# Meeting point of magnets

Physics Asked on February 18, 2021

Suppose, we leave two magnets in space close to each other with no other force acting on them apart from the attractive force between them. One is stronger than the other but, they have the same mass. Would they meet at the middle or closer to the stronger?

It’s important to clarify that although both magnets have the same mass we have to assume that a percentage of the material of the weaker one is not magnetizable, otherwise we could assume that the stronger magnet would induce a field in the weaker one making the strength of both fields similar.

Newton's Third Law says that the force exerted on Magnet #1 by Magnet #2 is equal in magnitude to the force exerted on Magnet #2 by Magnet #1. Since they have the same mass, they will therefore accelerate at the same rate, and they will therefore meet exactly in the middle.

The relative strength of the magnets is a red herring. One way to see this is to note that the magnetic force between two magnetic dipoles is proportional to the product of the dipole moments similar to how the force between two charges is proportional to the product of the charges. Thus, the stronger magnet creates a stronger magnetic field at the location of the weaker magnet than vice versa; but the stronger magnet also responds more strongly to a given field than the weaker magnet does. These two factors cancel each other out, and the force ends up being the same.

This does neglect the effects of electromagnetic radiation, which could conceivably carry a small amount of momentum off to infinity. I would expect, from similar calculations involving electric dipoles, that the radiation would be proportional to the square of the jerk $$j$$. A bit more dimensional analysis shows that the net momentum flux at infinity due to the radiation fields should be something like $$frac{d P_text{rad}}{d t} sim frac{ mu_0 m^2}{c^6} j^2.$$ The radiation reaction force must therefore also be proportional to this quantity. If we denote $$v$$ as the velocity scale of the dipoles during their motion, $$R$$ as the length scale of the motion, and $$T$$ as the time scale, we have $$j sim frac{v}{T^2} sim frac{v}{(R/v)^2} sim frac{v^3}{R^2}$$ and so the radiation reaction force on the dipoles should be (to within a few orders of magnitude $$frac{d P_text{rad}}{d t} sim frac{mu_0 m^2}{R^4} frac{v^6}{c^6}.$$ Since the actual dipole force between the magnets will be roughly proportional to $$mu_0 m^2/R^4$$, we conclude that the effects of the radiation reaction force will smaller than the those of the radiation reaction force by a factor of $$(v/c)^6$$ (to within a few orders of magnitude.) So long as the magnets remain non-relativistic, this effect would of course be utterly negligible.

Answered by Michael Seifert on February 18, 2021

If you consider the two magnets together as a system, there is no external force acting on it. Since there is no external force acting on it, the centre of mass of the system remains unchanged. As a result, no matter how the magnets move, they will do so in such a fashion as to keep the centre of mass constant. From this, it should be easy to see that the two magnets will meet at their centre of mass. If the magnets have the same mass, they will meet at the middle If one of the magnets is much more massive than the other, the centre of mass will be shifted towards that magnet, and so they will meet closer to the "heavier" magnet.

Note: This is independent of the strength of the magnets.

Answered by Philip on February 18, 2021

As you probably know, there's no magnetic monopoles thus it would be difficult to compare the magnitude of the force between two magnets unlike in the case of two electrically charged particles where its pretty straightforward.

If you leave two opposite charged particles like electron and proton and if you compare the gravitational and electromagnetic force between them, you can get the following ratio

$$frac{F_g}{F_e} approx 10^{-40}$$

If you take the electron and proton example into consideration, and if you consider everything classically you can say they will meet somewhere closer to the proton. But this picture is flawed.

Answered by Monopole on February 18, 2021

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