33.3 Content
33.3.1 Torque on a Current Loop
A rectangular coil carrying current in a magnetic field experiences a torque:
\[\tau = nIAB\sin\theta\]
where: - \(\tau\) = torque (N·m) - \(n\) = number of turns - \(I\) = current (A) - \(A\) = area of coil (m²) - \(B\) = magnetic field strength (T) - \(\theta\) = angle between coil normal and field
- Maximum torque when \(\theta = 90°\) (coil plane parallel to field)
- Zero torque when \(\theta = 0°\) (coil plane perpendicular to field)
33.3.2 Interactive: DC Motor Operation
Visualise torque and rotation in a DC motor:
33.3.3 The Commutator
The commutator is a split-ring device that:
- Reverses current direction every half rotation
- Maintains torque direction so the motor keeps spinning
- Converts DC input to the alternating current needed by the coil
Without a commutator, the motor would oscillate back and forth instead of rotating continuously.
33.3.4 Ways to Increase Motor Torque
From \(\tau = nIAB\sin\theta\), torque can be increased by:
| Method | Effect | Practical Consideration |
|---|---|---|
| More turns (n) | Linear increase | Adds weight and resistance |
| Higher current (I) | Linear increase | More heating (I²R losses) |
| Larger coil area (A) | Linear increase | Larger motor size |
| Stronger magnet (B) | Linear increase | Stronger/heavier magnets |
| Radial field design | Keeps \(\theta = 90°\) | Standard in real motors |
33.3.5 Back EMF in Motors
When a motor spins, the rotating coil acts as a generator, inducing an EMF that opposes the supply voltage:
\[V = IR + \varepsilon_{back}\]
or
\[I = \frac{V - \varepsilon_{back}}{R}\]
When a motor starts, \(\varepsilon_{back} = 0\) (no rotation), so starting current is high: \[I_{start} = \frac{V}{R}\] As the motor speeds up, back EMF increases and current decreases.
33.3.6 Interactive: Back EMF Effects
33.3.7 DC Generator
A DC generator is a motor run in reverse:
- Mechanical energy rotates the coil
- Changing flux through the coil induces an EMF
- The commutator converts AC output to pulsating DC
- Brushes transfer current to the external circuit
The induced EMF varies with angle: \[\varepsilon = nBAω\sin(ωt)\]
33.3.8 AC Generator (Alternator)
An AC generator uses slip rings instead of a commutator:
- Output is sinusoidal AC: \(\varepsilon = \varepsilon_0\sin(ωt)\)
- No sparking at brushes (smoother contact)
- Simpler construction than DC generator
33.3.9 Interactive: Generator Comparison
33.3.10 Lenz’s Law and Energy Conservation
In a generator, Lenz’s law ensures energy conservation:
- Turning the generator coil requires mechanical work
- This induces a current that creates a magnetic force opposing rotation
- More current drawn → more force to overcome → more mechanical energy needed
- Electrical energy out = Mechanical energy in (minus losses)
If the induced current aided rotation (instead of opposing it): - The generator would accelerate itself - Energy would be created from nothing - This violates conservation of energy