understand and explain experiments that demonstrate: • that a changing magnetic flux can induce an e.m.f. in a circuit • that the induced e.m.f. is in such a direction as to oppose the change producing it • the factors affecting the magnitude of the

Electromagnetic Induction – Cambridge IGCSE / A‑Level (9702)

1. Overview

When the magnetic flux linked with a closed conducting circuit changes, an electromotive force (e.m.f.) is induced in that circuit. The induced e.m.f. always acts so as to oppose the change that produced it (Lenz’s law). This principle underlies generators, transformers, induction heating, magnetic braking and many other modern devices.

2. Key Concepts

• Magnetic flux, Φ

  For a uniform magnetic field \(B\) passing through a single‑turn loop of area \(A\) whose normal makes an angle \(\theta\) with the field direction, the flux is

  \[

  \Phi = BA\cos\theta

  \]

  • Maximum flux (\(\Phi_{\max}=BA\)) occurs when the field is perpendicular to the plane of the loop (\(\theta =0^{\circ}\)).
  • Zero flux occurs when the field is parallel to the plane of the loop (\(\theta =90^{\circ}\)).

• Flux linkage, \(N\Phi\)

  For a coil of \(N\) turns the total flux linked is the sum of the flux through each turn:

  \[

  \text{Flux linkage}=N\Phi

  \]

• Faraday’s law (general form)

  \[

  \mathcal{E}= -\frac{d(N\Phi)}{dt}

  \]

  The magnitude of the induced e.m.f. is proportional to the rate of change of flux linkage; the minus sign expresses Lenz’s law.

• Lenz’s law (qualitative statement)

  The induced current creates a magnetic field that opposes the change in the original flux. This determines the direction of the e.m.f.

• Motional e.m.f. (straight conductor)

  A conductor of length \(\ell\) moving with velocity \(v\) perpendicular to a uniform field \(B\) experiences an e.m.f.

  \[

  \boxed{\mathcal{E}=B\ell v}

  \]

  (derived from the magnetic force on charge carriers \(F=qvB\)).

• Rotating‑coil e.m.f.

  For a coil of \(N\) turns, area \(A\), rotating at angular speed \(\omega\) in a uniform field \(B\):

  \[

  \mathcal{E}(t)=NAB\omega\sin(\omega t)

  \]

  Peak (maximum) value: \(\displaystyle \mathcal{E}_{\max}=NAB\omega\).

3. Experiments Demonstrating Electromagnetic Induction

4. Factors Affecting the Magnitude of the Induced e.m.f.

5. Determining the Direction of the Induced e.m.f. (Lenz’s Law)

1. Identify the change in magnetic flux (increase, decrease, or reversal).
2. Ask: *What magnetic field would oppose that change?* This is the direction of the induced field.
3. Apply the right‑hand rule for a coil:

   • Point your thumb in the direction of the opposing magnetic field.
   • Your curled fingers give the direction of the induced current (and hence the polarity of the e.m.f.).

4. For a straight moving conductor, use Fleming’s right‑hand rule:

   • First finger – magnetic field direction (\(B\)).
   • Second finger – motion of the conductor (\(v\)).
   • Thumb – direction of induced e.m.f. (conventional current).

6. Key Equations (quick reference)

7. Links to Other Syllabus Topics

• Inductors (self‑induction) – A coil with a changing current produces its own changing flux, giving an induced e.m.f. \(\mathcal{E}= -L\frac{dI}{dt}\). The inductance \(L\) depends on \(N^2A/l\) and the core material.
• Transformers – Two coils sharing a common magnetic flux. The e.m.f. ratio \(\displaystyle \frac{\mathcal{E}s}{\mathcal{E}p}= \frac{Ns}{Np}\) follows directly from Faraday’s law applied to each winding.
• Energy considerations – Electrical power delivered by an induced e.m.f. is \(P=\mathcal{E}I\). In the falling‑magnet experiment this power appears as heat (eddy‑current heating) in the tube.

8. Sample A‑Level Exam Questions & Marking Points

1. Magnetic damping – Explain why a magnet falling through a copper tube experiences a magnetic drag.

   • State that the falling magnet changes the magnetic flux through each loop of the tube (flux ↓ as the magnet approaches, ↑ as it leaves).
   • Apply Lenz’s law: induced eddy currents create a magnetic field opposing the magnet’s motion.
   • The resulting upward magnetic force reduces the acceleration – magnetic damping.

2. Generator calculation – A coil of 200 turns, area \(5.0\times10^{-3}\,\text{m}^2\), rotates at \(300\ \text{rev s}^{-1}\) in a uniform field of \(0.20\ \text{T}\). Calculate the peak e.m.f.

   • \(\omega = 2\pi f = 2\pi(300)=1.88\times10^{3}\ \text{rad s}^{-1}\)
   • \(\mathcal{E}_{\max}=NAB\omega = 200 \times 5.0\times10^{-3} \times 0.20 \times 1.88\times10^{3}=3.76\ \text{V}\) (to 2 sf).

3. Increasing induced e.m.f. (moving‑magnet set‑up) – Describe two ways to increase the e.m.f. without changing the magnet.

   • Increase the number of turns \(N\) in the coil – \(\mathcal{E}\propto N\).
   • Increase the speed at which the magnet is inserted/removed – larger \(\frac{d\Phi}{dt}\).
   • (Other acceptable answers: enlarge the coil area \(A\), use a stronger external magnetic field, or rotate the coil instead of moving the magnet.)

4. Motional e.m.f. calculation – A rod 0.15 m long moves at 4.0 m s⁻¹ through a magnetic field of 0.25 T, perpendicular to both the rod and its velocity. Find the induced e.m.f.

   • \(\mathcal{E}=B\ell v = 0.25 \times 0.15 \times 4.0 = 0.15\ \text{V}\).

9. Practical Tips for Laboratory Work

• Ensure the coil leads are firmly attached to the measuring instrument; loose connections add unwanted resistance.
• When using a galvanometer, remember that the deflection is proportional to the induced current, not directly to \(\mathcal{E}\). Use a known shunt resistance to convert deflection to voltage if required.
• In the rotating‑coil experiment, keep the angular speed constant (use a motor with regulated voltage) to obtain a clean sinusoidal waveform.
• For the falling‑magnet demonstration, use a tube of high conductivity (copper or aluminium) and a strong neodymium magnet to maximise the observable effect.
• Record the direction of deflection (or polarity) as well as magnitude; this provides direct evidence of Lenz’s law.