# Energy density in an E field: Does it contribute to inertial mass?

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Will Learn
Hi.

I'm not sure where to put this question, it concerns particles, mass-energy equivalence and various things. Classical electromagnetism seems to be as sensible a place as any.

There is energy stored in an E field.
Energy density (at position r, time t) = $$\frac{1}{2} {\epsilon} |E(\vec{r},t)|^2$$

(Minor note: I can't find any guides on how this forum supports LaTex, I hope the above prints. If there is a guide and anyone can give a link to the guide that would be a help).

So, a charged particle should create a E field throughout all of space. Does that energy density contribute to the mass of the particle? To say this another way, is some of the inertial mass of the particle non-local to the particle?
For example, if you changed the permittivity of space somewhere, which you might do by flooding a region with some dielectectric material, can you affect the resistance that the particle would show to an applied force (i.e. change it's inertial mass) even though you have been changing something that is some distance away from the particle?

Anyway, the basic question is: Does the energy stored in the E field contribute to the inertial mass of a charged particle? Moving the particle clearly forces the E field to change throughout space. So that if the particle is accelerated, then some energy located in space (which would seem to be equivalent to a mass) is also accelerated. That mass/energy in the E field seems to be dragged around with the particle, so you might expect the increased inertia to be noticed or evident at the point of contact by an applied force on the particle.

I don't know. I can't get to any place with Physicists at the moment, so I'd be grateful for any answers from the forum. I've got a copy of Electrodynamics by Griffiths at home and of course, the internet, if anyone wants to suggest a reference.

Thanks for your time.

## Answers and Replies

Mentor
I'm not sure where to put this question
Moderator's note: Thread has been moved to the relativity forum.

Mentor
I can't find any guides on how this forum supports LaTex
Look for the "LaTeX Guide" link at the bottom left of each post window.

Will Learn
Will Learn
Hi again,

Would it help if we restrict attention to a solenoid or capacitor? A capacitor is the obvious choice since that uses an E Field. For an ideal capacitor, the E filed is confined to the region of space between the plates. At least here, the E field doesn't extend all over the universe but is confined to a region you can consider as being part of the macroscopic object that is the charged capacitor.

Does the Capacitor then have more mass when it is charged rather than in a discharged state? The usual amount of energy $$\frac {1}{2} C V^2$$ does seem to be "in" the capacitor. Has anyone tried pushing a charged capacitor with a fixed force and seeing if its inertial mass is different while charged and discharged?

Since this has now been located in the SR and GR section, I can reasonably suggest that the equivalent of "gravitational mass" for the macroscopic object (the capacitor) has changed. The energy density you would assign to this location for GR does seem to have changed - because the electromagnetic stress-energy tensor should contribute to the overall stress-energy tensor. So the curvature of space around the macroscopic object should be different and exactly like the object has just been assigned more mass.

Thanks and sorry for writing on my own thread, just keen to get some opinions.

Mentor
Does the Capacitor then have more mass when it is charged rather than in a discharged state?
Yes, because it contains more energy and the additional energy it contains has inertia.

I can reasonably suggest that the equivaent of "gravitational mass" for the macroscopic object (the capacitor) has changed.
Yes, the stress-energy tensor of the capacitor will also have changed (since the square of the EM field appears directly in the stress-energy tensor and the EM field has changed).

vanhees71 and Will Learn
Mentor
Has anyone tried pushing a charged capacitor with a fixed force and seeing if its inertial mass is different while charged and discharged?
I don't think so, and I don't think our current measurement technology would be accurate enough to detect the difference. The difference is very small in ordinary terms.

Will Learn
Will Learn
Hi.

I don't think our current measurement technology would be accurate enough
Yes. I'm sure that's true, if a shame. I'd still be interested in the theory, even if the experiment isn't available yet.

Best Wishes.

Will Learn
Hi.
Thanks for your time and advice @anorlunda .
I've only scanned through those pages at the moment. It seems very general but this section seems most relevant:

Einsteins 1905 derivation:
... This was tackled by Einstein in his paper "Does the inertia of a body depend upon its energy content?"...
...Einstein concluded that the emission reduces the body's mass by E/c2, and that the mass of a body is a measure of its energy content...

That seems to be in agreement with what we might reasonably do for the mass of a charged capacitor, where the energy is fairly easily seen to be contained within the object. However, it's less clear how you handle a point charge where the Energy in the E field is spread out over all of space and not contained in any finite region you might describe as the location of the charged particle.

It's possible to re-phrase the original question, if that helps: Is the energy contained in the E field created by a charged particle considered to be energy "in" the particle, does it influence the inertial mass of the particle?

Thanks again.

Staff Emeritus
https://en.wikipedia.org/wiki/Shell_theorem
Is the energy contained in the E field created by a charged particle considered to be energy "in" the particle, does it influence the inertial mass of the particle?
Yes it is part of the invariant mass, and it gravitates. In the same way that a bottle of warm air has more invariant mass than the same bottle with cold air. But the air is not contained in a point.

My guess is that you could apply Newton's Shell Theorem to the gravitational attraction of the energy of an E field partially contained within a shell to objects outside that shell.

https://en.wikipedia.org/wiki/Shell_theorem

However, direct measurements of such tiny mass differences are probably not possible.

Will Learn
Mentor
Yes. I'm sure that's true, if a shame. I'd still be interested in the theory, even if the experiment isn't available yet.
You will likely find it interesting to calculate about how sensitive of a measurement we're talking about. Start with a reasonable estimate of the mass of a reasonable-sized capacitor, say one microfarad. Figure how much energy is required to charge it to 12 volts (easy to find the relationships between charge, voltage, energy, and capacitance online), then use ##E=mc^2## to compute the mass increase from charging the capacitor as a fraction of the mass of the capacitor.

I don't think anyone has done that, but somewhere around we have an old thread in which we calculated the mass increase from charging a full-sized lead-acid automobile battery (tens of kilograms). If I recall rightly, charging the battery increased its mass by only a few nanograms.

Will Learn
jartsa
It's possible to re-phrase the original question, if that helps: Is the energy contained in the E field created by a charged particle considered to be energy "in" the particle, does it influence the inertial mass of the particle?
Energy is in the field. Particle with highly energetic field has large inertia.

Will Learn
Will Learn
Hi and thanks for everyone's time.
The least I can do is try to reply to all of the above comments.

Yes it is part of the invariant mass
That's a nice clear answer. I would very much like it to be this way - but the consequences are significant and I hope you'll understand that I have to slow down and consider it carefully.

If it is that simple then most of the things mentioned in post #1 apply. For example, some of the mass of a charged particle is non-local to the particle, it can be changed by adjusting something that is a distance away.

My guess is that you could apply Newton's Shell Theorem to the gravitational attraction of the energy of an E field partially contained within a shell to objects outside that shell.
Yes, seems reasonable. Using the idea that inertial mass = gravitational mass, this suggests the mass grows as you consider a larger sphere around the particle. So there is no finite spatial region where you can draw a surface and say - that's it, that is "the body". The only system that contains all the energy is the entire universe. We can't disconnect the E field created by the particle at the boundary of some sphere and only move that around, even if we wanted to.

You are effectively saying that there is no convenient finite body that can be identified and would have the mass of the particle + the energy in the E field it created. That may be true.

Calculating the gravitational effect is interesting but it's a bit further downstream from the issue in the first post which is the inertia or resistance to a force applied "at" or "on" what we would normally describe as the particle. Classically, there is something we usually describe as "the particle" - even if we might be accidentally assigning inertial mass to more than just the particle (i.e. even if its E field unavoidably comes along with it).

Skipping ahead to gravity, we don't have to approximate the effect with the shell theorem. Fortunately GR doesn't care what caused the E field to be there, we can just use the fact that it is there and count that as energy density. The only thing I would like to avoid is accidentally counting the contribution from the E field twice over, once in the matter component of the stress-energy tensor (from the particle mass) and again in the e-m component of the tensor.

You will likely find it interesting to calculate about how sensitive of a measurement we're talking about... ...easy to find the relationships between charge, voltage, energy, and capacitance online

Most of this was already in post #4. That doesn't matter, you can't read everything and I'm grateful for your time. The difference certainly is small, it's just that it should be there and that matters. The experimentalists probably do want something they can measure.

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@jartsa , that was an interesting and relevant thread, thank you.
The capacitor and its E field was well discussed there. There seems to have been a clear answer for this: Yes, the energy stored in the E field is included in the inertial mass of the system you would call the (charged) capacitor.

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I'm not going to discuss anything else in this post, it's already too long. I'll just thank everyone again for their time.

Best Wishes to everyone.

Staff Emeritus
Anyway, the basic question is: Does the energy stored in the E field contribute to the inertial mass of a charged particle?
You might want to do some reading and/or research on the idea of "electromagnetic mass" from classical pre-relativity physics.

See for instance https://en.wikipedia.org/wiki/Electromagnetic_mass

wiki said:
Electromagnetic mass was initially a concept of classical mechanics, denoting as to how much the electromagnetic field, or the self-energy, is contributing to the mass of charged particles. It was first derived by J. J. Thomson in 1881 and was for some time also considered as a dynamical explanation of inertial mass per se. Today, the relation of mass, momentum, velocity, and all forms of energy – including electromagnetic energy – is analyzed on the basis of Albert Einstein's special relativity and mass–energy equivalence.

The short version, I would say, is that historically the idea of electromagnetic mass came up first, even before special relativity. I'm not very familiar with the pre-relativity work though.

Max Jammer's book, "Concepts of Mass in Classical and Modern Physics", https://www.amazon.com/dp/0486299988/?tag=pfamazon01-20, was where I first read about the history of this concept. But I don't remember what I read well enough to discuss it intelligently.

I'd add that I'm more interested in the modern concepts than the pre-relativity ones. The historical concepts can be interesting, but I think the modern ideas are more important. An oversimplification of my personal view is that mass has largely been replaced by "rest mass" aka "covariant mass" for point particles in special relativity, and for non-point particles in a continuum, the relevant concept is the rank 2 stress-energy tensor. The key concept her is covariance, in my opinion.

Will Learn
Will Learn
Hi.

You might want to do some reading and/or research on the idea of "electromagnetic mass" from classical pre-relativity physics.
Thanks. I have now read that Wiki article.

I had just recently found some information about the classical Compton radius for an electron and some of the older work done by Thomson on the subject.

Ultimately, most sources of information I've seen so far seem to end their discussion of the topic as you might have expected: The E field is only a classical approximation, it's not (part of) a quantum theory. An electron must be assigned a non-zero radius for the classical E-field energy calculations to yield finite values. Overall, the usefullness of determining energy in the E field has its limits.

Best Wishes.

Will Learn
Thanks. Sorry it's taken me a while to reply. I'm not going to be following this thread as much and may not reply to people quickly from now onwards.