Power produced by an induction coil?

Question

I know that you can get a decent amount of power with a coil and a magnet. I've seen flashlights powered by this. I'm having difficulty finding a succinct equation to describe this.

I think its going to be a function of the number of coils, the diameter of the coils, the diameter of the wires, the strength of the magnet and the average speed of the magnet. But that's about as far as I've gotten. I know its probably going to have some messy derivatives in it with the induction but I think my math skills aren't quite advanced enough yet to understand derive them myself.

Thanks!

Answer 1

You start with the induction law: $$U_{ind} = - \int \vec{E} \ d \vec{s} = - \frac{ \partial }{\partial t} \int {\vec B}\ d \vec{A}$$

This law says, that every change of the magnetic field in the cross section area of the closen wire winding leads to a certain induction voltage. The power is given by $$ P = \frac{U^2}{R} $$ where \$R\$ is the resistance of your consumer. Regarding to Lenz' law, the induced current works "against its cause". Therefore a higher resistance of the load increases the electrical power but takes more kinetic energy from the generator. The quanitative relation is given by inductivity $$ L = \frac{N^2}{R_M} $$ where \$N\$ is the number of windings and \$R_M\$ is the magnetic resistance of your magnetic circuit, which can be determined with some geometry. Then you need the time-dependency of the magnetic field through the coil. To aim a description of the process, you will not outflanks maths. But for your motivation you will find some easy experiments.