Wouldn't this violate the Law of Conservation of Energy?

Question

If you ran an electric current through two neighboring electromagnetic coils so that the force between them repelled them, the two coils would accelerate away from each other. If you doubled the energy input to double the strength of the repelling fields, the force would be quadrupled. If work = force x distance, wouldn't the potential mechanical energy also quadruple, despite the fact that you only doubled the energy input?

Answer 1

Yes, it would.  So, you just explained why you have to quadruple the energy input to double the magnetic field strength.

Answer 2

basic mistake there.  the idea of fields is based on a need to explain the force experienced.  the two quantities are inextricable.  therefore if you double the field, you double the force.

So no, this wouldn't violate conservation of energy.  Just like nothing else, ever, in the world, or in the universe, for now and forever will ever violate conservation of energy.

Answer 3

The setup you describe is found in a series-wound DC motor, whose field and armature windings comprise the two coils. While doubling the current in both coils quadruples the static torque, you need to consider the energy ratio, or more conveniently the power ratio (power out/power in, or force*velocity / current*voltage). Assuming superconducting coils (so we can disregard resistive losses), it will take double the voltage to push this doubled current through. Why? Because the moving coil (armature) generates a "back EMF" that is proportional to the product of field-winding current and speed. This EMF must be equaled by the power source to send current through the armature. So assuming constant speed and double the field current, you need to double the voltage as well. Thus there is no violation of energy conservation.

2*current*2*voltage = 4*power in = 4*force*1*speed = 4*power out.

P.S. This "double the voltage and double the # of coil windings, it will double the field strength" you mentioned applies to ohm's law. In fact with a coil that has resistance, you can double the voltage and do anything you like to the number of turns and you double the field strength, since the ampere-turns remain constant at twice the previous value. I assumed superconductivity above to avoid that rather irrelevant discussion.