[Physics] Does an iron bar become heavier when magnetized

Question

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.

Answer 1

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:

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.