Physics › Magnetic fields › Electromagnetic induction: Faraday and Lenz
Electromagnetic induction: Faraday and Lenz
Change the flux linking a circuit and the circuit answers with an emf. Faraday sets its size and Lenz sets its direction, with energy conservation explaining why that direction could never have been otherwise. Spin a coil in a field and out comes the sine wave of mains electricity.
Pick your board and the few notes written for the other boards quietly fold away, here and in the practice players. Nothing is deleted: every folded piece reopens on a tap.
Builds on Magnetic flux and flux linkage and Force on a moving charge.
IN THIS TOPIC
- Use Faraday's law: the induced emf equals the rate of change of flux linkage.
- Use Lenz's law for direction, and justify it by energy conservation.
- Apply both to a moving conductor and to a uniformly rotating coil.
COMMON MISCONCEPTION
The emf is biggest when the most flux passes through the coil.
Faraday: change makes emf
A steady flux through a circuit does nothing at all. Change the flux linkage and an emf appears, lasting exactly as long as the change does. Faraday's law sets the size. The induced emf equals the rate of change of flux linkage,
and the moving rod shows the mechanism from underneath. Slide a conductor across a field, at right angles to it, and its free charges ride along, so each one feels F = BQv along the rod. Charge drifts to one end only while the separation is building: the piled-up charge pulls back, and the drift stops the instant QE balances BQv. What remains is an emf and no current: a battery on open circuit for as long as the rod moves. A sustained current needs a closed loop, which the next section's coils provide. Counting the flux swept per second, area lv, gives Faraday's answer: ε = Blv. Derive it once and keep it as a rate of sweep; it is not in the booklet.
WORKED EXAMPLE
An emf from a collapse
The flux through each turn of a 150-turn coil falls steadily from 2.4 × 10−3 Wb to zero in 0.20 s. Find the induced emf.
Faraday counts the linkage, so ε = NΔΦ/Δt.
ε = 150 × 2.4 × 10−3/0.20 = 1.8 V, steady for as long as the collapse lasts.
More turns, more flux, or a faster collapse would each raise the emf in direct proportion.
Lenz: the direction that keeps the books
Lenz's law gives the direction. The induced current flows so as to oppose the change producing it. Push a north pole towards a coil and the induced current builds a north face towards the magnet, resisting the approach; pull it away and the face flips to south, resisting the retreat. The justification is energy conservation, and it is examinable. Suppose the induced current ever aided the change. The magnet would accelerate, generating ever more current and ever more kinetic energy out of nothing at all. Induced currents therefore always cost work to make, and the work you do against the opposition is the electrical energy that comes out.
GUIDED PRACTICE
Which way does it flow?
A magnet's north pole approaches the face of a coil. Use Lenz's law to decide which magnetic pole the coil's near face becomes, and hence the current direction seen from the magnet.
Show the working
The induced current must oppose the approach, so the coil's near face becomes a north pole, repelling the incomer.
Get the direction from the grip rule, never from memory. For that face to act as a north pole its field must point out of the face towards the magnet, so the current circulates anticlockwise as seen by the approaching magnet. The energy story seals it, since pushing against the repulsion is the work that becomes the electrical output.
INDEPENDENT PRACTICE
The reluctant magnet
A strong magnet dropped down a copper pipe falls dramatically slower than an identical unmagnetised weight, without touching the walls. Explain, and account for the energy.
Show the working
The falling magnet sweeps changing flux past every ring of the pipe, inducing circulating currents; by Lenz's law their fields oppose the change, pulling back on the magnet whichever way it moves.
The magnet reaches a gentle terminal speed. The lost gravitational energy has not vanished. Induced currents dissipate it as internal energy in the copper, and the pipe grows measurably warm. Lenz's law is energy conservation deciding a direction.
The rotating coil
Spin a coil at steady angular speed ω in a uniform field and its flux linkage is BANcosωt, sweeping through the cosine forever. Faraday's law turns that steady rotation into an alternating emf,
a sine wave of peak BANω. Spin faster and the peak grows with ω, because the same flux gets swept in less time. Emf tracks the rate of change of linkage, so it sits at zero at the face-on instants, exactly where the linkage itself is greatest, and it peaks edge-on, where the linkage is momentarily zero but changing fastest. Maximum flux and maximum emf are a quarter-turn apart. This is the generator, and every power station on the grid ends in this equation.
ASSESSMENT FOCUS
- Faraday for magnitude, Lenz for direction. Name the law you are using, because the naming itself carries marks.
- The energy-conservation justification of Lenz is a stock three-marker. An aiding current would accelerate the magnet and create energy from nothing, so the current has to oppose.
- For the moving rod either route scores. Take the force on the charges through BQv, or the flux swept per second giving ε = Blv. Show one of them cleanly.
- In ε = BANω sinωt the peak is BANω, so doubling the rotation rate doubles the peak emf and halves the period. Both come from the one ω.
- Zero emf face-on, peak emf edge-on. Quote the quarter-turn offset between maximum linkage and maximum emf whenever those graphs appear.
CHECK YOURSELF
A 200-turn coil of area 1.5 × 10−3 m² rotates at 50 revolutions per second in a 0.20 T field. Find the peak emf, and state the emf at the instant the coil is face-on to the field.
Show a hint
Convert revolutions to radians first; then the peak is everything in front of the sine.
Show the answer
ω = 2π × 50 = 314 rad s−1.
Peak ε = BANω = 0.20 × 1.5 × 10−3 × 200 × 314 = 19 V.
Face-on the linkage is maximal but momentarily unchanging, so the emf is zero. That is the quarter-turn offset in action.
Emf is the rate of change of flux linkage.
Lenz turns the current against whatever caused it, to keep the energy books straight.
Spin a coil and out comes BANω times a sine.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
Or read them with their mark schemes on the electromagnetic induction: faraday and lenz questions page.
WHERE TO GO NEXT
- Gradients and areas under graphs is the maths this lesson leans on, worked through from GCSE.
CHECK YOUR PROGRESS
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- Use Faraday's law: the induced emf equals the rate of change of flux linkage.
- Use Lenz's law for direction, and justify it by energy conservation.
- Apply both to a moving conductor and to a uniformly rotating coil.
Open the full revision checklist to track your progress across the whole unit.