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Superficially, the force between two magnets would seem to be fully analogous to gravity because each is the product of two factors divided by the square of the distance between the objects being attracted.

In other words, if we imagine two celestial bodies with both mass and magnetic poles, it would seem that we could, by adjusting the mass and the magnitude of the magnetic poles, make the forces equal to each other, and as the bodies drew together the two forces would remain equal. In other words, if we were to graph the two forces as a function of distance, the two curves would be identical. Thus, the forces, though they are of a completely different nature, are fully analogous to each other.

Is this correct, or is there some essential difference between the two forces that will always make them behave in a different way?

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Let's use electric forces instead of magnetic forces because magnetic monopoles are not proven to exist (in other words, you can't just adjust the magnitude of the "magnetic poles").

Although Coulomb's force law is very similar to Newton's law of gravitation, there's one big difference in the two. Electric forces can attract and repel, because there's both positive and negative charge. Gravity is only ever attractive since negative mass is also not proven to exist.

If we go further and invoke relativity, then there's a further difference because relativity ties space to time, and the so-called "force" is just due to curved space. Nothing of this sort happens in electricity & magnetism.

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  • $\begingroup$ One subtle but often overlooked point I’ll add: while all forces act on mass, gravity is the only force that depends on mass. So there’s conceptually two types of mass: “gravitational force inducing mass” and “a measure of how difficult it is to accelerate something”. It’s often taken for granted that these conceptually distinct notions of mass turn out to be the same thing. $\endgroup$ – Ryan Franz Mar 19 at 3:35

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