understand and use the concept of magnetic flux linkage

Electromagnetic Induction – Magnetic Flux Linkage

1. Magnetic Flux ( Φ )

• Definition: the amount of magnetic field that passes through a given surface.
• Mathematical expression

  \[

  \Phi = B\,A\cos\theta

  \]

  • \(B\) – magnetic flux density (tesla, T)
  • \(A\) – area of the surface (m²)
  • \(\theta\) – angle between the field direction and the normal to the surface

• Unit: weber (Wb), where 1 Wb = 1 T·m².

2. Flux Linkage ( Λ )

• When a coil of exactly \(N\) turns (an integer) is considered, the total flux linked with the coil is called the flux linkage.
• Formula

  \[

  \Lambda = N\Phi = N\,B\,A\cos\theta

  \]

  • \(N\) must be an integer – this is a specific requirement of the Cambridge syllabus.

• Scalar quantity (no direction).
• Unit: weber‑turns (Wb·turn).

3. Faraday’s Law of Electromagnetic Induction

The induced electromotive force (emf) in any closed conducting loop equals the negative rate of change of its flux linkage:

\[

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

\]

• The minus sign is the mathematical statement of Lenz’s law: the induced emf always opposes the change that produces it.
• For a coil rotating at a constant angular speed \(\omega\) in a uniform field,

  \[

  \mathcal{E}(t)= N B A \,\omega \sin(\omega t)

  \]

  (maximum when \(\sin\omega t = 1\)).

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

• If the magnetic flux through a coil increases, the induced current creates a magnetic field that opposes the increase.
• If the flux decreases, the induced current creates a field that tries to maintain the original flux.
• In practice the direction is obtained by combining the right‑hand rule for a magnetic field with the sign convention in Faraday’s law (the “‑” sign).

5. Factors that Influence Flux Linkage ( Λ )

6. Self‑Inductance (Optional but part of the syllabus)

• A coil can produce an emf in its own circuit when the current through it changes. This is called self‑induction.
• Self‑inductance \(L\) is defined by

  \[

  \mathcal{E}_{\text{self}} = -L\frac{dI}{dt}

  \]

  where \(I\) is the current in the coil.

• Units: henry (H), where 1 H = 1 Wb·A\(^{-1}\).
• In many A‑Level questions the focus is on mutual induction (transformers), but recognising the self‑induction term helps when analysing circuits containing inductors.

7. Worked Example – Rotating Multi‑Turn Loop

Problem: A circular coil of radius \(r = 0.08\ \text{m}\) has \(N = 150\) turns. It rotates at an angular speed \(\omega = 120\ \text{rad s}^{-1}\) in a uniform magnetic field of magnitude \(B = 0.45\ \text{T}\). Find the peak (maximum) induced emf.

1. Area of one turn: \(A = \pi r^{2} = \pi (0.08)^{2} = 2.01\times10^{-2}\ \text{m}^{2}\).
2. Flux through one turn: \(\Phi(t)=B A\cos(\omega t)\).
3. Flux linkage for the whole coil: \(\Lambda(t)=N\Phi(t)=N B A\cos(\omega t)\).
4. Differentiate (Faraday’s law):

   \[

   \mathcal{E}(t)= -\frac{d\Lambda}{dt}= N B A \,\omega \sin(\omega t)

   \]

5. Maximum emf occurs when \(\sin(\omega t)=1\):

   \[

   \mathcal{E}_{\max}= N B A \omega

   = 150 \times 0.45 \times 2.01\times10^{-2} \times 120

   \approx 1.63\ \text{V}

   \]

8. Common Applications (Cambridge‑aligned wording)

• AC generators (alternators) – a coil of \(N\) turns rotates in a uniform magnetic field, producing a sinusoidal emf \(\mathcal{E}=N B A \omega \sin\omega t\).
• Transformers – a time‑varying current in the primary creates a changing flux linkage, which induces an emf in the secondary according to \(\mathcal{E}{s}= -N{s}\frac{d\Phi}{dt}\).
• Induction cookers – a high‑frequency alternating current in a coil generates a rapidly changing flux, inducing eddy currents in metal cookware.
• Electric‑guitar pick‑ups – vibrating strings disturb the magnetic field of a coil, producing a small induced emf that is then amplified.

9. Experimental Skills (relevant to the syllabus)

• Design a simple apparatus to measure the induced emf in a rotating coil (e.g., using a slip‑ring and a galvanometer).
• Vary one parameter at a time (‑ \(N\), \(B\), \(A\), \(\omega\) ‑) and record the corresponding emf.
• Plot \(\mathcal{E}_{\max}\) against the varied quantity to verify the linear relationships predicted by theory.
• Estimate uncertainties (instrumental and random) and propagate them to the final result.
• Discuss sources of systematic error such as non‑uniform magnetic field, friction in the rotation mechanism, and contact resistance in the slip‑ring.

10. Summary Checklist

• Magnetic flux: \(\displaystyle \Phi = B A \cos\theta\) (unit = Wb).
• Flux linkage: \(\displaystyle \Lambda = N\Phi = N B A \cos\theta\) (unit = Wb·turn; \(N\) must be an integer).
• Faraday’s law: \(\displaystyle \mathcal{E}= -\frac{d\Lambda}{dt}= -N\frac{d\Phi}{dt}\).
• Lenz’s law – the induced current always opposes the change that produces it (sign convention in Faraday’s law).
• Increasing any of \(N\), \(B\), \(A\) or the rate of change of \(\theta\) (or of \(B\) or \(A\)) increases the magnitude of the induced emf.
• Self‑induction: \(\displaystyle \mathcal{E}_{\text{self}} = -L\,\frac{dI}{dt}\) (optional syllabus item).

11. Practice Questions

1. A coil of \(N = 200\) turns, each of area \(5.0\times10^{-3}\ \text{m}^{2}\), is placed in a magnetic field that rises uniformly from \(0\) to \(0.80\ \text{T}\) in \(0.25\ \text{s}\).

   Calculate the average induced emf.

2. A rectangular loop \(0.15\ \text{m}\times0.30\ \text{m}\) rotates at \(60\ \text{rev s}^{-1}\) in a \(0.25\ \text{T}\) field.

   Determine the peak emf.

3. Explain qualitatively how the direction of the induced current changes when the rotating loop passes the position where the magnetic flux is maximum.
4. A transformer has 500 turns on the primary and 100 turns on the secondary. The primary is connected to a \(230\ \text{V}\) (rms) AC supply at \(50\ \text{Hz}\).

   Find the rms voltage induced in the secondary.

5. In a laboratory set‑up a single‑turn loop of area \(4.0\times10^{-3}\ \text{m}^{2}\) is rotated at \(\omega = 100\ \text{rad s}^{-1}\) in a field of \(0.60\ \text{T}\).

   Predict the peak emf and compare it with a measured value of \(1.45\ \text{V}\). Discuss possible reasons for any discrepancy.

12. Suggested Diagram