# Does a Static EM Field Acquire Mass Due to Stored Energy?

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Responding today to the thread in Speculations, it struck me I don't know how to treat the stored energy in a static EM field, according to  E² = (mc²)² + p²c².

Since, unlike the situation with EM radiation, there is no motion involved, I presume the second term does not apply. But does a static field gain rest mass, as its stored energy increases? Seems weird if true.

I've a feeling I'm missing something here. Can anyone help?

Edited by exchemist
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I think it does. Why do you think it's weird?

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1 hour ago, exchemist said:

Responding today to the thread in Speculations, it struck me I don't know how to treat the stored energy in a static EM field, according to  E² = (mc²)² + p²c².

Since, unlike the situation with EM radiation, there is no motion involved, I presume the second term does not apply. But does a static field gain rest mass, as its stored energy increases? Seems weird if true.

I've a feeling I'm missing something here. Can anyone help?

I think first you  would have to explain what a static EM field looks like.

I have heard of static E fields and static H fields separately.

But aren't all EM fields dynamic ?

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4 hours ago, exchemist said:

But does a static field gain rest mass, as its stored energy increases?

Could you further specify precisely which quantity you wish to obtain? Is it the total energy stored in an EM field? If so, then you can work this out using Poynting’s theorem - this should help:

This formalism can be straightforwardly generalised to the relativistic case, where it is written in terms of invariants of the EM energy-momentum tensor - meaning it applies no matter what reference frame you work from.

I don’t think looking at this in terms of mass and momentum is very helpful. But maybe that’s just me

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7 hours ago, Markus Hanke said:

Could you further specify precisely which quantity you wish to obtain? Is it the total energy stored in an EM field? If so, then you can work this out using Poynting’s theorem - this should help:

This formalism can be straightforwardly generalised to the relativistic case, where it is written in terms of invariants of the EM energy-momentum tensor - meaning it applies no matter what reference frame you work from.

I don’t think looking at this in terms of mass and momentum is very helpful. But maybe that’s just me

Quote
Let us now derive this more generally. Consider some distribution of charges and currents.
In small time d
t
a charge will move
v
d
t
and, according to the Lorentz force law, the work
done on the charge will be

which brings us straight back to my comment about static v dynamic.

This needs clearing up before looking at the question of how to increase the stored energy , without motion of something.

And of course there are no relativistic effects when there is no motion.

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This calculation cannot be done classically. You've hit the same wall that a generation of physicists (Abraham, Einstein, Lorentz) and a notorious mathematician (Poincaré) hit a century ago*.

If the electron is point-like and static (static cannot be, we know this from QM), then the Poynting-vector approach (that @Markus Hanke referred to) gives you infinity, as the integral of $$\frac{\varepsilon_{0}}{2}\int_{0}^{\infty}\left\Vert \boldsymbol{E}\left(r\right)\right\Vert ^{2}dV$$ is divergent. So the classical calculation is nonsense.

The possibility that the electron is a little sphere of charge is even worse, as it is impossible to make it relativistically consistent. The pedestrian way of seeing it is that discontinuous charge densities in space-time do not bode well for relativistic invariance. You need fields that are smooth everywhere.

The modern way of dealing with it is using QED (the fully-relativistic, quantum-mechanical version) and attribute part of the energy to self interaction of the electron. We could phrase it as 'the electron tries to move, emits a quantum of radiation, and suffers radiation reaction.' These virtual processes contribute to the energy.

Unfortunately, for all I know, nobody has come up with a way of plugging in the fields (involving the electron's charge), and deriving from there the mass of the electron. The mass of the electron has to be plugged in by hand.

13 hours ago, exchemist said:

I've a feeling I'm missing something here. Can anyone help?

This, I think, is an outstandingly-good question. By no means just a good question.

Edit:

* A century-odd, which was also an odd century.

Edited by joigus
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20 minutes ago, joigus said:

he modern way of dealing with it is using QED (the fully-relativistic, quantum-mechanical version) and attribute part of the energy to self interaction of the electron. We could phrase it as 'the electron tries to move, emits a quantum of radiation, and suffers radiation reaction.' These virtual processes contribute to the energy.

Quite right, here is a lovely chapter from Frank Wilczek  "The Lightness of Being."

You will need to get the book for the previous chapter containing the calculations.

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Now that I think about it, @exchemist wasn't necessarily talking about the electron...

For some reason, I was thinking about the electron.

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1 minute ago, joigus said:

Now that I think about it, @exchemist wasn't necessarily talking about the electron...

For some reason, I was thinking about the electron.

Yes, I wondered where the subject has changed from 'field' to 'electron'. And with no input from the OPer.

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47 minutes ago, studiot said:

Quite right, here is a lovely chapter from Frank Wilczek  "The Lightness of Being."

You will need to get the book for the previous chapter containing the calculations.

Thanks for confirming this.

In the case that a classical calculation were valid, you would have to add the EM contribution to the mass as $$\triangle m_{\textrm{EM}}=\frac{\varepsilon_{0}}{2c^{2}}\int_{0}^{\infty}\left(\left\Vert \boldsymbol{E}\right\Vert ^{2}+\left\Vert \boldsymbol{B}\right\Vert ^{2}\right)dV$$

Yes, I wondered where the subject has changed from 'field' to 'electron'. And with no input from the OPer.

Well, it's because it's really when you go down to something as elementary as an electron that the question becomes really intractable classically.

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Interesting... I was under impression that we generally believe that static field indeed gains mass due to stored energy - and this mass is distributed exactly as you would expect from the energy distribution. Are there any theoretical reasons to doubt this?

I do know that we don't have any experimental confirmation of this (or I couldn't find any).

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14 minutes ago, Danijel Gorupec said:

I do know that we don't have any experimental confirmation of this (or I couldn't find any).

Somebody would think it's a weak confirmation, but here's one: No charged particle with zero rest energy (relativistic mass) has been found. So the EM field has inertia. Always.

To which I will add a prediction: No charged particle with zero rest energy (relativistic mass) will ever be found.

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19 minutes ago, joigus said:

Somebody would think it's a weak confirmation, but here's one: No charged particle with zero rest energy (relativistic mass) has been found. So the EM field has inertia. Always.

To which I will add a prediction: No charged particle with zero rest energy (relativistic mass) will ever be found.

It would work for me. But, how it reconciles with the SM where particles get their mass from interaction with the Higgs field?

Edit: Retracted.

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20 minutes ago, joigus said:

Somebody would think it's a weak confirmation, but here's one: No charged particle with zero rest energy (relativistic mass) has been found. So the EM field has inertia. Always.

To which I will add a prediction: No charged particle with zero rest energy (relativistic mass) will ever be found.

I was looking up Earnshaw's original paper.

The contents list of the 7th volume of the Proceedings of the Cambridge Philosophical Society is quite stunning.

Earnshaw (Paper 5 page 97) is in the company of Green, Stokes, De Morgan, Airy, Challis and O'Brien to list but a few and the subjects covered as well as the presentations could be offered today.

And yes any force field has a source, which itself has inertia.

So much can indeed be done classically, but the Lagrangian calculus of variations refers to a conservative field using continuous linear mathematics so obviously has limitations in a quantum sense.

However I am waiting for exchemist to spell out exactly what he wants to achieve.

Edited by studiot
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It would work for me. But, how it reconciles with the SM where particles get their mass from interaction with the Higgs field?

Edit: Retracted.

OK. I don't know whether you retracted from your question or from a previous point. But keep in mind the Higgs mechanism is an ad hoc mechanism. Brilliantly insightful, to be sure, but ad hoc nonetheless. IOW: We don't really understand where mass comes from. It's nice to have a multiplet of particles that gives mass to everything else. The Higgs floats around while the Goldstone bosons from the multiplet (not the Higgs, this is not faithfully reflected in the literature) provide mass to all fermions and short-range bosons. But where do the completely disparate mass spectrum comes from? I don't think we've developed a picture in the way that the OP seems to suggest, that mass differences could be explained by means of field self-interaction, and self-interaction alone.

21 minutes ago, studiot said:

However I am waiting for exchemist to spell out exactly what he wants to achieve.

That's a fair point.

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5 minutes ago, joigus said:

OK. I don't know whether you retracted from your question or from a previous point. But keep in mind the Higgs mechanism is an ad hoc mechanism. Brilliantly insightful, to be sure, but ad hoc nonetheless. IOW: We don't really understand where mass comes from. It's nice to have a multiplet of particles that gives mass to everything else. The Higgs floats around while the Goldstone bosons from the multiplet (not the Higgs, this is not faithfully reflected in the literature) provide mass to all fermions and short-range bosons. But where do the completely disparate mass spectrum comes from? I don't think we've developed a picture in the way that the OP seems to suggest, that mass differences could be explained by means of field self-interaction, and self-interaction alone.

(I retracted an erroneous edit only.) It is correct that all these masses have to be put in by hand. Finding the Higgs boson supports the mechanism, nevertheless. (I don't think the OP had any of this in mind. It asked about a mass of the field.)

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(I retracted an erroneous edit only.) It is correct that all these masses have to be put in by hand. Finding the Higgs boson supports the mechanism, nevertheless. (I don't think the OP had any of this in mind. It asked about a mass of the field.)

I didn't mention the Higgs. I didn't mean to mention the Higgs. I don't think the Higgs has any bearing on OP. Why the Higgs came up at all is a mystery to me.   EM weighs, that's all I meant to say. And that, I said.

Edited by joigus
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One thing that gives me pause is the phrasing "acquire mass"

If you have a charge, the field is already there. Nothing is "acquired"

If you are creating a field by rearranging a charge configuration that has no field into one that does, you are doing work. In that sense mass is "acquired" because you are adding energy, and thus mass, which is stored in that new configuration.

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58 minutes ago, joigus said:

Somebody would think it's a weak confirmation, but here's one: No charged particle with zero rest energy (relativistic mass) has been found. So the EM field has inertia. Always.

To which I will add a prediction: No charged particle with zero rest energy (relativistic mass) will ever be found.

It is something. Although it does not show that the distribution of mass/energy is as expected from the energy density formula.

I have personal interest in confirming the density formula, so I searched quite a lot. All measurement proposals seems out of our reach (for many years to come).

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32 minutes ago, swansont said:

One thing that gives me pause is the phrasing "acquire mass"

If you have a charge, the field is already there. Nothing is "acquired"

If you are creating a field by rearranging a charge configuration that has no field into one that does, you are doing work. In that sense mass is "acquired" because you are adding energy, and thus mass, which is stored in that new configuration.

Agreed. Once you have any gauge field, you have mass. It's a package deal. It's a contribution to total mass.

The Higgs is different, I think. But that's another topic.

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15 hours ago, studiot said:

This needs clearing up before looking at the question of how to increase the stored energy , without motion of something.

Well, there simply won’t be a B field if the situation is a static one; the energy then is just the integral given by joigus. You can’t increase this without adding more electric charge, which in the real world implies moving charges into the spatial region in question (and thus the temporary existence of B).

15 hours ago, studiot said:

And of course there are no relativistic effects when there is no motion.

Sure. Nonetheless, it is often helpful to go to the full covariant formalism simply to illuminate the underlying physics. In this case, the point is that splitting the EM field into E and B fields is an arbitrary (mostly historical) choice; in reality though there is just one electromagnetic field that permeates all of spacetime, and the energy stored in it, as captured by the energy-momentum tensor, does not in any way depend on which observer defines it. Thus, whether the source distribution is static or not relative to any given observer is irrelevant for the underlying physics. The only way to increase total field energy is to move extra charges into it, ie increase its source density - which requires work to be done. Relative motion alone doesn’t qualify.

Quote

Does a Static EM Field Acquire Mass Due to Stored Energy?

I think while it may be formally possible to attribute some notion of ‘mass’ to an EM field (as an equivalent to its total energy), in practice this would be a fairly useless quantity, since it would be a global property of the entire field - which stretches into infinity. No observer could ever measure this mass. This is why in relativistic EM theory (the OP did reference the energy-momentum relation) we use the energy-momentum tensor of the EM field instead - which is a local quantity.

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10 hours ago, Markus Hanke said:

Well, there simply won’t be a B field if the situation is a static one; the energy then is just the integral given by joigus. You can’t increase this without adding more electric charge, which in the real world implies moving charges into the spatial region in question (and thus the temporary existence of B).

Sure. Nonetheless, it is often helpful to go to the full covariant formalism simply to illuminate the underlying physics. In this case, the point is that splitting the EM field into E and B fields is an arbitrary (mostly historical) choice; in reality though there is just one electromagnetic field that permeates all of spacetime, and the energy stored in it, as captured by the energy-momentum tensor, does not in any way depend on which observer defines it. Thus, whether the source distribution is static or not relative to any given observer is irrelevant for the underlying physics. The only way to increase total field energy is to move extra charges into it, ie increase its source density - which requires work to be done. Relative motion alone doesn’t qualify.

I think while it may be formally possible to attribute some notion of ‘mass’ to an EM field (as an equivalent to its total energy), in practice this would be a fairly useless quantity, since it would be a global property of the entire field - which stretches into infinity. No observer could ever measure this mass. This is why in relativistic EM theory (the OP did reference the energy-momentum relation) we use the energy-momentum tensor of the EM field instead - which is a local quantity.

Since we have not yet heard from exchemist, I hope he is OK.

However there is a clue in the name.
I suspect he is more interested in the field of the nucleus than the fields of astrophysics at the scale of the universe.
Particularly as he specified 'static' fields.

If the universe is actually infinite then the global mass integral must also be infinite so adding mass in any way does not change that result.

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23 hours ago, joigus said:

Now that I think about it, @exchemist wasn't necessarily talking about the electron...

For some reason, I was thinking about the electron.

No indeed. My question was far more basic, simply whether a static electric or magnetic field has mass as a consequence of its stored energy.

I realise now that my question was at one level a bit stupid, since if a battery gains mass when charged (albeit to an unmeasurably small degree ), it means the energy in the chemical bonding goes up and gains mass - and the energy of that bonding is a sum of the electrostatic potential and kinetic energy of the electrons. So it seems to me now that a static field must indeed have an associated mass, even though this feels unintuitive when one thinks of the magnetic field of a solenoid for example.

Also, from the other (very interesting) replies, it dawns on me that I should not find this idea of fields having mass unintuitive, since (as I understand it, very vaguely) fundamental particles are considered in QED to be excitations of a field.  I probably need to let go of this rather 6th form idea of mass applying to things called "particles" of matter, as distinct from insubstantial things called "fields".

17 minutes ago, studiot said:

Since we have not yet heard from exchemist, I hope he is OK.

However there is a clue in the name.
I suspect he is more interested in the field of the nucleus than the fields of astrophysics at the scale of the universe.
Particularly as he specified 'static' fields.

If the universe is actually infinite then the global mass integral must also be infinite so adding mass in any way does not change that result.

You are psychic! I was just replying. (I was out most of yesterday.)

There is some excellent food for thought in this thread and I'm glad I left it long enough for those replies to come in before responding.

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24 minutes ago, exchemist said:

I realise now that my question was at one level a bit stupid, since if a battery gains mass when charged (albeit to an unmeasurably small degree ), it means the energy in the chemical bonding goes up and gains mass - and the energy of that bonding is a sum of the electrostatic potential and kinetic energy of the electrons. So it seems to me now that a static field must indeed have an associated mass, even though this feels unintuitive when one thinks of the magnetic field of a solenoid for example.

If you want a classical answer you could consider the decrease in mass due to changing a silicon atom for a boron one in a silicon lattice, since this would decrease the charge by 1,thus reducing the field slightly. Since we are then talking about a solid lattice, momentum would not be involved the simple e = mc2 would suffice.

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3 hours ago, exchemist said:

I realise now that my question was at one level a bit stupid

On the contrary, it was an interesting question which generated quite a few interesting answers.

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