Physics › Magnetic fields › Electromagnetic induction: Faraday and Lenz
Electromagnetic induction: Faraday and Lenz
Faraday's law sets the induced emf equal to the rate of change of flux linkage, and Lenz's law gives its direction: the current opposes the change. Conservation of energy requires that opposition. A rod moving across a field gives ε = Blv, and a uniformly rotating coil gives ε = BANω sin ωt.
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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.
Flux itself induces nothing: the emf equals the rate of change of flux linkage, so it is greatest where the flux is changing fastest and zero at maximum flux.
Faraday: change makes emf
A steady flux through a circuit induces no emf. A change in flux linkage induces an emf, which lasts as long as the change does. Faraday's law gives its magnitude: the induced emf equals the rate of change of flux linkage,
Describe the drawing
A rod slides rightward across an into-the-page field. The magnetic force pushes its free positive charges up the rod, so positive charge collects at the top and negative at the bottom, and the electric field of that separated charge pushes back. The pile-up stops the moment the two forces balance, leaving an emf of B l v between the ends and no current at all, because the rod is not part of a circuit.
A moving rod shows the mechanism directly. For a straight conductor of length l moving at speed v perpendicular to a uniform field B, the free charges in the rod each experience F = BQv along the rod. Charge accumulates at the ends only while the separation is building: the accumulated charge exerts an opposing electric force, and the drift stops when QE balances BQv. The result is an emf and no current, as with a cell on open circuit, for as long as the rod moves. A sustained current requires a closed loop. Counting the flux swept per second, area lv, gives the induced emf ε = Blv. This is not printed in the booklet, so derive it as a rate of sweep.
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's law and the direction of the induced current
Lenz's law states that the induced current is in the direction that opposes the change in magnetic flux that produces it. Push a north pole towards a coil and the induced current makes the near face of the coil a north pole, opposing the approach; withdraw it and the near face becomes a south pole, opposing the withdrawal. This direction is required by conservation of energy, and the justification is examinable: if the induced current aided the change, the magnet would accelerate, generating more current and more kinetic energy from no energy source. Work must therefore be done against the opposition, and that work is the source of the electrical energy produced.
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.
Use the grip rule to obtain the direction. 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. As conservation of energy requires, the work done pushing the magnet against the repulsion is the source of the electrical output.
INDEPENDENT PRACTICE
A magnet falling down a copper pipe
A strong magnet dropped down a copper pipe falls much more slowly than an identical unmagnetised weight, without touching the walls. Explain this, 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 low terminal speed. The gravitational potential energy lost is transferred by the induced currents to internal energy in the copper, and the pipe becomes measurably warmer. Lenz's law is conservation of energy determining a direction.
The rotating coil
A coil rotating at steady angular speed ω in a uniform field has flux linkage BAN cos ωt. By Faraday's law, that steady rotation induces an alternating emf,
Describe the drawing
The coil turns at a steady rate; drawn face-on it is a circle, edge-on it closes to a line. The size of its flux linkage (amber bar, which shows magnitude only; the linkage itself changes sign twice a turn) is greatest face-on, yet that is exactly where the emf is zero, because there the flux is momentarily not changing. The emf peaks a quarter turn later, edge-on, where the linkage sweeps through zero fastest. In the second half of the loop the white line replays the trace in step with the coil.
a sine wave of peak value BANω. Increasing the rotation rate increases the peak in proportion to ω, because the same flux is swept in less time. The emf is the rate of change of linkage, so it is zero at the face-on positions, where the linkage itself is greatest, and maximum edge-on, where the linkage is momentarily zero but changing fastest. Maximum flux and maximum emf are a quarter of a turn apart. This is the principle of the ac generator.
ASSESSMENT FOCUS
- Faraday's law gives the magnitude and Lenz's law the direction. Name the law you are using, since the naming carries credit.
- The energy-conservation justification of Lenz's law is a standard three-mark answer: a current that aided the change would accelerate the magnet and create energy from nothing, so the induced current must oppose the change.
- For the moving rod, either route is acceptable: the force on the charges through BQv, or the flux swept per second giving ε = Blv. Set out one of them fully.
- In ε = BANω sin ωt the peak is BANω, so doubling the rotation rate doubles the peak emf and halves the period. Both follow from 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 per second to rad s⁻¹ first; the peak emf is the coefficient 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 at its maximum but is momentarily not changing, so the emf is zero. Maximum linkage and maximum emf are a quarter of a turn apart.
Emf is the rate of change of flux linkage.
Lenz's law states that the induced current opposes the change in magnetic flux that produces it. This direction is required by conservation of energy.
A coil rotating uniformly in a field gives ε = BANω sin ωt.
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 step by step.
USED LATER IN
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.