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Faraday's and Lenz's Laws

Key idea: Use Faraday’s law ε = −d(NΦ)/dt and Lenz’s law to find induced e.m.f. and predict its direction from changing flux linkage (A Level Physics).

  • GCE A-Level H2 Physics 2027
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Learning objectives

  • Apply Faraday's and Lenz's laws to induced e.m.f. and direction.

1. Definitions (Must Know)

A. Faraday’s law (magnitude + sign)

An e.m.f. is induced when the flux linkage through a circuit changes.

Faraday’s law (including the Lenz sign) is:

ε = -d(NΦ)/dt

B. Lenz’s law (direction idea)

The negative sign means the induced e.m.f. acts in a direction that opposes the change in flux linkage that produces it.

2. Key Ideas (What Earns Marks)

  • Flux linkage: NΦ = NBA cos θ
  • Induced e.m.f. comes from changing NΦ:
    • change B (e.g. move a magnet),
    • change A (e.g. deform or rotate a loop),
    • change θ (rotate the coil).
  • A larger number of turns or a faster change produces a larger e.m.f. magnitude.
  • Use Lenz’s law as an “opposes the change” sentence, not as a sign-guess.
Most common workflow
  1. Write NΦ. 2) Decide how it changes. 3) Use |E| = |d(NΦ)/dt| for magnitude. 4) Use Lenz’s law and a direction rule to state the polarity or current direction.
Exam pitfall: saying Lenz's law opposes the field

State that the induced current opposes the change in flux linkage, not simply the external magnetic field. This wording keeps your direction explanation precise.

3. Detailed Explanations

A. What Faraday’s law means in words

  • The bigger the rate of change of flux linkage, the bigger the induced e.m.f.
  • If flux linkage is constant, ε = 0 even if a magnetic field is present.

B. Using Lenz’s law safely

Lenz’s law is about opposing the change, not opposing the field itself.

Faraday's and Lenz's laws for a moving magnetA north pole approaches a coil connected to a galvanometer. Arrows show increasing magnetic flux, induced current and a repulsive north pole on the near face of the coil.SNmagnet approachesrightward flux increasesN facegalvanometer
Scroll diagram horizontally to read all labels.
As the north pole approaches, flux through the coil increases. The induced current makes the near face a north pole, opposing that increase; reversing the change reverses the current.

Example phrasing:

  • “Flux into the coil is increasing, so the induced current produces a field that reduces flux into the coil.”
  • “Flux is decreasing, so the induced current produces a field that increases flux in the original direction.”

C. What the standard experiments show

Move a magnet into and out of a coil connected to a sensitive voltmeter or galvanometer:

  • a reading occurs only while flux linkage changes;
  • reversing the motion reverses the polarity;
  • moving faster gives a larger peak reading;
  • increasing the number of linked turns gives a larger induced e.m.f.

With an open circuit, an e.m.f. can still be measured across the terminals, but there is no sustained induced current because there is no complete conducting path.

4. Common Mistakes

  • Using ε = -dΦ/dt when the question is about a coil (usually needs NΦ).
  • Treating the negative sign as “ε is negative” without defining a polarity or direction.
  • Using degrees for θ but forgetting your calculator mode (when doing sinusoidal problems later).

5. Exam Tips

  • If the question asks for “magnitude of induced e.m.f.”, use absolute values and state direction separately.
  • If NΦ changes linearly with time, d(NΦ)/dt is constant and the induced e.m.f. is constant.

6. Worked Examples

Modelled example 1

Constant rate of change of flux linkage

Core

Problem

Flux linkage changes uniformly from 0.060 to 0.015 Wb turn in 0.020 s. Find the induced e.m.f. magnitude and explain what information would be needed for a signed value.
Study the worked solution

  1. Find linkage change

    Method

    Δ(NΦ) = -0.045 Wb turn.

    Reason

    Change is final minus initial.

    Working

    0.015-0.060 = -0.045 Wb turn

  2. Find e.m.f. magnitude

    Method

    |ε| = 2.25 V.

    Reason

    Average magnitude is the absolute linkage-change rate.

    Working

    |ε| = |-0.045/0.020| = 2.25 V

  3. Interpret sign

    Method

    A signed result requires a chosen positive linkage and circuit-polarity convention.

    Reason

    The Lenz minus sign is meaningful only relative to defined directions.

    Working

    Use Lenz’s law after defining polarity.

Guided practice 2

Rotating-coil style change in flux linkage

About 6 min

Problem

A 200-turn coil of area 3.0 × 10⁻⁴ m² in 0.80 T rotates from 0° to 90° between field and normal in 0.050 s. Estimate the average e.m.f. magnitude.

Try this before viewing the solution

Unit: V

Hints

Hint 1: calculate both endpoint linkages
Use NΦ = NBA cos θ at 0° and 90°.
View solution step by step

  1. Find initial linkage

    Method

    NΦᵢ = 4.8 × 10⁻² Wb turn.

    Reason

    The normal begins parallel to the field.

    Working

    NΦᵢ = (200)(0.80)(3.0 × 10⁻⁴) cos 0° = 4.8 × 10⁻²

  2. Find final linkage

    Method

    NΦ_f = 0.

    Reason

    At 90° the field has no component along the normal.

    Working

    NΦ_f = NBA cos 90° = 0

  3. Find average e.m.f.

    Method

    |ε| = 0.96 V.

    Reason

    Divide the magnitude of the linkage change by 0.050 s.

    Working

    |ε| = 4.8 × 10⁻²/0.050 = 0.96 V

Common misconception 3

Induced e.m.f. from changing magnetic flux density

Find and correct the mistake

Learner claim

A perpendicular 500-turn coil of area 2.0 × 10⁻⁴ m² sees B rise from 0.10 to 0.70 T in 0.040 s. A learner uses 0.70 T rather than Δ B. Diagnose and calculate.

Try this before viewing the solution

Unit: V

View solution step by step

  1. Use the field change

    Method

    Δ B = 0.60 T.

    Reason

    Induction depends on change, not the final nonzero field alone.

    Working

    Δ B = 0.70-0.10 = 0.60 T

  2. Find linkage change

    Method

    Δ(NΦ) = 6.0 × 10⁻² Wb turn.

    Reason

    Orientation stays perpendicular, so Δ(NΦ) = NAΔ B.

    Working

    Δ(NΦ) = (500)(2.0 × 10⁻⁴)(0.60) = 6.0 × 10⁻²

  3. Find e.m.f.

    Method

    |ε| = 1.5 V.

    Reason

    Divide linkage change by elapsed time.

    Working

    |ε| = 6.0 × 10⁻²/0.040 = 1.5 V

Examiner practice 4

Find the time needed for a target e.m.f.

2 marks

Examination question

Flux linkage changes by 0.090 Wb turn. Find the time required for an average induced e.m.f. magnitude of 3.0 V. [2 marks]

Try this before viewing the solution

View solution step by step

  1. Rearrange Faraday's law

    1 mark

    Method

    Δ t = |Δ(NΦ)|/|ε|.

    Reason

    The data give average magnitude rather than polarity.

    Working

    Δ t = |Δ(NΦ)|/|ε|

  2. Evaluate

    1 mark

    Method

    Δ t = 3.0 × 10⁻² s.

    Reason

    Divide 0.090 linkage units by 3.0 V.

    Working

    Δ t = 0.090/3.0 = 3.0 × 10⁻² s

Challenge 5

Sinusoidal flux linkage (calculus)

Minimal support

Independent transfer

Flux linkage varies as NΦ = 0.040 sin(200π t). Derive ε(t), find its peak magnitude, and state the phase relation between linkage and e.m.f.

Try this before viewing the solution

Hints

Hint 1: differentiate then apply Lenz sign
Differentiate sine to cosine and retain the leading minus sign in Faraday’s law.
View solution step by step

  1. Differentiate linkage

    Method

    ε(t) = -(8.0π) cos(200π t) V.

    Reason

    d[sin(200π t)]/dt = 200π cos(200π t) and Faraday’s law adds a minus sign.

    Working

    ε = -d(NΦ)/dt = -(0.040)(200π) cos(200π t)

  2. Find peak magnitude

    Method

    ε₀ = 8.0π V ≈ 25 V.

    Reason

    The largest magnitude of cosine is one.

    Working

    ε₀ = 8.0π ≈ 25 V

  3. State phase relation

    Method

    E.m.f. is a quarter-cycle shifted from linkage, with sign fixed by Lenz’s law.

    Reason

    Differentiation changes sine to cosine and the negative sign reverses polarity.

    Working

    ε ∝ - cos(200π t).

7. Mind Stretchers

Mind stretcher 1: Why the induced e.m.f. must oppose the changeExtension

Explain why Lenz’s law is consistent with conservation of energy.

Show Answer

If the induced current aided the change in flux, the system would amplify itself and create energy from nothing.

Opposing the change means an external agent must do work to change the flux, and that work is converted into electrical energy (and usually heat), conserving energy.

Mind stretcher 2: Direction reasoning (Lenz’s law)Extension

A magnet’s north pole approaches a coil along its axis. The magnetic flux through the coil increases (into the coil). State the direction of the induced magnetic field produced by the coil.

Show Answer

By Lenz’s law, the induced effect opposes the increase in flux into the coil.

So the coil produces a magnetic field out of the coil (opposite to the increasing flux direction).

8. Optional (Enrichment)

A. A quick demo video

Continue with the next resource in this course.

Course and syllabus information
Course
GCE A-Level H2 Physics
Edition
GCE A-Level H2 Physics 2027