Faraday's Law of Induction
On 29 August 1831, Michael Faraday at the Royal Institution, London, wound two separate coils around an iron ring and connected one to a battery and the other to a galvanometer. When he connected and disconnected the battery, the galvanometer briefly deflected — showing that only a changing magnetic flux induces an emf. He recorded the effect precisely in his notebook: the rate of change of flux (dΦ/dt) determines the magnitude. This single discovery underpins every generator, transformer, induction cooktop, and wireless charger ever built.
Practise this lesson
Four printable worksheets that build from the foundations up to exam-style questions — start at whatever level suits you.
A magnet is dropped through a copper tube and through a plastic tube of the same dimensions.
- Which tube will the magnet fall through faster? Why?
- What would happen if you used a stronger magnet?
- What would happen if the copper tube were cut lengthwise into a C-shape?
Warm-up — which of the following is required for electromagnetic induction to occur?
Know — Faraday's Law
- The induced emf is proportional to the rate of change of magnetic flux
- $\varepsilon = -N\,\dfrac{\Delta\Phi}{\Delta t}$ for a coil of N turns
- The negative sign indicates the induced emf opposes the change (Lenz's Law)
Understand — Rate of Change Matters
- Fast flux change gives large emf; slow flux change gives small emf
- Constant flux gives zero emf regardless of its magnitude
- More turns multiply flux linkage and increase induced emf
Can Do — Calculate and Predict
- Calculate induced emf given N, $\Delta\Phi$, and $\Delta t$
- Predict the direction of induced current using Lenz's Law
- Apply Faraday's Law to generators and transformers
Core Content
The faster the flux changes, the bigger the induced emf
Hold a bar magnet above a coil connected to a galvanometer — the needle reads zero. Now plunge the magnet in quickly: the needle kicks sharply to one side, then returns to zero once the magnet is stationary inside the coil. Pull the magnet out quickly: the needle kicks the other way. The faster you move the magnet, the larger the deflection. This is Faraday's Law in action: it is the rate of flux change, not the flux itself, that drives an emf — and the quantitative relationship is:
$$\varepsilon = -N \frac{\Delta \Phi}{\Delta t}$$
- $\varepsilon$ — induced emf (V)
- $N$ — number of turns in the coil
- $\Delta\Phi$ — change in magnetic flux (Wb)
- $\Delta t$ — time interval over which flux changes (s)
- The negative sign indicates the direction opposes the change (Lenz's Law)
Key insight: A small flux changing quickly can induce a large emf. A very large flux that is constant induces no emf at all. This is why alternating currents are essential in transformers — they ensure flux is always changing.
For a single loop ($N = 1$): $\varepsilon = -\Delta\Phi/\Delta t$. For $N$ turns, the total flux linkage is $N\Phi$, so $\varepsilon = -N\,\Delta\Phi/\Delta t$. Every extra turn adds the same contribution, which is why generator coils and transformer windings use many turns.
A coil of 100 turns experiences a flux change from 0.020 Wb to 0.050 Wb in 0.10 s. Calculate the average induced emf. What would the emf be if the same flux change occurred in 0.010 s?
Faraday's Law: $\varepsilon = -N\,\Delta\Phi/\Delta t$ (V). Larger $N$, larger $\Delta\Phi$, or smaller $\Delta t$ → larger $|\varepsilon|$. Constant flux → zero emf. Units: Wb/s = V. The negative sign signals the direction is given by Lenz's Law.
Pause — copy the highlighted Faraday's Law formula and key proportionalities into your book before moving on.
A 50-turn coil has flux changing from 0.10 Wb to 0.30 Wb in 0.40 s. The average induced emf is:
Where this law appears in everyday technology
We just saw that $\varepsilon = -N\Delta\Phi/\Delta t$ quantifies induced emf. That raises a question: which real devices actually run on this principle? This card answers it → generators, transformers, induction cooktops, wireless chargers, and magnetic brakes all rely on changing magnetic flux.
Faraday's Law is the foundation of modern electromagnetic technology. Every device that converts between electrical and mechanical energy — or transfers electrical energy between circuits — relies on it.
- Generators: A coil rotates in a magnetic field, continuously changing flux. The rate of change produces an alternating emf that drives the grid.
- Transformers: An alternating current in the primary coil creates a changing magnetic field. This changing field passes through the secondary coil, inducing an emf proportional to the turns ratio.
- Induction cooktops: A rapidly alternating current in a coil beneath the cooktop creates a rapidly changing magnetic field, inducing eddy currents in the metal pot that heat it directly.
- Wireless charging: A transmitter coil creates a changing magnetic field that induces current in a receiver coil in the device.
- Magnetic braking: A moving magnet induces eddy currents in a conductor, creating an opposing magnetic field that slows the magnet — used in train brakes and roller coaster systems.
All of these applications share the same requirement: the magnetic flux through a conductor must be changing. Engineers design systems to maximise the rate of flux change to increase induced emf and efficiency.
Faraday applications: Generator (rotating coil → changing $\Phi$ → AC emf). Transformer (AC primary → changing $B$ → induced secondary emf). Induction cooktop / wireless charging (changing $B$ → eddy currents in target). Magnetic braking (eddy currents → opposing force on moving conductor). All require changing flux.
Add the highlighted application list to your notes before the check below.
A transformer works with a steady (DC) current in the primary coil.
Magnetic braking relies on induced eddy currents opposing the motion that causes them.
Induction cooktops heat the pot by directly applying current to it via electrical contacts.
Use the interactive tool. When you increase N (number of turns) while keeping $\Delta\Phi$ and $\Delta t$ constant, the induced emf:
Calculate induced emf in a multi-part scenario
We just saw where Faraday's Law appears in technology. That raises a question: how do we apply $\varepsilon = -N\Delta\Phi/\Delta t$ step-by-step in a calculation with multiple parts? This card answers it → compute $\Phi_i$, then $\Delta\Phi$, then divide by $\Delta t$ and multiply by $N$.
A worked example is the best way to practise the algorithm: identify knowns, calculate $\Delta\Phi$, apply $\varepsilon = -N\,\Delta\Phi/\Delta t$, and interpret the magnitude and direction.
A coil of 200 turns and area $5.0 \times 10^{-3}$ m² is perpendicular to a uniform magnetic field of 0.40 T.
- (a) Calculate the initial magnetic flux through the coil.
- (b) The field is reduced to zero in 0.20 s. Calculate the average induced emf.
- (c) If the same change occurred in 0.020 s, what would the average emf be?
Using $\Phi = BA$ with the coil perpendicular to the field ($\theta = 0°$, so $\cos\theta = 1$):
$$\Phi = BA = (0.40)(5.0 \times 10^{-3}) = 2.0 \times 10^{-3} \text{ Wb}$$
The flux changes from $2.0 \times 10^{-3}$ Wb to $0$, so $\Delta\Phi = 0 - 2.0 \times 10^{-3} = -2.0 \times 10^{-3}$ Wb.
$$\varepsilon = -N\frac{\Delta\Phi}{\Delta t} = -(200)\frac{(-2.0 \times 10^{-3})}{0.20} = 2.0 \text{ V}$$
The positive result confirms the induced emf opposes the decrease in flux (Lenz's Law).
$$\varepsilon = -(200)\frac{(-2.0 \times 10^{-3})}{0.020} = 20 \text{ V}$$
Ten times faster flux change produces ten times larger emf. The magnitude of $\varepsilon$ is inversely proportional to $\Delta t$.
Step order: (1) $\Phi_i = BA\cos\theta$; (2) $\Delta\Phi = \Phi_f - \Phi_i$; (3) $\varepsilon = -N\Delta\Phi/\Delta t$. Worked result: $\Phi_i = 2.0\times10^{-3}$ Wb; $\varepsilon = 2.0$ V (at 0.20 s), $\varepsilon = 20$ V (at 0.020 s). $\varepsilon \propto 1/\Delta t$.
Pause — write the highlighted step order and worked results into your book before moving on.
Quick drill — a coil of 80 turns has flux changing from 0.040 Wb to 0.080 Wb in 0.20 s. The average induced emf (in V, as a positive number) is _____.
Faraday's Law: $\varepsilon = -N\,\dfrac{\Delta\Phi}{\Delta t}$
Flux: $\Phi = BA\cos\theta$ (Wb)
Units check: Wb/s = V
Key rule: Constant flux → zero emf. Changing flux → non-zero emf. Faster change → larger emf.
Three of these statements about Faraday's Law are correct. Pick the odd one out.
Practise calculating induced emf in different scenarios
- A 100-turn coil experiences a flux change from 0.020 Wb to 0.050 Wb in 0.10 s. Calculate the average induced emf. What would the emf be if the same flux change occurred in 0.010 s?
- Set N = 50, $\Delta\Phi$ = 20 mWb, $\Delta t$ = 0.10 s in the interactive tool. Calculate the emf by hand and verify with the tool. Then halve the time to 0.05 s — what happens to the emf?
- A student claims that if flux is constant at 0.50 Wb, the induced emf is large because the flux is large. Is this correct? Explain the error.
The negative sign in Faraday's Law $\varepsilon = -N\,\Delta\Phi/\Delta t$ is associated with:
Explain the physics behind a real-world demonstration
A bar magnet is dropped through a long copper tube. As it falls, it reaches a terminal velocity much slower than free-fall. Using Faraday's Law and Lenz's Law, explain:
- Why eddy currents are induced in the copper tube as the magnet falls.
- Why these eddy currents create an upward force on the magnet.
- Why the magnet reaches a terminal velocity rather than continuing to accelerate.
Misconceptions — final check
Copy into your books
Key Law
- $\varepsilon = -N\,\Delta\Phi/\Delta t$
- Negative sign = Lenz's Law direction
- Units: Wb/s = V
What Increases emf?
- More turns (larger N)
- Larger flux change ($\Delta\Phi$)
- Shorter time ($\Delta t$)
Applications
- Generators (rotating coil)
- Transformers (AC in primary)
- Induction cooktops, wireless charging
Key Condition
- Constant flux → zero emf
- Changing flux → non-zero emf
- Rate of change matters, not amount
A fresh five-question set drawn from this lesson's bank — feedback shown immediately. +5 XP per correct · +25 XP all correct
Pick your answer, then rate your confidence — that tells the system what to drill next.
ApplyBand 4(3 marks) 1. A coil of 80 turns experiences a flux change from 0.040 Wb to 0.080 Wb in 0.20 s. (a) Calculate the average induced emf. (b) The same flux change now occurs in 0.050 s — calculate the new average emf. (c) Explain why the emf is larger in the second case.
1 mark: correct emf for (a) · 1 mark: correct emf for (b) · 1 mark: explanation links to faster rate of change
AnalyseBand 5(4 marks) 2. A bar magnet is dropped through a vertical copper tube and reaches a terminal velocity much lower than free-fall. Using Faraday's Law and Lenz's Law, explain: (a) why eddy currents are induced as the magnet moves, and (b) why the magnet reaches a terminal velocity rather than continuing to accelerate under gravity.
1 mark: identifies changing flux as the cause of induction · 1 mark: applies Lenz's Law to predict opposing current direction · 1 mark: induced currents create upward (opposing) force · 1 mark: terminal velocity when gravitational force equals opposing magnetic braking force
Show all answers
Multiple choice
MC answers and full explanations are shown inline as you complete each question. Use the retry button to attempt a fresh set drawn from the lesson bank.
Short Answer — Model Answers
Q1 (3 marks): (a) $\Delta\Phi = 0.080 - 0.040 = 0.040$ Wb. $\varepsilon = N\,\Delta\Phi/\Delta t = 80 \times 0.040/0.20 = 16$ V (1 mark). (b) $\varepsilon = 80 \times 0.040/0.050 = 64$ V (1 mark). (c) The emf is larger because the same flux change occurs in a shorter time, giving a greater rate of change of flux $\Delta\Phi/\Delta t$, and since $\varepsilon \propto 1/\Delta t$, the emf is proportionally larger (1 mark).
Q2 (4 marks): (a) As the magnet moves through the copper tube, the magnetic flux through any section of copper changes with time. By Faraday's Law, $\varepsilon = -N\,\Delta\Phi/\Delta t$, this changing flux induces an emf and hence eddy currents in the copper (1 mark + 1 mark). (b) By Lenz's Law, the induced eddy currents create a magnetic field that opposes the change in flux — that is, they create an upward force on the descending magnet (1 mark). As the magnet accelerates, the rate of flux change increases, the induced force increases, until the upward braking force equals the gravitational force. At this point the net force is zero and the magnet moves at constant (terminal) velocity (1 mark).
Five timed questions on Faraday's Law and electromagnetic induction. Beat the boss to bank a tier — gold (perfect + fast), silver (80%+), or bronze (cleared).
Enter the arenaAt the start you were asked about Faraday's 1831 iron ring experiment at the Royal Institution — and what it has in common with a magnet falling through a copper tube versus a plastic tube.
The common thread is Faraday's Law: both require a changing flux to induce a current. In the ring experiment, switching the battery on/off changed the flux through the secondary coil. In the copper tube, the falling magnet continuously changed the flux through the tube wall — inducing eddy currents that (by Lenz's Law) opposed the motion and slowed the magnet dramatically. The plastic tube has no free electrons to carry current, so no eddy currents form and the magnet falls freely.
A stronger magnet increases the flux change rate and therefore the induced eddy currents and braking force. A C-shaped (slit) copper tube breaks the current path — eddy currents cannot complete a circuit, so the braking force largely disappears and the magnet falls much faster again.