Cambridge Syllabus Notes

Electromagnetic Induction – 4.5.1

Objective

State the factors that affect the magnitude of an induced e.m.f. and describe how the direction of the induced e.m.f. is determined (Lenz’s law).

Definition

Electromagnetic induction occurs when

a conductor moves across magnetic field lines, or
the magnetic flux linking a stationary conductor changes.

Either situation produces an induced e.m.f. in the conductor.

Faraday’s Law (Quantitative relationship)

The magnitude of the induced e.m.f. in a coil of N turns is proportional to the rate of change of magnetic flux linking the coil:

\[ \mathcal{E}= -N\frac{d\Phi}{dt} \]

N – number of turns
Φ = BA\cos\theta – magnetic flux through one turn
B – magnetic field strength (T)
A – area of the coil (m²)
θ – angle between the field direction and the normal to the coil surface

Note for exam candidates: the syllabus expects the proportional form
\[ \mathcal{E}\propto N\frac{\Delta\Phi}{\Delta t} \] Use \(\mathcal{E}=N\Delta\Phi/\Delta t\) for calculations with discrete changes.

Lenz’s Law (Direction of the induced e.m.f.)

“The induced e.m.f. always acts so as to oppose the change in magnetic flux that produced it.”

Consequences

If the flux through a coil is increasing, the induced e.m.f. drives a current that creates a magnetic field opposing the increase.
If the flux is decreasing, the induced current tries to maintain it.

Determining the direction of the induced current

Right‑hand grip (generator) rule – for a rotating coil:
- Grip the coil with the right hand so that the fingers follow the direction of the windings.
- Point the thumb in the direction the side of the coil is moving into the magnetic field.
- The thumb then points in the direction of the induced e.m.f. (conventional current).
Fleming’s right‑hand rule – for a straight conductor moving in a uniform field:
- First finger – direction of the magnetic field \(B\) (N → S).
- Second finger – direction of the induced current \(I\).
- Thumb – direction of motion \(v\) of the conductor.

See the suggested diagram (a) for a quick visual reference of a rotating coil.

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

The induced e.m.f. depends on how rapidly the magnetic flux through the circuit changes. The Cambridge syllabus lists the factors in the order shown below. The three core factors most frequently examined are highlighted in **bold**.

Number of turns (N) – \(\mathcal{E}\propto N\).
Magnetic field strength (B) – \(\mathcal{E}\propto B\).
Area of the coil (A) – \(\mathcal{E}\propto A\).
Rate of change of magnetic flux (i.e. \(dB/dt\) or \(dA/dt\) or \(d\theta/dt\)).
Length of a moving conductor (\(l\)) – for a straight rod \(\mathcal{E}=Blv\).
Velocity of the conductor relative to the field (\(v\)) – \(\mathcal{E}=Blv\).
Angle between the field and the coil normal (\(\theta\)) – \(\Phi = BA\cos\theta\); a rapid change of \(\theta\) increases \(\mathcal{E}\).

Summary Table

Factor	Effect on \(\mathcal{E}\)
Number of turns (N)	\(\mathcal{E}\propto N\) – double \(N\), double \(\mathcal{E}\)
Magnetic field strength (B)	\(\mathcal{E}\propto B\)
Coil area (A)	\(\mathcal{E}\propto A\)
Rate of change of flux (\(d\Phi/dt\))	\(\mathcal{E}\propto d\Phi/dt\)
Length of moving conductor (l)	\(\mathcal{E}=Blv\) – proportional to \(l\)
Velocity of motion (v)	\(\mathcal{E}=Blv\) – proportional to \(v\)
Angle \(\theta\) (or its rate of change)	\(\Phi = BA\cos\theta\); rapid change of \(\theta\) ⇒ larger \(\mathcal{E}\)

Safety Checklist (Laboratory)

Keep strong magnets away from electronic devices, credit‑card strips, and pacemakers.
Secure the coil and rotating apparatus to prevent it from slipping.
Use insulated wires and avoid touching live contacts.
Never place metal objects near the magnet while it is moving rapidly.
Turn off power supplies before making adjustments to the set‑up.

Experimental Demonstration (Classic IGCSE Lab)

Goal: Show that a changing magnetic flux induces an e.m.f.

Apparatus: rectangular coil (≈ 50 turns, area ≈ 0.03 m²), wooden frame, bar magnet with uniform vertical field, galvanometer (or sensitive voltmeter), sliding contacts (brushes), stand and clamp.
Connect the coil to the galvanometer using the sliding contacts so the coil can rotate freely.
Place the coil between the pole faces of the magnet.
Rotate the coil rapidly through 180° and observe the galvanometer deflection. Reverse the sense of rotation – the deflection reverses, illustrating Lenz’s law.
Repeat the experiment while varying:
- Number of turns – use coils with different N and note the proportional change in deflection.
- Speed of rotation – rotate faster to obtain a larger peak deflection.
- Angle of rotation – rotate through 90° to obtain half the maximum deflection.

Record‑keeping (AO3 skill)

Title, aim and hypothesis.
Table of observations: N, rotation speed (rev s⁻¹), angle turned, galvanometer reading (deflection or voltage).
Calculate \(\mathcal{E}=N\Delta\Phi/\Delta t\) for each trial and compare with the measured value.
Include uncertainties (e.g., timing with a stopwatch ±0.1 s) and comment on sources of error (friction, non‑uniform field, contact resistance).

Numerical Examples

Example 1 – Rotating coil (average e.m.f.)

Problem: A coil of 20 turns, area \(0.04\;\text{m}^2\), is placed in a uniform magnetic field of \(0.5\;\text{T}\). It is rotated from \(\theta =0^{\circ}\) to \(\theta =90^{\circ}\) in \(0.20\;\text{s}\). Find the average induced e.m.f.

Solution:

\[ \Phi_i = BA\cos0^{\circ}= (0.5)(0.04)(1)=0.020\;\text{Wb} \] \[ \Phi_f = BA\cos90^{\circ}= (0.5)(0.04)(0)=0\;\text{Wb} \] \[ \Delta\Phi = \Phi_f-\Phi_i = -0.020\;\text{Wb} \] \[ \mathcal{E}_{\text{avg}} = -N\frac{\Delta\Phi}{\Delta t}= -20\frac{-0.020}{0.20}=2.0\;\text{V} \]

Example 2 – Straight conductor moving in a field

Problem: A metal rod 0.10 m long moves at \(2.0\;\text{m s}^{-1}\) perpendicular to a uniform magnetic field of \(0.30\;\text{T}\). Find the induced e.m.f.