recall and use Faraday’s and Lenz’s laws of electromagnetic induction

Electromagnetic Induction – Faraday’s & Lenz’s Laws (AS/A‑Level Physics 9702)

1. Syllabus Context

Where this note fits: It provides the core theory needed for all A‑level questions that involve a changing magnetic flux – e.g. generators, transformers, inductive heating, and electromagnetic wave generation.

2. Learning Objectives

• Define magnetic flux (Φ) and flux linkage (NΦ) and state their units.
• State Faraday’s law and use it to calculate the magnitude of an induced emf.
• Apply Lenz’s law to determine the direction of the induced current.
• Identify the physical quantities that affect the size of the induced emf.
• Analyse a practical set‑up (moving coil, rotating coil, changing field) and predict the emf and its direction.

3. Key Concepts

• Magnetic flux, Φ = ∫ B·dA (weber, Wb)
• Flux linkage, NΦ (for a coil of N turns)
• Faraday’s law, 𝓔 = ‑N dΦ/dt
• Lenz’s law – the induced emf always opposes the change that produced it.
• Right‑hand rule for the sense of induced current.
• Conservation of energy in electromagnetic systems.

4. Magnetic Flux

4.1 General definition

\$\Phi = \int \mathbf{B}\!\cdot\! d\mathbf{A}\$

Units: 1 Wb = 1 T·m².

4.2 Uniform magnetic field

When B is uniform over a planar surface of area A,

\$\Phi = B\,A\cos\theta\$

θ is the angle between the field direction and the surface normal.

4.3 Flux linkage for a coil

\$\text{Flux linkage}=N\Phi\$

The induced emf in a coil depends on the rate of change of this quantity.

4.4 Worked line‑area example

A rectangular loop of width w = 0.08 m and height h = 0.12 m lies in a uniform field B = 0.40 T, with the field perpendicular to the plane of the loop.

\$\Phi = B\,A = 0.40\;\text{T}\times(0.08\times0.12)\;\text{m}^2 = 3.84\times10^{-3}\;\text{Wb}\$

5. Faraday’s Law of Electromagnetic Induction

5.1 Statement

\$\boxed{\mathcal{E}= -\dfrac{d\Phi}{dt}} \qquad\text{or}\qquad \boxed{\mathcal{E}= -N\dfrac{d\Phi}{dt}}\$

The negative sign is the mathematical expression of Lenz’s law.

5.2 Derivation for a uniform field

For Φ = B A cosθ, the total time derivative is

\$\$\frac{d\Phi}{dt}= B\frac{dA}{dt}\cos\theta

+ A\frac{dB}{dt}\cos\theta

- BA\sin\theta\;\frac{d\theta}{dt}\$\$

In most exam questions only one term is non‑zero, giving three useful shortcut forms:

• Changing area (rod pulled into a field) 𝓔 = B l v
• Changing field (solenoid with varying current) 𝓔 = A (dB/dt)
• Rotating coil (generator) 𝓔 = NBA ω sin ωt

6. Lenz’s Law – Direction of the Induced Current

Lenz’s law: the induced emf (and thus the induced current) creates a magnetic field that opposes the change in the original flux.

6.1 Right‑hand rule (loop version)

1. Determine whether the magnetic flux through the loop is increasing or decreasing.
2. Visualise the magnetic field that the induced current would have to produce to oppose that change.
3. Point the thumb of your right hand in the direction of this imagined field (for a solenoid the thumb points along the axis of the induced field). The curled fingers then give the direction of the induced current around the loop.

7. Factors Affecting the Induced emf

• Speed of the conductor relative to the magnetic field (v).
• Magnetic‑field strength (B).
• Number of turns in the coil (N).
• Area of the loop or the rate at which the area changes (dA/dt).
• Rate of change of the magnetic field itself (dB/dt).
• Angle between B and the normal to the area (θ); a rotating coil changes θ with time.

8. Common Situations & Shortcut Formulas

1. Straight conductor moving perpendicular to a uniform field (v ⊥ B ⊥ l)

   \$\mathcal{E}=Blv\$

2. Rotating coil (AC generator)

   Flux per turn: \$\Phi = NBA\cos\theta\$, \$\theta=\omega t\$

   \$\mathcal{E}=NBA\omega\sin\omega t\qquad\text{(peak emf }=NBA\omega)\$

3. Changing magnetic field with fixed area

   \$\mathcal{E}= -A\frac{dB}{dt}\$

4. Changing area with fixed field (e.g., a loop being pulled into a field)

   \$\mathcal{E}= -B\frac{dA}{dt}\$

9. Worked Example (Maximum emf for a moving rectangular loop)

Problem: A rectangular loop of width 0.10 m and height 0.20 m moves at 2.0 m s⁻¹ into a region where a uniform magnetic field of 0.50 T points into the page. The loop has N = 5 turns. Find the maximum induced emf.

1. Maximum rate of change of flux occurs while the leading edge is entering the field.
2. Rate at which area enters the field:

   \$\frac{dA}{dt}= \text{width}\times v = 0.10\;\text{m}\times2.0\;\text{m s}^{-1}=0.20\;\text{m}^2\!\!/\text{s}\$

3. Change of flux per turn:

   \$\frac{d\Phi}{dt}=B\frac{dA}{dt}=0.50\;\text{T}\times0.20\;\text{m}^2\!\!/\text{s}=0.10\;\text{Wb s}^{-1}\$

4. Induced emf for N turns (AO2):

   \$\mathcal{E}_{\max}=N\Bigl|\frac{d\Phi}{dt}\Bigr|=5\times0.10=0.50\;\text{V}\$

10. Practical Investigation (Paper 5 style)

Objective: Verify 𝓔 = Blv for a straight conductor moving through a uniform magnetic field.

1. Set up a U‑shaped magnet with pole pieces creating a uniform field B (measure with a gauss‑meter).
2. Pass a straight copper rod of known length l on low‑friction rails so it can slide perpendicular to B.
3. Connect the rod to a digital voltmeter (or galvanometer) and record the induced emf for several speeds v (measured with a photogate).
4. Plot 𝓔 against v; the slope should equal Bl (AO3 – data analysis, error estimation).
5. Discuss sources of systematic error (e.g., non‑uniform B, contact resistance) and how they affect the verification of Faraday’s law.

11. Key Equations Summary

12. Common Misconceptions (and how to avoid them)

• “Induced emf creates energy.” – The emf is a response that opposes the cause; any energy extracted comes from the mechanical work done to change the flux (energy conservation).
• Confusing external B‑field direction with current direction. – Lenz’s law, not the external field, determines the sense of the induced current.
• Assuming a static magnetic field can induce an emf. – A change in flux (via B, A, or θ) is essential; a constant field alone produces no emf.
• Neglecting the sign in Faraday’s law. – The minus sign encodes Lenz’s law; dropping it leads to the wrong current direction.

13. Practice Questions (with AO tags)

1. (AO2) A circular coil of radius 0.15 m and N = 10 turns rotates at 50 rad s⁻¹ in a uniform magnetic field of 0.30 T.

   • Write the expression for the instantaneous emf.
   • Calculate its maximum value.

2. (AO2) A solenoid 0.40 m long with 200 turns carries a current that is increasing at 3.0 A s⁻¹. The solenoid’s cross‑sectional area is 2.5 × 10⁻³ m². Determine the magnitude of the induced emf in a single loop placed around the solenoid.
3. (AO3) Explain qualitatively why a metal ring placed on a magnetic core that is being switched on experiences a repulsive force.

14. Integration with Other Syllabus Topics

Understanding induction is essential for:

• AC circuits (Module 12) – the sinusoidal emf from a rotating coil underpins alternating‑current generators.
• Transformers (Module 12) – Faraday’s law applied to primary and secondary coils explains voltage transformation.
• Electromagnetic waves (Module 13) – a time‑varying magnetic field produces an electric field, the reverse of induction.
• Energy concepts (Module 11) – mechanical work done in moving conductors is converted to electrical energy, illustrating conservation of energy.

15. Summary

Faraday’s law quantifies how a changing magnetic flux induces an emf, while Lenz’s law provides the direction of the induced current, guaranteeing that the induced effect always opposes its cause. Mastery of these principles enables analysis of generators, transformers, inductive sensors, and many modern technologies, fulfilling the AO1–AO3 requirements of the Cambridge AS/A‑Level Physics (9702) syllabus.