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Galaxies have associated magnetic fields. So does the Sun and so does the Earth. In the case of the Earth and Sun it seems clear that the energy content of the magnetic fields is much less than their mass.

Can the same be said of the Milky Way or other galaxies?

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    $\begingroup$ I'm glad my answer helped you, but I'd recommend unaccepting it for now - I don't want to discourage other, possibly better, answers. $\endgroup$
    – HDE 226868
    Jun 27 '21 at 17:53
  • $\begingroup$ @HDE226868 Done. One more thing. Would the magnetic field of the black hole in the center substantially change the outcome? Or are fields always less in content than the masses producing them? $\endgroup$ Jun 27 '21 at 18:12
  • $\begingroup$ No, taking Sgr A* into account wouldn't make a difference. $\endgroup$
    – HDE 226868
    Jun 27 '21 at 21:20
  • $\begingroup$ @DescheleSchilder Since there was no answer past 1 month, I don't think there will be any new answer in the near future. So, if you want you can accept HDE's answer. $\endgroup$ Aug 2 '21 at 3:02
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Yes, the energy of the Galactic magnetic field is substantially smaller than the mass-energy of the Milky Way. The total energy of a magnetic field $\mathbf{B}$ in a volume $\mathcal{V}$ is $$E=\frac{1}{2\mu_0}\int_{\mathcal{V}}|\mathbf{B}|^2\mathrm{d}V$$ You could make a very rough estimate of the magnetic energy content of the Milky Way's field by taking any average magnetic field strength and a rudimentary measure of the volume of the galaxy. The field strength varies, with a peak towards the center and other fluctuations based on local structures and spiral arms, but the average is in the range of a few $\mu$G. It doesn't really make sense to speak of the volume of a galaxy because boundaries aren't well-demarcated, but we could say that much of it lies in within a sphere 50,000-100,000 light-years from the center. This rough estimate gives me an energy corresponding to about $500M_{\odot}$, which is 9 or 10 orders of magnitude below the total mass of the Milky Way. You could rightly quibble about factors of a few or even orders of magnitude in a couple places, but we still wouldn't get any higher than a fairly insignificant figure.

Dense spiral arms may have field strengths of $\simeq15\;\mu\text{G}$, and starburst regions could reach as high as $50\;\mu\text{G}$ (Beck 2003) in other spiral galaxies. Sure, the $E\sim|\mathbf{B}|^2$ relation would gain you a couple orders of magnitude, as would larger galaxies, but it won't be anywhere near comparable to the mass of luminous baryonic and dark matter. You could also sidestep some of the issues in volume estimation by comparing the magnetic energy density to the total energy density of a galaxy; I'm sure the result will be basically the same.

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