field and \(B\), \(v\) and \(L\) are mutually perpendicular. This comes straight from Faraday — the rod sweeps out area at a rate \(Lv\), so the flux changes at a rate \(BLv\). If the rod is part of a complete circuit a current flows, and then a force \(BIL\) acts on the rod opposing its motion, which is Lenz again.

Eddy currents are induced currents in bulk metal. They dissipate energy as heat and oppose relative motion, which is why a magnet dropped down a copper pipe falls in slow motion, and why induction hobs and magnetic braking work.

✏️Worked example

A coil of 250 turns and area \( 3.2 \times 10^{-3} \) m\(^{2}\) lies with its plane perpendicular to a magnetic field. The field falls steadily from 0.48 T to 0.12 T in 0.15 s.

(a) Initial and final flux linkage. With the field perpendicular to the plane, \( \theta = 0 \) and \( \cos\theta = 1 \). Initially \( N\Phi = 250 \times 0.48 \times 3.2 \times 10^{-3} = 0.384 \) Wb. Finally \( N\Phi = 250 \times 0.12 \times 3.2 \times 10^{-3} = 0.096 \) Wb.

(b) Induced emf. \( \varepsilon = \Delta(N\Phi)/\Delta t = (0.384 - 0.096)/0.15 = 0.288/0.15 = 1.9 \) V.

(c) If the coil has resistance 6.0 Ω, what current flows? \( I = \varepsilon/R = 1.92/6.0 = 0.32 \) A.

(d) In which direction? The flux through the coil is decreasing, so by Lenz’s law the induced current flows so as to maintain it — that is, it circulates in the direction that produces a field in the same direction as the original. If the field had been increasing, the current would reverse.

The trap in (d). Students often answer “opposite to the field” because Lenz’s law contains the word “opposes”. It opposes the change, not the field. A falling flux is opposed by supporting it.

🔭See it happen

Drop a small strong magnet down a copper pipe and time it against a non-magnetic slug of the same size. The magnet takes several seconds where the slug takes a fraction of one. Nothing is touching the magnet — the eddy currents it induces in the copper create fields that oppose its fall, exactly as Lenz requires.

📝Practise

Work through these, then reveal the answer. Each question targets a different objective from the list above.

1. A coil of area 0.025 m\(^{2}\) sits with its plane perpendicular to a magnetic field of 0.30 T. Find the magnetic flux through it.

With the field perpendicular to the plane it is parallel to the normal, so \( \theta = 0 \): \( \Phi = BA\cos\theta = 0.30 \times 0.025 = 7.5 \times 10^{-3} \) Wb.

2. The same coil is tilted so the field is at 30° to the normal of the coil. Find the new flux.

\( \Phi = BA\cos 30^\circ = 0.30 \times 0.025 \times 0.866 = 6.5 \times 10^{-3} \) Wb. The angle is measured to the normal, not the plane — getting that wrong turns every cosine into a sine.

3. A 400-turn coil has the flux through it fall from \( 8.0 \times 10^{-3} \) Wb to \( 2.0 \times 10^{-3} \) Wb in 0.050 s. Find the magnitude of the induced emf.

\( \varepsilon = N\dfrac{\Delta\Phi}{\Delta t} = 400 \times \dfrac{6.0 \times 10^{-3}}{0.050} = 48 \) V.

4. A rod 0.25 m long moves at 4.0 m s\(^{-1}\) perpendicular to a 0.60 T field. Find the emf induced across it.

\( \varepsilon = BvL = 0.60 \times 4.0 \times 0.25 = 0.60 \) V.

5. A magnet is pushed north-pole-first towards a coil. State the polarity the near face of the coil develops, and explain why in terms of energy.

The near face becomes a north pole, repelling the approaching magnet. If it attracted instead, the magnet would accelerate, inducing a larger current, accelerating it further — energy from nothing. Lenz’s law is conservation of energy in the language of induction: the work you do pushing against the opposition is exactly the electrical energy that appears.

6. Explain why a strong magnet dropped down a vertical copper pipe falls far more slowly than a non-magnetic slug of the same size and mass.

The moving magnet changes the flux through each ring of copper it passes, inducing circulating eddy currents. By Lenz’s law those currents produce magnetic fields that oppose the change — that is, they oppose the magnet’s motion, retarding its fall. Nothing touches the magnet; the braking force is entirely electromagnetic, and the lost gravitational energy appears as heat in the copper.

🔗Go deeper — other people’s work

These are external resources, not mine. If one stops working, tell me and everything above it on this page still stands.

HyperPhysics — Faraday’s law and Lenz’s law
The Physics Hypertextbook — electromagnetic induction