Induction is the topic that closes IB Physics HL Theme D — and it's where every single concept from D.1 to D.3 finally pays off. Topic D.4 is also notorious for two things: the angle convention in the flux equation $\Phi = BA\cos\theta$ (the most common arithmetic trap in HL physics) and the conceptual demand of Lenz's law (which the IB examiners love to combine with a sketching question). The good news: once you internalise that EMF responds to the change in flux, every D.4 question collapses into the same algorithm.
This cheatsheet condenses every formula, graph, trick and exam-mark trap from D.4 (AHL) into a one-page revision tool. Scroll to the bottom for the printable PDF, the full Notes, the Tutorial booklet, and the marked-up solutions in the gated full library.
§1 — Induced EMF & Magnetic Flux D.4 AHL
Key formulas
EMF in moving wire:$\displaystyle \varepsilon = Blv$
$N$-turn coil:$\displaystyle \varepsilon = BvlN$ (entering / leaving a field)
Magnetic flux:$\displaystyle \Phi = BA\cos\theta$
Flux density:$\displaystyle B = \frac{\Phi}{A}$
Unit:$1\;\mathrm{Wb} = 1\;\mathrm{T \cdot m^2}$
Note$\theta$ is the angle between $\vec{B}$ and the normal to the surface, NOT between $\vec{B}$ and the surface itself.
TrickIf a question states "the field makes angle $\alpha$ with the surface", convert: $\theta = 90° - \alpha$, so $\Phi = BA\sin\alpha$. Always re-read the question to confirm which angle is given.
TrapWhen a field reverses direction, $\Delta B = 2B$, so $\Delta\Phi = 2BA$, NOT $BA$. This factor-of-2 trap appears in nearly every Lenz/Faraday calculation.
§2 — Faraday's Law & Lenz's Law D.4 AHL
Faraday's law:$\displaystyle \varepsilon = -N\frac{\Delta\Phi}{\Delta t}$
Flux linkage:$\displaystyle \Lambda = N\Phi$ [unit: Wb]
Lenz's law — quick reference
- The induced current opposes the change in flux (not the flux itself).
- It is a direct consequence of the conservation of energy.
- An approaching magnet $\to$ the coil repels it; a retreating magnet $\to$ the coil attracts it.
- Use the right-hand rule to convert "required induced $\vec{B}$" $\to$ "direction of induced current".
TrickFor rotating coils, always compute $\Phi_i$ and $\Phi_f$ separately (with their cosines), then $\Delta\Phi = \Phi_f - \Phi_i$. Do NOT assume $\Delta\Phi = BA$.
Trap$\varepsilon$ is maximum when $\Phi = 0$ (the flux is changing fastest at zero crossings). $\varepsilon = 0$ when $\Phi$ is at a maximum or minimum (flux is momentarily not changing). The two graphs are 90° out of phase — sine vs cosine.
§3 — AC Generator D.4 AHL
Sinusoidal EMF:$\displaystyle \varepsilon = \varepsilon_0 \sin(\omega t)$, $\varepsilon_0 = NBA\omega$
Angular velocity:$\displaystyle \omega = 2\pi f = \frac{2\pi}{T}$
Effect of doubling rotation speed
| Quantity | Original | Doubled speed |
|---|---|---|
| Frequency $f$ | $f$ | $2f$ |
| Period $T$ | $T$ | $T/2$ |
| Peak EMF $\varepsilon_0$ | $NBA\omega$ | $2NBA\omega$ |
Doubling the rotation rate increases both amplitude AND frequency — the graph becomes "taller and narrower".
TrickOn a generator graph, if at $t = 0$ you have $\varepsilon = \varepsilon_0$ (maximum), then the coil plane is parallel to $\vec{B}$ (the normal is $\perp$ to $\vec{B}$, so $\Phi = 0$ at that instant). If $\varepsilon = 0$ at $t = 0$, the plane is $\perp$ to $\vec{B}$ (and $\Phi$ is at its maximum).
§4 — RMS Values & AC Power D.4 AHL
RMS current:$\displaystyle I_{\mathrm{rms}} = \frac{I_0}{\sqrt{2}} \approx 0.707\,I_0$
RMS voltage:$\displaystyle V_{\mathrm{rms}} = \frac{V_0}{\sqrt{2}} \approx 0.707\,V_0$
Average power:$\displaystyle \bar{P} = \tfrac{1}{2}I_0 V_0 = I_{\mathrm{rms}} V_{\mathrm{rms}} = I_{\mathrm{rms}}^2 R = \frac{V_{\mathrm{rms}}^2}{R}$
Peak power:$\displaystyle P_0 = I_0 V_0 = 2\bar{P}$
TrapIf you are given the peak voltage on a transformer's primary, find the peak secondary voltage first (using the turns ratio), and only THEN divide by $\sqrt{2}$ to get the RMS. Doing it in the wrong order is a common arithmetic slip.
NoteThe instantaneous power graph $P = I_0 V_0 \sin^2(\omega t)$ has twice the frequency of the $V$-$t$ graph and is always $\geq 0$. The mean of $\sin^2$ over a full cycle is $\tfrac{1}{2}$ — that's where the $\sqrt{2}$ in RMS comes from.
§5 — Self-Induction D.4 AHL
Instantaneous current:$\displaystyle I_{\mathrm{inst}} = \frac{V_{\mathrm{ext}} - \varepsilon_{\mathrm{back}}}{R}$ (inductor + resistor)
Key facts
- The back EMF opposes the change in current (Lenz's law).
- At switch-on: back EMF $= V_{\mathrm{ext}}$, so $I = 0$.
- Steady state: back EMF $= 0$, so $I = V_{\mathrm{ext}}/R$.
- The current follows an exponential growth curve toward $V/R$.
- Application: spark plugs — a sudden current break gives huge $\mathrm{d}I/\mathrm{d}t$ and a very large back EMF.
TrickIn IB exams, self-induction questions almost always ask you to explain the shape of the $I$-$t$ graph. Always mention all four points: (1) back EMF opposes the change in current, (2) as $I$ increases, $\mathrm{d}I/\mathrm{d}t$ decreases, (3) so the back EMF decreases, (4) hence $I$ approaches $V/R$ asymptotically.
§6 — Exam Attack Plan All sections
When you see this in the question — reach for that:
| Question trigger | Reach for |
|---|---|
| "EMF in a wire moving through $\vec{B}$" | $\varepsilon = Blv$ (or $BvlN$ for $N$ turns) |
| "Flux through a coil at angle $\alpha$ to surface" | $\theta = 90° - \alpha$, so $\Phi = BA\sin\alpha$ |
| "Average EMF over a time interval" | $\varepsilon = N\,\Delta\Phi/\Delta t$ — compute $\Phi_i$ and $\Phi_f$ separately |
| "Direction of induced current" | Lenz: induced $\vec{B}$ opposes the change in flux; right-hand grip gives current direction |
| "Field reverses direction" / coil flipped | $\Delta B = 2B$, so $\Delta\Phi = 2BA$ — don't lose the factor of 2 |
| "Peak EMF of generator" | $\varepsilon_0 = NBA\omega$ where $\omega = 2\pi f$ |
| "Effect of doubling rotation rate" | Frequency doubles AND amplitude doubles ($\omega \to 2\omega$) |
| "Mean power in AC circuit" | $\bar{P} = I_{\mathrm{rms}}V_{\mathrm{rms}} = V_{\mathrm{rms}}^2/R$ |
| "Convert peak to RMS" | Divide by $\sqrt{2}$ — but apply transformer ratio first if relevant |
| "Why does inductor current rise gradually?" | Back EMF opposes change — quote all four steps |
Worked Example — IB-Style Rotating Coil
Question (HL Paper 2 style — 7 marks)
A rectangular coil of $N = 200$ turns and area $A = 80\;\mathrm{cm^2}$ rotates with frequency $f = 50\;\mathrm{Hz}$ about an axis perpendicular to a uniform magnetic field $B = 0.040\;\mathrm{T}$.
(a) Calculate the angular velocity $\omega$. (b) Calculate the peak EMF $\varepsilon_0$ generated. (c) Calculate the RMS voltage. (d) State the instant during the rotation when the EMF is zero, and explain why.
Solution
- Angular velocity (a): $\omega = 2\pi f = 2\pi (50) = 314\;\mathrm{rad\,s^{-1}}$ (M1)(A1)
- State peak EMF formula (b): $\varepsilon_0 = NBA\omega$ (M1)
- Convert area to SI: $A = 80\;\mathrm{cm^2} = 80 \times 10^{-4}\;\mathrm{m^2} = 8.0 \times 10^{-3}\;\mathrm{m^2}$ (R1)
- Substitute (b): $\varepsilon_0 = (200)(0.040)(8.0\times 10^{-3})(314) = 20.1\;\mathrm{V}$ (A1)
- RMS (c): $V_{\mathrm{rms}} = \dfrac{\varepsilon_0}{\sqrt{2}} = \dfrac{20.1}{\sqrt{2}} = 14.2\;\mathrm{V}$ (A1)
- Zero EMF instant (d): $\varepsilon = 0$ when the coil's plane is perpendicular to $\vec{B}$ — equivalently, when $\Phi$ is at its maximum (or minimum). At that instant $\mathrm{d}\Phi/\mathrm{d}t = 0$, so by Faraday's law $\varepsilon = 0$. (A1)
Examiner's note: Two punishing errors. (1) Forgetting to convert $\mathrm{cm^2}$ to $\mathrm{m^2}$ — a factor of $10^{-4}$ that destroys every subsequent answer. (2) Saying "EMF is zero when the flux is zero". That is the OPPOSITE of the truth: EMF is zero when the flux is maximum (instantaneously not changing), and EMF is maximum when the flux is zero (changing fastest). Sketch a sine and cosine on the same axes to fix this in muscle memory.
Common Student Questions
What is the angle $\theta$ in $\Phi = BA\cos\theta$?
$\theta$ is the angle between $\vec{B}$ and the normal (perpendicular) to the surface, NOT between $\vec{B}$ and the surface itself. So if a question says "the field makes an angle $\alpha$ with the surface", then $\theta = 90° - \alpha$ and $\Phi = BA\sin\alpha$. Misreading this is the single most common sign/factor error in D.4.
When is the induced EMF a maximum in a rotating coil?
EMF is maximum when the flux is changing fastest — which happens when the flux itself is zero. That is, when the coil plane is parallel to $\vec{B}$ (the normal to the coil is perpendicular to $\vec{B}$). Conversely, the EMF is zero when $\Phi$ is at a maximum or minimum, because at those instants $\Phi$ is momentarily not changing. Sketch a sine and cosine on the same axes — they are 90° out of phase.
What does Lenz's law actually say, and why does it matter?
Lenz's law: the induced current flows in the direction that opposes the change in flux (not the flux itself). It's a direct consequence of conservation of energy — if the induced current reinforced the change, you would get a runaway perpetual-motion machine. Practically, you use it to find the direction of the induced current: figure out which way $\Phi$ is changing, then the induced $\vec{B}$ must point opposite to that change, and the right-hand grip rule gives the current direction.
How do I convert between peak and RMS values?
For a sinusoidal AC signal: $I_{\mathrm{rms}} = I_0/\sqrt{2} \approx 0.707 I_0$, and similarly $V_{\mathrm{rms}} = V_0/\sqrt{2}$. To convert RMS to peak, multiply by $\sqrt{2}$. The average power equation $\bar{P} = I_{\mathrm{rms}} V_{\mathrm{rms}}$ is the practical reason RMS values exist — they let you treat AC like DC for power calculations. Singapore mains voltage of "230 V AC" is the RMS value; the peak is about 325 V.
What happens to the induced EMF if a magnetic field reverses direction?
If $\vec{B}$ reverses (e.g. flipping a coil over, or $B$ going from $+B$ to $-B$), then $\Delta B = 2B$, not $B$. So $\Delta\Phi = 2BA$, doubling the EMF compared to a same-magnitude change without reversal. This "factor of 2" trap appears in almost every Lenz/Faraday calculation paper question — write down both $\Phi_{\mathrm{initial}}$ and $\Phi_{\mathrm{final}}$ explicitly with signs.
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