Misunderstanding these particular energy transfers

Question

Given 2 identical iron bars “A” and “B” in deep space. At some distance from them are 2 coils. “A” is moving with constant velocity “V”, “B” stands.

Same currents pass through the coils separately, creating same magnetic fields. They pull the bars. Those currents pass until the bars cover same distance “x”. As the bars covered the distances, each bar together with its coil created same force “F” (constant, for simplicity).

From the “point of view” of the bars they gained same energy F*x. On the other hand, “A” was moving, so it covered the distance faster. Accordingly, the current through its coil passed less time than the current passed the coild of “B”. Thus, less energy was spent to support that current than to support the current through the other coil.

Question: how can it be that at the same time different amounts of energy are spent from one "point of view", and same amounts of energy are gained from other "point of view"?

PS: Sorry for bad English.

Answer 1

The energy delivered to the magnetic field of the coil, and the work that this magnetic field does on the iron bars, do not depend on the amount of time it takes the bars to move the distance $x$. For the following I will ignore the nonlinear permeability of the iron bars for simplicity, so that the energy stored in the magnetic field is well-defined.

The work-energy balance for the problem can be written as follows: $$ W + \Delta E_{field} = E_{supp} $$ where $W$ is the work done on the iron bar by the magnetic field, $E_{field}$ is the energy in the magnetic field and $E_{supp}$ is the energy supplied to the coil by the power source that keeps the current constant. Note that power must be supplied to the coils, otherwise the current would decrease as the iron bars are pulled toward the coils. Expressing quantities in terms of the current $I$ along the coil and the flux $\Phi$ through the coil,

$$ W + \Delta\left(\frac{1}{2}\Phi I\right) = \Delta \Phi I $$ $$ W + \frac{1}{2} \Delta\Phi I = \Delta \Phi I $$ $$ W = \frac{1}{2}\Delta \Phi I. $$

Now, with the quasi-static approximation, the change in the flux only depends on the position of the bar and not the velocity, so the same work will be done on bars A and B. During the process, the energy in the field increases by the same amount $W$, and an energy of $2W$ is supplied by the power source.

Does it take more energy to keep the constant current through the coil flowing for a longer period of time? Realistically, yes, but this has nothing to do with the energy delivered to the magnetic field or work done on the iron bar. The coil will likely have a non-zero internal resistance $R$, dissipating power as heat at a rate of $\frac{1}{2}RI^2$. The longer you run the current through the coil, the more energy will be dissipated, but this energy is converted to heat and not to magnetic energy.