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I've had argument on a forum about this. People think, that: if the same object is magnetised, it has less energy. so by $E = mc^2$ it weighs less than an object that is not magnetised (hence spin of its particles is chaotic).

I've thought about experiment :)

Let's say we have square of 4 protons and another square of 4 antiprotons. If we magnetise each square and annihilate them, do we get the same energy that we would get if only the antiprotons were magnetised?

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(1) Magnetized objects typically have higher energy than before they were magnetized. There is energy in the magnetic field, and the magnetic field for a magnetized object is larger.

One way to see this is to imagine a large collection of tiny magnets. If you piled them into a mound they wouldn't organize themselves into all pointing in the same direction. Instead, they'd energetically prefer to be pointing in random directions so their magnetic fields cancel better.

(2) So magnetizing an object increases its energy and therefore increases its mass. But with the usual magnets available to mankind, the change in mass is so small as to be undetectable.

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