Describe a simple form of a.c. generator (rotating coil or rotating magnet) and the use of slip rings and brushes where needed

4.5.2 The a.c. generator

Objective

Describe a simple alternating‑current (a.c.) generator – either the rotating‑coil (armature) type or the rotating‑magnet type – and explain when and why slip rings and brushes are required.

1. Principle of electromagnetic induction

Two equivalent ways of producing an induced emf are required by the Cambridge syllabus:

1. Moving conductor in a static magnetic field – a coil (or part of a coil) rotates while the magnetic field remains fixed.
2. Changing magnetic field around a stationary conductor – the magnetic field varies with time while the coil stays still.

Both situations are described by Faraday’s law

\$\mathcal{E}= -\frac{d\Phi}{dt},\$

where \(\Phi = BA\cos\theta\) is the magnetic flux through the coil. The minus sign expresses Lenz’s law: the induced emf always produces a current whose magnetic field opposes the change in flux.

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

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

This sinusoidal variation is the origin of the alternating voltage produced by an a.c. generator.

2. Simple rotating‑coil (armature) generator

2.1 Components

2.2 Operation (step‑by‑step)

1. The coil is turned at a constant angular speed \(\omega\). The angle between the coil’s normal and the magnetic field is \(\theta = \omega t\).
2. Magnetic flux through the coil varies as \(\Phi = BA\cos\theta\); therefore \(\displaystyle \mathcal{E}= -\frac{d\Phi}{dt}=NAB\omega\sin\theta\).
3. Every half‑turn the sign of \(\sin\theta\) changes, so the polarity of the induced emf reverses – an alternating voltage is produced.
4. According to Lenz’s law, the induced current flows in a direction that creates a magnetic field opposing the change in flux. Using Fleming’s right‑hand rule (thumb = motion of conductor, fore‑finger = field direction, middle‑finger = induced current) the direction of the current can be predicted. (Insert diagram of right‑hand rule showing the coil at \(\theta=45^\circ\)).
5. Slip rings rotate with the axle, keeping the electrical connection to the coil continuous. The brushes stay stationary, sliding on the rings and delivering the alternating voltage to the external circuit.

2.3 Slip rings and brushes

2.4 emf‑time graph and frequency

The induced emf varies sinusoidally:

The frequency of the alternating voltage is

\$f=\frac{\omega}{2\pi}\quad\text{(revolutions per second)}.\$

The rms (root‑mean‑square) value, which is used when comparing with dc voltages, is

\$\mathcal{E}{\text{rms}}=\frac{\mathcal{E}{\max}}{\sqrt{2}}.\$

2.5 Practical demonstration (lab activity)

1. Mount a rectangular coil (≈ 200 turns, 10 cm × 10 cm) on a low‑friction axle.
2. Place two bar magnets on a wooden base so that a uniform field of about 0.5 T passes through the coil.
3. Connect the coil to a sensitive galvanometer via slip rings and carbon brushes.
4. Turn the crank at a steady speed (e.g. 30 rev s⁻¹). Observe the galvanometer needle deflect alternately, confirming the sinusoidal emf.
5. Record the number of reversals per second and compare with the calculated frequency \(f=\omega/2\pi\).

3. Simple rotating‑magnet generator

3.1 Components

3.2 Operation

1. Rotate the magnets at angular speed \(\omega\). The magnetic flux through the stationary coil changes as \(\Phi = BA\cos\omega t\).
2. Faraday’s law gives the same sinusoidal emf as for the rotating‑coil generator: \(\mathcal{E}(t)=NAB\omega\sin\omega t\).
3. Because the coil does not move, the output can be taken directly from the fixed terminals; slip rings and brushes are unnecessary.

3.3 emf‑time graph

The same sine‑wave graph shown for the rotating‑coil generator applies here; only the mechanical motion is transferred to the magnets.

4. Factors that affect the magnitude of the induced emf

• Number of turns (N) – emf ∝ N.
• Coil area (A) – larger area intercepts more flux.
• Magnetic field strength (B) – stronger field gives larger flux change.
• Angular speed (ω) – faster rotation increases the rate of flux change.
• Frequency (f = ω/2π) – determines how many cycles per second; the shape of the graph remains sinusoidal.
• RMS value – for a sinusoidal emf, \(\mathcal{E}{\text{rms}} = \mathcal{E}{\max}/\sqrt{2}\).

All of these appear in the peak‑emf expression \(\mathcal{E}_{\max}=NAB\omega\).

5. Comparison of the two simple generators

6. Key points to remember

• Both types of simple a.c. generators rely on Faraday’s law; the emf varies as \(\mathcal{E}(t)=\mathcal{E}_{\max}\sin\omega t\).
• Lenz’s law (or Fleming’s right‑hand rule) tells the direction of the induced current.
• Slip rings and brushes are required only when the emf‑producing coil rotates.
• Peak emf \(\mathcal{E}_{\max}=NAB\omega\); increasing any of \(N, A, B,\) or \(\omega\) raises the output.
• The rms value of a sinusoidal emf is \(\mathcal{E}{\text{rms}}=\mathcal{E}{\max}/\sqrt{2}\); frequency \(f=\omega/2\pi\) determines how many cycles occur each second.

7. Sample exam‑style question

Question: A rectangular coil of 200 turns, each side 10 cm long, rotates at 300 rev min⁻¹ in a uniform magnetic field of 0.5 T. Calculate the maximum induced emf and the rms value of the output voltage.

Solution:

1. Convert the speed to angular velocity:

   \[

   \omega = 2\pi\;\frac{300}{60}=10\pi\ \text{rad s}^{-1}

   \]

2. Area of one turn:

   \[

   A = (0.10\ \text{m})^{2}=0.01\ \text{m}^{2}

   \]

3. Peak emf:

   \[

   \mathcal{E}_{\max}=NAB\omega = 200 \times 0.01 \times 0.5 \times 10\pi = 10\pi\ \text{V}\approx 31.4\ \text{V}

   \]

4. RMS emf:

   \[

   \mathcal{E}{\text{rms}}=\frac{\mathcal{E}{\max}}{\sqrt{2}} \approx \frac{31.4}{1.414}=22.2\ \text{V}

   \]