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Let me go step-by-step.

First I will outline a simple transformer. There are two circuits next to each other (not touching). One is an electromagnet set-up (with AC supply) and the other has a galvanometer. Turning on the electromagnet induces an EMF and a current in the circuit next to it. Increasing the number of turns in the electromagnet increases the strength of the magnetic field and the induced EMF in the second circuit falls while its current rises. This I've been told is due to the fact that $P$ remains constant and $P=VI$.

Now I'd like to outline another setup. There is a coil in a circuit with no power supply. There is a galvanometer. I take a magnet and continuously thrust it into and out of the coil, inducing an alternating EMF and current. The speed with which I thrust the magnet is constant, and I believe that I do a certain amount of work by thrusting it. Therefore I have a constant power. Now I take a stronger magnet (so as to increase the strength of the magnetic field) and I repeat the experiment with the same speed of thrust (therefore same power). Both the induced EMF and the induced current increase.

Now correct me if I'm wrong. A moving magnetic field and a changing magnetic field both have the same effect for inducing EMF and current. My power input is constant in both setups. I increase the magnetic field strength in both setups. Both setups are virtually the same. The EMF falls while current increases in the first setup. Both the EMF and the current increase in the second setup. Why this difference in the results?

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    $\begingroup$ I recommend searching Amazon or other booksellers for books with titles like " simple electronics experiments for kids" . I found some of those when I was a kid, and they can be quite enjoyable and educational. $\endgroup$ – Carl Witthoft Dec 31 '15 at 13:41
  • $\begingroup$ Energy conservation does, indeed, give you the correct result. Where you are a little bit wrong is that you would be moving the stronger magnet with the same speed. If you wanted to keep the exerted power constant, then the amplitude of the motion could be smaller. You can see this in the design of electric machines. We have been able to make them quite a bit smaller over time because we have learned to make magnetic materials that can produce stronger fields, so less motion (either axial or radial) is necessary to achieve the same power conversion. $\endgroup$ – CuriousOne Dec 31 '15 at 21:41

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