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 – 20.5)

1. Magnetic Flux and Flux Linkage

  • Magnetic flux (Φ) – the amount of magnetic field that passes through a surface of area \(A\).

    \[

    \Phi = B A \cos\theta

    \]

    where

    \(B\) = magnetic field strength (T),

    \(A\) = area of the loop (m²),

    \(\theta\) = angle between the field direction and the normal to the surface.

    The cosine term shows that the flux is maximum when the field is perpendicular to the loop (\(\theta =0^\circ\)) and zero when the field is parallel (\(\theta =90^\circ\)).

  • Flux linkage – for a coil of \(N\) turns the total flux linked is the sum of the flux through each turn:

    \[

    \text{flux linkage}=N\Phi .

    \]

2. Faraday’s Law and Lenz’s Law (Sign Convention)

  • Faraday’s law (quantitative)

    \[

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

    \]

    \(\mathcal{E}\) is the induced e.m.f. (V). The minus sign is the mathematical statement of Lenz’s law.

  • Lenz’s law (qualitative) – the induced current always flows in a direction that creates a magnetic field opposing the change in magnetic flux that produced it.

  • Why the minus sign?

    The negative sign ensures that the induced e.m.f. (and thus the induced current) produces a magnetic field whose effect is to *reduce* the original change in flux, i.e. it enforces the opposition described by Lenz’s law.

3. Determining the Direction of the Induced Current

  1. Identify whether the magnetic flux through the loop is increasing or decreasing.

  2. Apply Lenz’s law: the induced magnetic field must oppose that change.

  3. Use the right‑hand rule for a current‑carrying loop:

    • Point the thumb of the right hand in the direction of the required induced magnetic field (the one that opposes the change).

    • Curl the fingers – they give the direction of the induced current around the loop.

  4. For a straight conductor moving in a magnetic field, use Fleming’s left‑hand rule (thumb = motion, fore‑finger = field, middle‑finger = induced current).

4. Quantitative Relationships

4.1 General dependence of \(\mathcal{E}\) on experimental variables

FactorEffect on \(\mathcal{E}\) (from \(\mathcal{E}= -N\,d\Phi/dt\))How to vary in the laboratory
Number of turns, \(N\)\(\mathcal{E}\propto N\)Wind more turns on the same core or swap to a coil with a different \(N\).
Magnetic field strength, \(B\)\(\mathcal{E}\propto B\)Use a stronger permanent magnet or an electromagnet with adjustable current.
Area of the loop, \(A\)\(\mathcal{E}\propto A\)Use larger coils or change the effective area with a sliding rod.
Angular (or linear) speed, \(\omega\) (or \(v\))\(\mathcal{E}\propto \omega\) (or \(v\))Vary motor speed, or drop the magnet faster.
Angle \(\theta\) between field and loop normal\(\Phi = BA\cos\theta\); varying \(\theta\) changes \(\Phi\) sinusoidally, giving an alternating e.m.f.Rotate the coil continuously (see generator experiment).
Resistance of the circuit, \(R\)Does not affect \(\mathcal{E}\); however \(I = \mathcal{E}/R\) so the measured current changes.Insert known resistors and measure the current with an ammeter.

4.2 Derivation for a Rotating Coil (Generator)

Consider a single‑turn rectangular coil of area \(A\) rotating at a constant angular speed \(\omega\) in a uniform magnetic field \(B\) (field direction perpendicular to the axis of rotation).

  1. Instantaneous magnetic flux: \(\displaystyle \Phi(t)=BA\cos\theta(t)=BA\cos(\omega t)\).

  2. Differentiate with respect to time:

    \[

    \frac{d\Phi}{dt}= -BA\omega\sin(\omega t).

    \]

  3. Apply Faraday’s law for a coil of \(N\) turns:

    \[

    \mathcal{E}(t)= -N\frac{d\Phi}{dt}= NAB\omega\sin(\omega t).

    \]

  4. The e.m.f. varies sinusoidally; its maximum (peak) value is

    \[

    \boxed{\mathcal{E}_{\max}=NAB\omega }.

    \]

5. Experiments Demonstrating Electromagnetic Induction

ExperimentPurposeApparatusKey ProcedureObservations & Interpretation
Moving Magnet Through a CoilShow that a *changing* magnetic flux induces an e.m.f.Coil (~100 turns), galvanometer, bar magnet, stand.

  1. Connect the coil to the galvanometer.

  2. Insert the magnet into the coil at a constant speed; note the deflection.

  3. Withdraw the magnet and observe the reversal of deflection.

  4. Repeat with different speeds.

  • Deflection appears only while the magnet moves → flux is changing.

  • Direction reverses on withdrawal → sign of \(d\Phi/dt\) reverses.

  • Higher speed → larger \(|\mathcal{E}|\) (greater rate of change).

Rotating Coil in a Uniform Magnetic Field (Generator)Quantify how \(N\), \(A\), \(B\) and \(\omega\) affect \(\mathcal{E}\).Square coil (adjustable \(N\) and \(A\)), wooden frame, horseshoe magnet, variable‑speed motor, voltmeter.

  1. Mount the coil on the motor shaft so it rotates between the magnet poles.

  2. Connect the coil leads to the voltmeter.

  3. Run the motor at a low speed; record the peak voltage \(\mathcal{E}_{\max}\).

  4. Increase the speed and repeat.

  5. Change \(N\) and/or \(A\) and repeat the measurements.

  • Peak voltage follows \(\mathcal{E}_{\max}=NAB\omega\) (derived above).

  • Doubling any of \(N\), \(A\), \(B\) or \(\omega\) doubles \(\mathcal{E}_{\max}\).

Falling Magnet Through a Conducting Ring (Lenz’s‑law demonstration)Show that the induced e.m.f. creates a magnetic field that opposes the motion that produced it.Copper ring, strong neodymium magnet, vertical stand, stop‑watch (or electronic timer).

  1. Hold the ring freely so it can move vertically.

  2. Drop the magnet through the centre of the ring.

  3. Measure the fall time and compare with a free‑fall test (no ring).

  • The magnet falls more slowly when the ring is present → a current is induced, its magnetic field repels the approaching magnet (Lenz’s law).

Current‑Carrying Loop Near a Magnet (Opposition to change)Demonstrate the direction of the magnetic force on a current‑carrying coil.Rectangular coil on a low‑friction air track, fixed bar magnet, DC power supply, ammeter.

  1. Place the coil parallel to the field lines of the magnet.

  2. Pass a known current through the coil and release it.

  3. Observe the direction of motion of the coil.

  • The coil moves away from the magnet, confirming that the induced magnetic force opposes any increase of flux through the coil.

6. Applications

  • AC Generator – A coil rotating in a uniform magnetic field continuously changes the flux linkage, producing an alternating e.m.f.

    \[

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

    \]

    Hand‑crank generators and large power‑station alternators operate on this principle.

  • Transformer – A time‑varying current in the primary coil creates a changing magnetic flux in a common iron core; this varying flux induces an e.m.f. in the secondary coil.

    \[

    \mathcal{E}p = -Np\frac{d\Phi}{dt},\qquad

    \mathcal{E}s = -Ns\frac{d\Phi}{dt}

    \]

    Hence the voltage ratio is

    \[

    \boxed{\frac{Vs}{Vp}= \frac{Ns}{Np}}

    \]

    (ideal transformer, neglecting losses).

7. Worked Example

Problem: A coil of \(N=200\) turns, each turn having an area \(A=0.015\;\text{m}^2\), rotates at \(\omega = 50\;\text{rad s}^{-1}\) in a uniform magnetic field of \(B=0.25\;\text{T}\). Find the maximum induced e.m.f.

Solution:

\[

\mathcal{E}_{\max}=NAB\omega

= (200)(0.015\;\text{m}^2)(0.25\;\text{T})(50\;\text{rad s}^{-1})

= 37.5\;\text{V}.

\]

The coil will therefore produce a sinusoidal e.m.f. of amplitude 37.5 V, i.e. \(\mathcal{E}(t)=37.5\sin(50t)\) V.

8. Summary of Key Points

  • Magnetic flux: \(\Phi = BA\cos\theta\); flux linkage for a coil: \(N\Phi\).

  • Faraday’s law: \(\displaystyle \mathcal{E}= -N\frac{d\Phi}{dt}\); the minus sign embodies Lenz’s law.

  • Lenz’s law (qualitative): the induced current creates a magnetic field that opposes the change in flux.

  • Direction of induced current is found by (i) identifying the change in flux, (ii) applying Lenz’s law, and (iii) using the right‑hand rule (or Fleming’s left‑hand rule for forces).

  • \(\mathcal{E}\) increases linearly with the number of turns \(N\), magnetic field strength \(B\), coil area \(A\) and the rate of change of flux (angular speed \(\omega\) or linear speed \(v\)).

  • Electrical resistance does not affect the induced e.m.f.; it only determines the resulting current \(I=\mathcal{E}/R\).

  • Generator: rotating coil → alternating e.m.f. \(\mathcal{E}=NAB\omega\sin\omega t\).

  • Transformer: \(Vs/Vp = Ns/Np\) (ideal case).