The Hand-Cranked Generator That Lights Nothing
Students in a higher secondary physics club build a simple AC generator from a rectangular coil of 200 turns and area 0.01 m², rotated at 3 revolutions per second in a uniform magnetic field of 0.15 T. The coil resistance is 3 Ω. They connect the ends through slip rings to a 6 V, 0.3 A torch bulb. On turning the handle at the design speed, the bulb glows a dull orange. The students have spare magnets, a variable-speed drill, extra wire of the same gauge, and a split-ring commutator. Their teacher asks them to decide, with calculations, whether the machine is faulty or simply under-designed, and which single change will make the bulb glow at its rated brightness without burning it out. They must also explain why a commutator is not automatically an improvement for this test.
1. Calculate the peak and rms emf generated at 3 rev/s. Then calculate the current and power in the bulb, taking the coil resistance into account, and explain the dull glow.
Read the worked answer
The angular frequency is ω = 2πf = 2π × 3 = 18.85 rad/s. For a coil of N turns, area A, in flux density B, the peak emf is E₀ = NABω = 200 × 0.01 × 0.15 × 18.85 = 5.65 V. The rms emf is E₀/√2 = 5.65/1.414 = 4.0 V. The bulb's resistance is R_bulb = V²/P = 6²/0.3? Careful: rated current 0.3 A, so R = 6/0.3 = 20 Ω, and rated power = 6 × 0.3 = 1.8 W. The total circuit resistance is 20 + 3 = 23 Ω. The rms current is 4.0/23 = 0.174 A. Power in the bulb is I²R = 0.174² × 20 = 0.61 W, about one-third of its 1.8 W rating. The terminal voltage is 4.0 − (0.174 × 3) = 3.48 V, well below 6 V. The machine is not faulty; it is under-designed for this bulb, and the coil's internal resistance wastes some of the generated emf.
2. Which single change would you recommend: doubling the number of turns, doubling the magnetic field, doubling the speed, or doubling the coil area? Support your choice with calculations, including the effect on coil resistance where relevant.
Read the worked answer
Doubling any of N, B, f or A doubles the peak emf to about 11.3 V and the rms to about 8.0 V. The differences lie in the resistance penalty and practical difficulty.
- Doubling B leaves the coil resistance at 3 Ω. Current = 8.0/23 = 0.348 A; bulb power = 0.348² × 20 = 2.42 W, above the 1.8 W rating, so the bulb may burn out. A 1.5× increase in B would give about 6.0 V rms and a safer power near 1.4 W.
- Doubling speed also leaves R at 3 Ω but may be mechanically difficult and increases bearing loss and vibration.
- Doubling the turns with the same wire gauge roughly doubles the wire length and therefore the coil resistance to about 6 Ω. Current = 8.0/26 = 0.308 A; bulb power = 1.90 W, very close to rated. This is electrically the best match, though it uses more copper.
- Doubling the area increases the length per turn and hence resistance, so the gain is less than proportional. Recommendation: double the turns if the winding window allows, because it gives rated brightness without the over-power risk of doubling B. If the magnet can be changed easily, a moderate increase in B with a series resistor is an alternative.
3. The students think that fitting the split-ring commutator will make the generator better. Explain what the commutator does, and whether it helps this particular test.
Read the worked answer
A commutator reverses the connection of the coil to the external circuit every half cycle, so the output at the brushes is unidirectional, though it is still pulsating rather than smooth DC. Slip rings, by contrast, give alternating output. For a torch bulb, the direction of current does not matter; the bulb heats on either half cycle and responds to the rms value. Fitting a commutator would not increase the generated emf or reduce the coil resistance, so it would not make the bulb brighter. It would add brush contact resistance and sparking losses, slightly reducing the available voltage. The commutator becomes necessary only if the students want to charge a battery, run a small DC motor, or demonstrate a DC generator. For this test, the better improvements are more turns, a stronger field, or a higher speed, together with checking that the bulb rating matches the generated rms voltage.