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So this is actually a two-part question. The graphics I've included are just rough ideas of what I'm thinking:

  1. Let's assume we have two bar magnets separated by an air gap of distance $g$ and with the top magnet exerting having a magnetic field 5x stronger than the bottom one. For now, let's also assume the poles of each magnet facing each other are the same (repelling). Is there a general formula for calculating the repulsive force felt by each magnet? Like what would be the downward force felt by the bottom magnet and the upward force felt by the top magnet?

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  1. Okay, so let's also now assume that the two magnets are held in the same physical system: Let's assume that both magnets are in a box - the top one nailed to the ceiling and the bottom one nailed to the floor. Would there be any net upward or downward force acting on the box?

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Somehow I think the answer is "no" but shouldn't there be a net downward force since the top magnet is substantially stronger than the bottom one?

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Is there a general formula for calculating the repulsive force felt by each magnet? Like what would be the downward force felt by the bottom magnet and the upward force felt by the top magnet?

To calculate the repulsive forces between the two magnets, we would need more information about the magnets. You can find more details about it in this Wikipedia article.

Would there be any net upward or downward force acting on the box?

Your intuition is correct: there won't be any net upward or downward force on the box due to the action of the magnets (of course, there will be a downward force due to gravity).

This is because the repulsion forces between the magnets and the resulting normal reaction forces between the box and the magnets are all internal forces in the system comprised of the magnets and the box and the internal forces always cancel each other, i.e., they always yield zero net force. If it was not the case, we would have a perpetual motion machine.

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  • $\begingroup$ Thanks for this answer! However, I'm still unclear regarding the normal forces particularly as it pertains to magnetism. If the upper magnet exerts a magnetic field that intersects with the field of the lower magnet, it will cause a repulsive magnetic force. But for that to cancel out, where would the equal and opposite reaction force come from? Would it be by the field of the lower magnet, the upper magnet itself, or from the container? $\endgroup$ – user3425451 Nov 5 '18 at 5:24
  • $\begingroup$ Okay, I think I get it now. The repulsive force is never unidirectional so the magnetic field's effect is always going to have equal and opposite forces on the two bodies it affects. I guess kind of like gravitational fields. $\endgroup$ – user3425451 Nov 5 '18 at 8:04

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