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
- Identify whether the magnetic flux through the loop is increasing or decreasing.
- Apply Lenz’s law: the induced magnetic field must oppose that change.
- 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.
- 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
| Factor | Effect 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).
- Instantaneous magnetic flux: \(\displaystyle \Phi(t)=BA\cos\theta(t)=BA\cos(\omega t)\).
- Differentiate with respect to time:
\[
\frac{d\Phi}{dt}= -BA\omega\sin(\omega t).
\]
- Apply Faraday’s law for a coil of \(N\) turns:
\[
\mathcal{E}(t)= -N\frac{d\Phi}{dt}= NAB\omega\sin(\omega t).
\]
- The e.m.f. varies sinusoidally; its maximum (peak) value is
\[
\boxed{\mathcal{E}_{\max}=NAB\omega }.
\]
5. Experiments Demonstrating Electromagnetic Induction
| Experiment | Purpose | Apparatus | Key Procedure | Observations & Interpretation |
|---|
| Moving Magnet Through a Coil | Show that a *changing* magnetic flux induces an e.m.f. | Coil (~100 turns), galvanometer, bar magnet, stand. | - Connect the coil to the galvanometer.
- Insert the magnet into the coil at a constant speed; note the deflection.
- Withdraw the magnet and observe the reversal of deflection.
- 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. | - Mount the coil on the motor shaft so it rotates between the magnet poles.
- Connect the coil leads to the voltmeter.
- Run the motor at a low speed; record the peak voltage \(\mathcal{E}_{\max}\).
- Increase the speed and repeat.
- 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). | - Hold the ring freely so it can move vertically.
- Drop the magnet through the centre of the ring.
- 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. | - Place the coil parallel to the field lines of the magnet.
- Pass a known current through the coil and release it.
- 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).