← Back to course

Faraday’s Law & Lenz’s Law

You’ll be able to

Magnetic flux: counting field through a loop

Magnetic flux measures how much magnetic field threads a surface: Φ_B = ∫ B·dA, which for a uniform field over a flat loop reduces to Φ_B = BA·cosθ, with θ the angle between B and the loop’s normal. Its unit is the weber (1 Wb = 1 T·m²). Flux can change in exactly three ways: the field strength B changes, the area A changes (a loop stretching or a bar sliding on rails), or the orientation θ changes (a rotating coil — the heart of every generator).

Faraday’s law
ε = −N·dΦ_B/dt, Φ_B = ∫ →B·d→A
N is the number of turns, each contributing the same flux. The derivative — not the flux itself — drives the EMF: a huge steady flux induces nothing.

Faraday’s law: changing flux makes EMF

Faraday’s law says a changing magnetic flux through a circuit induces an EMF equal to the rate of change: ε = −dΦ_B/dt (times N for a coil of N turns). This is a genuinely new physics law — the induced EMF exists whether or not a conductor is present to carry current. The calculus matters: given Φ(t), you differentiate to get ε(t). A special case you can also get from F = qv × B is motional EMF: a rod of length L sliding at speed v perpendicular to B sweeps out area at rate Lv, so |ε| = BLv.

Motional EMF
|ε| = BLv
A conducting rod moving across field lines acts like a battery: the magnetic force piles positive charge at one end until an internal E field balances it.

Lenz’s law: the minus sign has a message

Lenz’s law gives the direction: the induced current flows so that its own magnetic field opposes the change in flux that created it. Flux increasing into the page? The induced current circulates counterclockwise to push flux out of the page. Flux decreasing? The induced current fights to prop it up. Lenz’s law is energy conservation in disguise — if the induced current aided the change, it would amplify itself for free, and you could build a perpetual motion machine.

Worked example

A 50-turn coil of area 0.20 m² sits with its plane perpendicular to a field that grows as B(t) = 0.50t² (tesla, seconds). Find the induced EMF as a function of time and its value at t = 3.0 s.

  1. 1.Flux through one turn: Φ = B(t)·A = (0.50t²)(0.20) = 0.10t² Wb.
  2. 2.Differentiate: dΦ/dt = 0.20t (webers per second).
  3. 3.Faraday with N turns: |ε| = N·dΦ/dt = 50 × 0.20t = 10t volts.
  4. 4.At t = 3.0 s: |ε| = 10 × 3.0 = 30 V.
  5. 5.Note the EMF grows with time even though it is “induced”: what matters is the slope of Φ(t), and a quadratic flux has an ever-steeper slope.
Answer: |ε| = 10t volts; at t = 3.0 s, |ε| = 30 V
Worked example

The north pole of a bar magnet is pushed toward a conducting loop, approaching face-on. Viewed from the magnet’s side, which way does the induced current circulate?

  1. 1.Field lines leave a north pole, so the magnet’s field points away from the magnet, passing through the loop, and it strengthens as the magnet approaches: flux through the loop (directed away from the viewer) is increasing.
  2. 2.Lenz’s law: the induced current must oppose the increase, so its own field inside the loop must point back toward the magnet (toward the viewer).
  3. 3.Right-hand rule for a loop: for the field inside to point toward the viewer, the current must circulate counterclockwise as seen from the magnet’s side.
  4. 4.Consistency check: a counterclockwise current (seen from the magnet) makes the loop’s near face a north pole, which repels the incoming north pole — the loop resists the magnet’s approach, as energy conservation demands.
Answer: Counterclockwise as viewed from the approaching magnet; the loop’s near face becomes a north pole and repels the magnet
Watch out

The induced EMF opposes the change in flux, never the flux itself. A loop with a large, steady flux through it has zero induced EMF; a loop with tiny but rapidly changing flux can have a huge one. And when flux is decreasing, the induced current flows to maintain it — in the same rotational sense as the current that would create that flux — not against it.

On the exam

Faraday FRQs almost always follow one script: (1) write Φ_B symbolically as B·A·cosθ or an integral, (2) identify which factor depends on t, (3) differentiate to get ε = −N dΦ/dt, (4) if asked for current, divide by resistance, and (5) give the direction from Lenz’s law with an explicit “opposes the increase/decrease of flux” sentence. Write the derivative before plugging in numbers.

Checkpoint

A field of 0.40 T passes through a 0.50 m² loop at 60° to the loop’s normal. The magnetic flux is:

Checkpoint

The flux through a single loop varies as Φ(t) = 6.0t − 2.0t² (milliwebers, seconds). The magnitude of the induced EMF at t = 1.0 s is:

Answer the 2 checkpoints as you read.

Sign in to save your progress