# Electrostatic increase of particle mass

1. Jun 20, 2011

### johne1618

I understand that the electrostatic field around a charged particle contributes to its mass.

It is as though there is a cloud of mass/energy centred on the charged particle that travels with it contributing to its inertia.

I have found that the mass/energy of this cloud can be enhanced by the presence of other fixed charges in the vicinity of the charged particle. By calculating how the position of the center of mass of the mutual electrostatic energy changes as one moves the test particle, I have derived an increase of the mass/inertia of the charged particle.

For example:

Assume that one has a particle of charge q inside a charged (insulating) sphere with a voltage V.

The mass of the particle is increased by (2 / 3) * q * V / c^2.

If the particle is an electron then its mass can be doubled by putting a voltage of about 1000000 volts on the sphere.

The reason this effect has not been noticed before is that the charged sphere must not be a conductor. If it is then there won't be any mutual energy term outside the sphere between the particle's field and the sphere's field as the particle's field won't penetrate the sphere. The mass enhancement effect depends on this mutual energy term.

This change in mass should be measureable if one could get enough charge on an insulator surrounding a test charge. For example one could perform the classic experiment, inside the charged sphere, where one measures the ratio of particle's charge to mass by deflecting the moving particle using a magnet.

John

Last edited: Jun 20, 2011
2. Jun 20, 2011

### Andrew Mason

Why would the field of a charge contribute to mass? Are you suggesting that the field of a single isolated electron contains energy?

AM

3. Jun 22, 2011

### johne1618

Surely that is an accepted effect?

The energy density in an electrostatic field, u, is given by:

u = (1/2) * eps_0 * E^2

where E is the strength of the electric field given by

E = e / 4 pi eps_0 r^2.

The total energy around a spherical particle with charge e and radius R can be found by integrating this energy desnsity over all space around the particle.

Total energy = e^2 / 8 Pi eps_0 R

By the relationship E = m_e c^2 this translates to the mass

m_e = e^2 / 8 Pi eps_0 c^2 R

For an electron one could assume that R is given by the Compton length:

R = hbar / 2 m c

Thus for the electron the ratio of the mass of the electrostatic field to the total mass, m_e / m, is given by:

m_e / m = e^2 / 4 Pi eps_0 hbar c

Thus we find that

m_e / m = fine structure constant = 1 / 137

Interesting huh?

4. Jun 22, 2011

### Bill_K

Not really. For a particle of mass m one can write down three quantities with dimensions of length:

e2/mc2 - the classical electron radius
h/mc - the Compton wavelength
h2/me2 - the Bohr radius

Each may be obtained from the others by multiplying or dividing by the fine structure constant e2/hc. Means nothing. Except it's a handy way of remembering what they are!

5. Jun 22, 2011

### Andrew Mason

I think you will find that this is the energy density in a uniform charge distribution. The energy contained in the field of a single point charge has no physical meaning. There is only energy in relation to other charges.

AM

6. Jun 22, 2011

### harrylin

A point charge is a mathematical abstraction; I think that it has no physical meaning.

In contrast, I have a textbook* which illustrates that for an electron, the rest energy is approximately the energy of its electric field and its kinetic energy is approximately the energy of its magnetic field (but note that this only works for an electron). And of course, the total "electromagnetic mass" corresponds with the sum of the energies.

*Alonso and Finn, Fundamental University Physics, part 2 Electromagnetism.

7. Jun 22, 2011

### Bill_K

This is incorrect. I would have replied to your earlier remark on this if I'd had any idea you were even remotely serious. The energy density in an E field is always (1/8π) E·E, and is just as real regardless of whether it comes from one charge or two charges. Energy is a local quantity, and yes it is meaningful to say where it is located. What you cannot do is attempt to calculate the mass of a charged particle by integrating the E field all the way in to r = 0, since the integral diverges.
This is exactly what you cannot do. The textbook you're quoting is wrong.

8. Jun 22, 2011

### harrylin

It's what they did and it makes perfect sense to me. Why would you think that they could not do what they did? :tongue2:
And do you also claim that the total "electromagnetic mass" does not correspond to the sum of the energies?

9. Jun 22, 2011

### vanhees71

Well, this is a subtle question, that cannot be answered from first principles yet, let alone within classical theory. The electromagnetic self-mass of the electron came up with Lorentz's electron theory in the beginning of the 20th century when Lorentz tried to incorporate the electron as a "point charge" into classical (Maxwellian) electromagnetics. This issue comes up when you ask for the radiation reaction of an accelerated charge: Since the acceleration of charge leads to the creation of electromagnetic waves (Lienard-Wiechert potentials), this must affect the motion of the particle since the em. waves carry energy away from the accelerated charge. However, the energy of the em. field of a point charge (even when taken the charge at rest) diverges. On the other hand, this part of the particle's energy should be already contained in its mass, since it doesn't make sense to talk about the "bare" particle without its electric field since any charge always carries its em. field around it. Lorentz could, within a kind of perturbation theory, for the classical motion treat the self-interaction of the charge with its own em. field by subtracting the infinities by lumping it into the principally unobservable bare mass of the electron and derive an equation of motion for the particle, including the radiation damping to first order. However, this doesn't work at higher orders.

The same ideas came into play about 40-50 years when Tomonoga, Schwinger, and Feynman developed (perturbative) renormalization theory for QED: They and (not to forget!) Dyson conjectured that QED is renormalizable (or more precisely said Dyson renormalizable), i.e., all UV infinities can be removed at any order of perturbation theory by lumping infinities into a finite number of unobservable bare parameters and thus rearranging the perturbation series by expressing it in terms of the corresponding finite observable parameters (wave-function normalization, masses, coupling constants).

10. Jun 22, 2011

### Andrew Mason

It is not so much that the electron is physically dimensionless. Rather it is that it cannot be broken down into sub particles having charge and separated by a distance.

The concept of the self energy of an electron's electric field is itself controversial. Since electrical potential energy requires at least two charges separated by a distance, I don't see how the the field of fundamental charged particle can contain energy.

Also, I am not sure any answer you might get to this question can be tested. If it was physically possible to separate the electron from its electric field, one could determine whether the mass of the electron without its electric field was different than its mass without its field. How would you test your contention that the electric field of an electron contains energy and, therefore, contributes to its inertia?

AM

11. Jun 22, 2011

### harrylin

How would you test any contention that you can not directly test? There are too many of such to mention, for example the hypothesis that the far away stars have electrical charges. Typically what one does in such cases is apply Occam's Rasor and generalise, and then look if it works. For this case and in the field of classical physics (this forum as well as that textbook), that means to simply assume that the laws of Maxwell are valid for all fields.
And if then, as is the case here, the estimated sum of the energies and the corresponding EM inertia roughly correspond to the measured mass, then this is seen as support for that assumption (the "prediction" is confirmed). In other words, classical physics works reasonably well for this aspect of the electron.

Last edited: Jun 22, 2011
12. Jun 22, 2011

### Andrew Mason

There is a difference between drawing a reasonable inference based on evidence and a theory that has no basis in evidence and which, even in theory, cannot be falsified. The latter is beyond the realm of science.

Typically what one does in such cases is apply Occam's Rasor and generalise, and then look if it works. [/quote] Occam's Razor is not science. In fact it is sometimes quite wrong.

AM

13. Jun 22, 2011

### Antiphon

General Relativity has the electromagnetic field as a source of gravity including terms proportional to E^2, so the contribution of the (static) electric field to mass-energy is obvious here.

The field energy in an electrostatic field is well-defined so why would it not be so for the case of a single electron? The source of the field is irrelevant. The energy density is 1/2 epsilon E^2.

14. Jun 22, 2011

### Andrew Mason

That is the energy density of a charge distribution. If you apply the same analysis to the field of an electron, you get a different result - the energy is infinite. This problem was studied by Feynman who concluded that the concept of the field was an unnecessary abstraction. It is a little difficult to see how an unnecessary abstraction it can be a source of inertia, which is quite real. See http://nobelprize.org/nobel_prizes/physics/laureates/1965/feynman-lecture.html" [Broken]

AM

Last edited by a moderator: May 5, 2017
15. Jun 23, 2011

### atyy

There is a mass term in Feynman's equations, but is it the same mass that appears in a defining equation for inertial mass such as Newton's second law?

http://arxiv.org/abs/0905.2391" [Broken] say that their Eq 47 shows that the electromagnetic field contributes to the mass of a body.

Last edited by a moderator: May 5, 2017
16. Jun 23, 2011

### harrylin

Classical physics uses Maxwell's equations which can certainly be falsified; the way science works is that we push theories to the limit and for applications that they fail, we have to come up with something better.

Concerning estimations of the electron's field energy and electromagnetic mass, Alonso&Finn illustrated how classical electromechanics can be applied for v<<c if we assume a certain realistic charge distribution. Classical EM often fails for very high velocities (due to Newton's laws) and Alonso&Finn did not try that in their examples.

17. Jun 23, 2011

### harrylin

That looks like an improvement on the simplistic textbook examples - thanks.
Do you know by chance if that paper has been published (=reviewed)?

Last edited by a moderator: May 5, 2017
18. Jun 23, 2011

### harrylin

As far as we know, the electrostatic field of a charged particle contributes to its mass. I don't think that the field of a capacitor contributes to the mass of an electron in that field; and certainly this is never assumed in accelerator experiments. Wouldn't someone have noticed it if you were right?

Harald

19. Jun 23, 2011

### atyy

http://arxiv.org/abs/0905.2391" [Broken].

I think it's better to use Gralla and Wald's language, since the electrostatic energy is when the particle is non-accelerating, whereas the self-force is present when the particle is accelerating. The sort of language they use seems to be: "the electromagnetic self-energy of the body makes a finite, non-zero contribution to the mass, m, that appears in the Lorentz force equation (p5)", "the electromagnetic self-energy of the body makes a non-zero, finite contribution to the particle’s mass (p13)", "the electromagnetic self-energy contributes to the mass of the body, i.e., there is a finite “mass renormalization” of the body contributed by its electromagnetic field (p20)".

Last edited by a moderator: May 5, 2017
20. Jun 23, 2011

### Andrew Mason

The electrostatic energy contained in a capacitor contributes (very slightly, of course) to the mass of the capacitor. Exactly where that mass is located in the capacitor is not so easy to determine. In principle, it could be determined by measuring the moment of inertia of a charged and uncharged capacitor. But in practice this would be extremely difficult to do I would think.

AM