Bicycle Dynamo vs White LED: Will It Light?
At a school science fair a team clamps a bicycle dynamo so that its rubber drive wheel runs against the front tyre. The tyre has outer radius 0.35 m and the dynamo's drive wheel has radius 2.0 cm, so the dynamo spins far faster than the wheel. The dynamo coil has 400 turns, each of area 1.5 x 10^-4 m^2, and rotates in a radial field of 0.20 T. The team wants to run a white LED rated 3.0 V, 0.06 W, which needs about 2.5 V across it before it glows visibly, and they plan to pedal at 12 km/h on level ground. Their teacher warns that the output is alternating, and a friend suggests simply winding on more turns. On a slope the team manages only 6 km/h and the LED dims; when the dynamo is lifted off the tyre it goes out completely. The coil resistance is 8 ohm and the LED draws about 0.02 A at 3.0 V.
1. Will the LED glow visibly at 12 km/h? Support your answer with numbers, and state the minimum number of turns for the peak EMF to reach 3.0 V.
Read the worked answer
Step 1 - speeds of rotation. v = 12 km/h = 3.33 m/s. Wheel angular speed = v/r = 3.33/0.35 = 9.52 rad/s. The drive wheel is not slipping, so the surface speeds match: omega_coil x 0.02 = 9.52 x 0.35, giving omega_coil = 9.52 x (0.35/0.02) = 9.52 x 17.5 = 167 rad/s (about 26.5 rev/s).
Step 2 - peak EMF. For a coil rotating in a uniform field, e = NBAomegasin(omega t), so the peak is E0 = NBA*omega = 400 x 0.20 x 1.5e-4 x 167 = 2.0 V. The rms value is 2.0/sqrt(2) = 1.41 V.
Step 3 - verdict. The peak is 2.0 V, which is already below the ~2.5 V the LED needs before it glows visibly, so the LED will not light at 12 km/h. Even with a perfect rectifier and smoothing capacitor the best the capacitor can charge to is about the peak minus a diode drop, so it still fails.
Step 4 - turns needed. N = E0/(BA*omega) = 3.0/(0.20 x 1.5e-4 x 167) = 3.0/5.0e-3 = 600 turns. About 600 turns would give a 3.0 V peak (2.12 V rms), which is workable only with rectification and smoothing.
2. The friend says 'just add more turns'. Evaluate that advice quantitatively and list the practical reasons why 600 turns is not a straightforward fix.
Read the worked answer
The arithmetic works: 600 turns at 12 km/h gives E0 = 600 x 0.20 x 1.5e-4 x 167 = 3.0 V, so on paper the LED lights.
But the fix is not free:
- Coil resistance grows. Resistance is proportional to length and inversely to cross-section; fitting 50% more turns of thinner wire into the same body can roughly double the 8 ohm. At 0.02 A that is an extra 0.16-0.32 V lost inside the coil, eating much of the gain.
- Inductance and reactance. More turns mean more self-inductance L, so the inductive reactance 2pif*L rises at the operating frequency of about 26.5 Hz; the current lags and the current delivered is less than E/R predicts.
- Mechanical load and slip. The extra magnetic drag and the friction of a small drive wheel on a tyre sidewall must come from the rider; at low speed the drive wheel is likelier to slip, so the gear ratio is not reliably achieved.
- Frequency is low. At 26.5 Hz an unfiltered LED visibly flickers; a bridge rectifier plus a capacitor is needed, and the capacitor must hold up over the zero crossings.
A better engineered answer: increase B (stronger magnet) or coil area, use a rectifier and reservoir capacitor, or add a small boost stage rather than only adding turns.
3. Explain (a) why the LED dims at 6 km/h, (b) why it goes out completely when the dynamo is lifted off the tyre, and (c) what would be observed if the LED were connected to the coil with no rectifier.
Read the worked answer
(a) Speed dependence. EMF is proportional to the rate of change of flux, which is proportional to rotation rate and hence to road speed (no slip). At 6 km/h omega_coil halves to about 83 rad/s, so the peak EMF falls from 2.0 V to 1.0 V. That is far below the LED's ~2.5 V knee, so the diode barely conducts on each half-cycle and the light output collapses. The dependence is linear in speed, not quadratic - note that the power available falls faster, but the threshold behaviour is what you see.
(b) Lifting the dynamo. Faraday's law needs a changing flux. With the drive wheel off the tyre the coil stops rotating, flux through it is constant, induced EMF is zero, and the LED goes out. A common misconception is that a magnet 'stored in the dynamo' keeps the voltage; it does not - only relative motion between coil and field matters.
(c) No rectifier. The coil gives sinusoidal AC, so the LED conducts on only one half-cycle and is dark on the other; at 26.5 Hz the eye sees an obvious flicker. In this design the peak reverse voltage across the LED is only about 2 V, below typical reverse breakdown (about 5 V), so the LED survives - but with the 600-turn version the peak is 3 V and still marginal. A bridge rectifier with a capacitor gives steady light.