How to calculate induced energy? Suppose we have a fixed copper coil with $n$ turns. It has no iron core. We throw a magnet through it with speed $v$ and we don't need to account for gravity. How do we calculate the induced electrical energy in the coil and thus the movement energy loss and speed loss of the magnet? Is there a useful formula or something?
 A: (a) There will be no electromagnetic transfer of energy unless the ends of your coil are connected to a 'load' of some sort, or even just connected together so that there is a complete circuit.
(b) There will be two pulses of voltage (emf), in opposite senses, one as the magnet enters the coil, and the other as it leaves.
(c) To calculate the energy converted during these pulses is no easy matter. We'd need to know how the magnitude of the flux density, $B$ due to the magnet varied with distance $x$ along the axis of the magnet from its centre. Then if their wasn't much clearance between the magnet and the coil as the magnet passes through, the induced emf when the centre of the magnet is distance $x$ from the coil would be given roughly by
$$\mathscr E =\ -nvA\frac{dB}{dx}.$$
$A$ is the coil area. This equation wouldn't be easy to use, unless you had a tesla-meter so you could measure how $B$ varied along the axis of the magnet so you could calculate $\frac{dB}{dx}$ for the various values of $x$.
It would be easier to investigate the emf induced in the coil experimentally, perhaps using an oscilloscope. Once you know how the emf varies with time you can work out the energy transfer by finding the area under a graph of $\mathscr E^2$ against time and dividing by the circuit resistance. Good luck!
