Isn't the oscillation of the coil drawing the 'healing' power from the battery? IIRC, volts, watts, and amps are in a relationship such that volts x amps = watts. So, when the transformer (coil) is opened such that the electrons are released, the voltage sufficient to power the LED is available.
When the transformer is reconnected to the battery by the transistor, then it draws power from the battery, and the LED does not have sufficient voltage.
I reckon that the use of two batteries in this circuit should last twice as long as the single battery, and this would indicate that no ergs of energy are entering the system from somewhere besides the battery, but that the transformer is, well, transforming the electricity into 3 volts for half the time the battery is connected.
Have you done this experiment?
Wow!!! One of the most technical question I have received so far, I can see you are a learned man Sir, I salute you, and hopefully you have read already my previous two posts. Actually, when we try to understand the healing part of the cycle, we need to understand as to why it heals in the first place, because healing is the circuit's reaction when it is energized initially. There is always a reaction for every action, equal and opposite right?
I see this circuit's healing cycle as its reaction, but reaction to what? The inductor is just reacting to the fact that it is always being "nudged" by the battery's voltage/current/wattage components (meaning electric), and every time the battery nudged the inductor, it will always go back to its original state and position after the nudge, hence the magnetizing (the battery nudged the inductor firstly) and de-magnetizing (the inductor going back to its original state before it was nudged) is but the inductor's reaction to the battery's action.
You are very much correct that more batteries means more time to operate. I have done this experiment for the last six years and counting on the same and simplistic joulethief circuit, I just modified it so many times and it is when things will get exciting when we do that. If you can replicate this Sir when we will reach that point, I will be privileged as we can maybe teach the world how to do this as well. Regards.