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Say I have a piece of iron bar. Would the same iron bar weigh more when being magnetized, (assuming no outside forces like magnetic field, etc,) and the scale is non-magnetic? I think you know what I'm saying.

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  • $\begingroup$ A static magnetic field does have mass, just like any other form of mass-energy. $\endgroup$
    – CuriousOne
    Dec 27, 2015 at 4:00

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Yes, a magnetic field will contribute to the mass of an object because it has energy and in special relativity mass is the length of the fourvector:

mass

The static field has zero momentum , but it has energy, therefore from the above formula it has an invariant mass. The densest magnetic fields are of a few tesla. I copy the answer to a similar question which gives an estimate:

Let's say you have a lab magnet that fills 1000 cm3 of space with a very high field of 10^5 G. That field has an energy of (roughly) 10^12 ergs. That corresponds to a mass of 10^-9 grams. So for most purposes we just forget about it.

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  • $\begingroup$ So is mg field count as relativistic mass or rest mass? $\endgroup$
    – user6760
    Dec 27, 2015 at 7:25
  • $\begingroup$ If you are adding linearly the masses of all the atoms, it is a good approximation as the motion of the atoms is too low to make a difference to the relativistic mass. As the estimate shows the energy of a magnetic field has a mass density very small with respect to the atomic masses. So even though one should add all the energy and all the momentum vectors to get the invariant mass, which will be the correct mass for a system of atoms+fields, the difference will be infinitessimally small, that it is not worth the effort. One just adds the classical atomic masses to get the mass of an object. $\endgroup$
    – anna v
    Dec 27, 2015 at 7:45
  • $\begingroup$ @anna v, when you say "invariant mass", do you mean that it's an invariant quantity for an observer in any reference frame? $\endgroup$ Dec 27, 2015 at 15:54
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    $\begingroup$ @no_choice99 yes, it is invariant to Lorenz transformations. $\endgroup$
    – anna v
    Dec 27, 2015 at 17:28
  • $\begingroup$ An iron magnet is just a normal piece of iron with many of its atoms magnetic moment aligned. When not magnetized, each atoms magnetic moment points into random directions, cancelling each other, therefore, magnetizing an iron bar do not add mass to the resulting magnet. The energy and mass of the magnetic field was already present even when not magnetized. $\endgroup$
    – J. Manuel
    Nov 24, 2017 at 7:23

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