# How do electrical motors obey the conservation of energy?

Feynman introduces the concept of an electromagnetic motor with this diagram.

I believe I understand why the wire turns when there is current through it, because $$F = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})$$, but I do not understand how this is allowed under conservation of energy.

Supposing I had a perfectly conducting wire, it seems I could close the loop and have an electron travel continuously around it, causing perpetual motion.

What am I missing? I suspect that my idea of closing the loop is wrongheaded, but would appreciate a principled explanation of why.

• I’m not really well versed on this subject but it is my understanding that perpetual motion machines violate the second law of thermodynamics not the first,?applies to conservation of energy – Bob D Jan 19 '20 at 6:56
• @BobD If I have a motor that turns my wheels and never stops (against friction), it gives me free energy, violating the first law, right? – Eli Rose Feb 24 '20 at 15:19
• You mean after taking the motor away the wheels move forever? – Bob D Feb 24 '20 at 15:23
• Nope, just a motor that doesn't stop, attached to a little car with wheels. – Eli Rose Feb 24 '20 at 15:57

The reason you can't do that is as follows, which really doesn't have anything directly to do with energy conservation.

We apply a voltage to the wire, which pushes current through it which then produces a magnetic field that acts against the magnetic field produced by the permanent magnet, then that reaction force acts on the wire, causing it to rotate.

But as the moving wire cuts through the field lines produced by the permanent magnets, another voltage (called an electromotive force or EMF) is induced in the wire which opposes the voltage we are applying to wire. The faster the wire is moving, the greater this "back EMF" becomes. When the back EMF equals the source voltage, the current through the wire goes to zero (in the ideal case) and the motor can revolve no faster.

This back EMF effect prevents us from being able to make a perpetual motion machine out of an electric motor, because it acts to cancel the flow of current even in the case of a perfectly-conducting wire.

• Isn't there another issue, where we would need to have a voltage difference between the two ends of the wire in order to get a current, and this voltage difference wouldn't last forever? – Eli Rose Jan 19 '20 at 15:30
• thats true too, but it's the back EMF that does the trick. – niels nielsen Jan 19 '20 at 19:42