Bicycle dynamo

Problem

An old dynamo rests on the tyre and powers a $3 \ \mathrm{W}$ headlamp at $6 \ \mathrm{V}$. Pedalling faster makes it harder to pedal. Explain in five lines the energy balance, indicating how the EMF, current, useful power and resistive force on the pedal change when the speed doubles.

Solution

The dynamo is an alternator: its EMF is $\mathcal{E} = N B A\,\omega \propto \omega$, so it doubles when the speed doubles. The current $i = \mathcal{E}/R \propto \omega$ doubles. The electrical power delivered $P = \mathcal{E}\,i \propto \omega^2$ quadruples. $\mathcal{E} \propto \omega, \qquad i \propto \omega, \qquad P \propto \omega^2, \qquad F_{\text{brake}} \propto \omega$ The braking force felt at the pedal is linked to the resistive torque $\propto P/v \propto \omega$: it doubles. The energy balance is: the muscular work of the legs is converted into electrical energy in the dynamo, which the headlamp turns into light and heat through Joule heating. Pedalling harder is tiring because all the extra power delivered by the lamp must come from the muscles.

Links

Topics: Induzione elettromagnetica Concepts: Legge di Faraday-Neumann-Lenz · Corrente alternata · Effetto Joule · Forza elettromotrice