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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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    $\begingroup$ I don't think this is right. This argument suggests that aligned magnetic dipoles have lower potential energy than anti-aligned dipoles, but in fact the opposite is true. $\endgroup$
    – tparker
    Aug 31 '18 at 23:00
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    $\begingroup$ Let me try it this way: Piled into a mound they will cancel their overall magnetic field. As you note, this is a lower potential energy. Hence aligning them raises their potential energy and therefore raises their mass. $\endgroup$ Apr 6 '19 at 6:18
  • $\begingroup$ Arranging random placed dominoes to a coherent aligned structure will not change the mass of the dominoes nor their weight. The work you have done by arranging the dominoes is not stored as added energy into the dominoes but expelled as heat out to the environment. Similar process occurs when magnetizing ferromagnetic matter done by passing electric current through a coil. The EM energy in the coil is converted to kinetic energy for aligning the magnetic moments in the material therefore heat. After the magnetization process is stopped there is no more energy transfer. No energy is stored. $\endgroup$
    – Markoul11
    Apr 9 '21 at 17:45
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Since gravity as far as we know today seems totally uncoupled to the other fundamental forces (although gravity is not a force but anyway emulates the effect of a force) we can safely say that no, magnetization will not affect the weight of the object. The degree of electromagnetic coherence thus magnetization of ferromagnetic or paramagnetic matter does not change the mass of an object and therefore its weight inside the gravitational field of the Earth for a given altitude and latitude remains unaffected.

Another more technical explanation of the same phenomenon described above is as a paradigm, arranging random placed dominoes to a coherent aligned structure will not change the mass of the dominoes nor their weight. The work you have done by arranging the dominoes is not stored as added energy into the dominoes but expelled as heat out to the environment. Similar process occurs when magnetizing ferromagnetic matter done by passing electric current through a coil. Part of the EM energy in the coil is converted to kinetic energy for aligning the magnetic moments (i.e. magnetic domains) in the ferromagnetic material therefore heat. After the magnetization process is stopped there is no more energy transfer. No energy is stored.

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