Sketch and interpret graphs of e.m.f. against time for simple a.c. generators and relate the position of the generator coil to the peaks, troughs and zeros of the e.m.f.

4.5.2 The a.c. Generator

Objective

  • Sketch and interpret the e.m.f. versus time graph for a simple a.c. generator.

  • Relate the physical orientation of the rotating coil to the peaks, troughs and zero‑crossings of the e.m.f.

  • Identify the essential construction features (coil, magnetic field, slip‑rings & brushes) and apply Lenz’s law.

  • Distinguish briefly between a rotating‑coil and a rotating‑magnet generator.

1. Construction of a simple a.c. generator

  • Coil (armature): a rectangular loop of area A with N turns, mounted on a shaft that can rotate freely.

  • Magnetic field: a uniform field B produced by permanent magnets (or a field coil). The field is assumed uniform across the coil – a condition satisfied by the simple laboratory generator used in the syllabus.

  • Rotation: the shaft is driven mechanically (hand‑crank, motor, water turbine) at an angular speed ω (rad s⁻¹).

  • Slip‑rings and brushes: the two ends of the coil are connected to split‑ring (slip‑ring) contacts; carbon brushes press against the rings, allowing the alternating e.m.f. to be taken from the rotating coil while the coil continues to turn.

  • Rotating‑coil vs. rotating‑magnet:

    • Rotating‑coil generator (the diagram above) – the coil rotates in a stationary magnetic field.

    • Rotating‑magnet generator – the magnet rotates while the coil is fixed. The relative motion is the same, so the induced e.m.f. has the identical sinusoidal form.

2. Principle of operation

When the coil rotates, the magnetic flux Φ through it varies sinusoidally:

\[

\Phi(t)=N\,B\,A\cos(\theta)=N\,B\,A\cos(\omega t)

\]

Faraday’s law states that the induced e.m.f. is the negative rate of change of flux:

\[

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

\]

Writing the expression in the familiar textbook form gives

\[

\boxed{\mathcal{E}(t)=\mathcal{E}_{\max}\,\sin(\omega t)}\qquad

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

\]

Lenz’s law – the minus sign in Faraday’s law indicates that the induced e.m.f. always acts to oppose the change in magnetic flux. In the sinusoidal graph this means the e.m.f. is positive when the flux is increasing in the chosen positive direction and negative when the flux is decreasing.

3. Shape of the e.m.f. – time graph

  • The graph is a sine wave that repeats every full rotation of the coil.

  • Period: \(\displaystyle T=\frac{2\pi}{\omega}\) (one complete cycle corresponds to one revolution).

  • Maximum positive e.m.f. (peak) occurs when the coil’s plane is perpendicular to the magnetic field – the coil normal is parallel to B.

  • Maximum negative e.m.f. (trough) occurs when the coil is again perpendicular but the normal points opposite to B.

  • Zero e.m.f. occurs when the coil’s plane is parallel to the field – the flux is momentarily zero and changing most rapidly.

Sinusoidal e.m.f. vs. time graph. Four points are labelled: A (t=0, zero crossing upward), B (t=T/4, +Emax peak), C (t=T/2, zero crossing downward), D (t=3T/4, –Emax trough). The coil angles corresponding to each point are 0°, 90°, 180°, 270° respectively.

Typical e.m.f.‑time graph for a simple a.c. generator. Points A–D are used in the Quick‑Check question.

4. Coil orientation versus graph points

Coil angle θ (°)Orientation of coilMagnetic flux ΦInduced e.m.f. 𝓔Graph point
0° (or 360°)Plane ‖ B; normal ⟂ BΦ = 0𝓔 = 0 (rising through zero)A – upward zero crossing
90°Plane ⟂ B; normal ‖ BΦ = +N B A (maximum)𝓔 = +𝓔maxB – positive peak
180°Plane ‖ B again; normal ⟂ B (opposite direction)Φ = 0𝓔 = 0 (falling through zero)C – downward zero crossing
270°Plane ⟂ B; normal opposite to BΦ = –N B A (minimum)𝓔 = –𝓔maxD – negative trough

Quick‑Check

In the graph above, point A is marked. State the exact orientation of the coil (plane ‖ B or ⟂ B, and direction of the normal) when the generator is at point A.

5. Quantitative relationships

  • Amplitude of the sinusoid:

    \(\displaystyle \mathcal{E}_{\max}=N\,B\,A\,\omega\)

  • Frequency of the alternating e.m.f. equals the rotation frequency of the coil:

    \(\displaystyle f=\frac{\omega}{2\pi}\) and \(\displaystyle T=\frac{1}{f}\)

  • Both \(\mathcal{E}_{\max}\) and \(f\) are directly proportional to the rotation speed.

6. Sample quantitative problem (AO 2)

Given: \(N = 200\) turns, \(B = 0.30\;{\rm T}\), \(A = 5.0\times10^{-3}\;{\rm m^{2}}\), rotation speed = 300 rev min⁻¹.

  1. Convert to angular velocity:

    \(f = \dfrac{300}{60}=5\;{\rm Hz}\) → \(\omega = 2\pi f = 2\pi\times5 = 31.4\;{\rm rad\,s^{-1}}\).

  2. Maximum e.m.f.:

    \(\mathcal{E}_{\max}=N B A \omega = 200\times0.30\times5.0\times10^{-3}\times31.4 \approx 4.7\;{\rm V}\).

  3. Frequency of the a.c. output is the same as the rotation frequency: \(f = 5\;{\rm Hz}\) (period \(T = 0.20\;{\rm s}\)).

  4. Students should sketch a sine wave of period 0.20 s with peaks at \(\pm4.7\;{\rm V}\) and label the four characteristic points A–D.

7. Practical activity – Demonstrating the sinusoidal e.m.f. (AO 3)

Apparatus

  • Hand‑cranked a.c. generator (visible slip‑rings & brushes)

  • Oscilloscope or digital voltmeter with a fast‑response probe

  • Stopwatch

  • Variable resistor (optional, to load the generator)

Method

  1. Connect the generator terminals to the oscilloscope input.

  2. Rotate the crank at a steady speed; observe the sinusoidal trace.

  3. Measure the time for one complete cycle on the oscilloscope – this gives the period \(T\) and frequency \(f=1/T\).

  4. Increase the crank speed in steps; record the new period and the peak‑to‑peak voltage each time.

  5. Plot \(V{\max}\) against \(\omega\); the straight‑line graph confirms \(\mathcal{E}{\max}=N B A \omega\).

Safety notes

  • Keep fingers away from rotating parts and slip‑ring contacts.

  • Use a low‑voltage generator (typically a few volts) to avoid shock.

8. Summary of key points

  • The rotating coil in a uniform magnetic field produces a sinusoidal e.m.f. because the magnetic flux varies as \(\cos(\omega t)\); the e.m.f. is its time derivative, \(\sin(\omega t)\).

  • Peak e.m.f. occurs when the coil is perpendicular to the field; zero e.m.f. occurs when the coil is parallel.

  • Slip‑rings and brushes provide a continuous electrical connection to the rotating coil.

  • Lenz’s law determines the sign of the sine wave – the induced e.m.f. opposes the change in flux.

  • Amplitude \(\mathcal{E}_{\max}=N B A \omega\); frequency \(f = \omega/2\pi\) (directly proportional to rotation speed).

  • Students must be able to sketch the e.m.f.–time graph, label points A–D, and state the corresponding coil orientation.