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, Lenz sets its direction, and energy conservation explains why the direction could never have been otherwise. Spin a coil in a field and the answer is mains electricity's sine wave.

Year 13AQA 3.7.5.4CIE 20.5OCR A 6.3.3

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.

WHAT YOU PROBABLY THINK

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,

ε = NΔΦΔtON YOUR DATA SHEET
Move a conductor across a field and its own free charges feel F = BQv along the rod: an emf appears at its endsB into pagevthe rod's free charges feel F = BQvconventional current driven up the rod
FIG. 1A rod moving across a field: its own charges feel F = BQv along the rod, driving a current while the motion lasts.

and the moving rod shows the mechanism from underneath. Slide a conductor across a field and its free charges are carried through the field with it; each feels F = BQv along the rod, driving conventional current toward one end. The rod becomes a battery for as long as it moves. Counting the flux the rod sweeps per second gives the same answer wearing Faraday's clothes: the area swept per second is lv, so the emf is ε = Blv, worth deriving once and remembering as the rate-of-sweep in disguise.

Lenz: the direction that keeps the books

Lenz's law in one scene: the north pole approaches, the loop's induced current builds an opposing north face, and the magnet is pushed backNmoving inpushed backinduced current:in at the top,out at the bottomflux through the loop grows, so the loop objectsthe induced pole opposes the approach: energy is never free
FIG. 2The approaching north pole meets an induced north face: the current's direction always opposes the change that made it.

Lenz's law gives the direction: the induced current flows so as to oppose the change producing it. Push a north pole toward a coil and the induced current builds a north face toward 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: if the induced current ever aided the change, the magnet would accelerate, generating ever more current and kinetic energy from nothing. The opposition is the universe refusing free lunches, and it is why induced currents always cost work to create: the work you do against the opposition is the electrical energy out.

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:

ε = BANωsin ω tON YOUR DATA SHEET
the coil turns; the emf peaks where the flux sweeps fastestBflux linkageemf peaks here: coil edge-onflux max, emf zero: coil face-onquarter turn
FIG. 3The coil turns at a steady rate; drawn face-on it is a circle, edge-on it closes to a line. Its flux linkage (amber bar) 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 BANω: spin faster and the peak grows with ω, because the same flux is swept in less time. This graph is also where the opening claim fails. The emf tracks the rate of change of linkage, so it is 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, and this is the generator: every power station on the grid ends in this equation.

THE EXAM BIT

  • Faraday for magnitude, Lenz for direction: name the law you are using, since 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 must oppose.
  • For the moving rod, either route scores: force on the charges via BQv, or flux swept per second giving ε = Blv. Show one cleanly.
  • In ε = BANω sinωt the peak is BANω: doubling the rotation rate doubles the peak emf and halves the period. Both follow from one ω.
  • Zero emf face-on, peak emf edge-on: quote the quarter-turn offset between maximum linkage and maximum emf whenever the 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: the quarter-turn offset in action.

Emf is the rate of change of flux linkage; direction opposes.

Spin a coil and out comes BANω times a sine.

No animated video for this topic yet; these notes stand alone. InkPhysics on YouTube.