Electrostatic increase of particle mass

[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

 

[Andrew Mason]

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

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

AM

 

[johne1618]

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

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?

 

[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!

 

[Andrew Mason]

Surely that is an accepted effect?

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

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

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

 

[harrylin]

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

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.

 

[Bill_K]

The energy contained in the field of a single point charge has no physical meaning. There is only energy in relation to other charges.

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 into r = 0, since the integral diverges.

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)

This is exactly what you cannot do. The textbook you're quoting is wrong.

 

[harrylin]

This [calculating that the rest energy is approximately the energy of the electron's electric field and its kinetic energy the energy of its magnetic field] is exactly what you cannot do. The textbook you're quoting is wrong.

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?

 

[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).

 

[Andrew Mason]

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.

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

 

[harrylin]

[..] 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

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.

 

[Andrew Mason]

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.

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

 

[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.

 

[Andrew Mason]

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.

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"

AM

 

[atyy]

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"

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" say that their Eq 47 shows that the electromagnetic field contributes to the mass of a body.

 

[harrylin]

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. [...]

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.

 

[harrylin]

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

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

 

[harrylin]

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

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

 

[atyy]

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

http://arxiv.org/abs/0905.2391" .

As far as we know, the electrostatic field of a charged particle contributes to its mass.

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)".

 

[Andrew Mason]

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

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

 

[harrylin]

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

Right - the energy of the capacitor contributes to its mass. However, the OP claims that the electrostatic energy contained in a capacitor contributes to the mass of an electron that passes between the capacitor plates.

 

[Andrew Mason]

Right - the energy of the capacitor contributes to its mass. However, the OP claims that the electrostatic energy contained in a capacitor contributes to the mass of an electron that passes between the capacitor plates.

It is easier to see this mass in a proton, except that in the proton the source of inertia is predominantly the nuclear force/energy with only a small amount due to the coulomb force/energy.

The three quarks that comprise the proton have individual masses whose sum is much less than the mass of the proton. The inertia difference between a proton and the separate constituent quarks is from the energy due to the forces holding the quarks together.

Now, it seems that the mass difference is somewhere inside the proton. Exactly where inside (in relation to the position of the quarks) I don't think anyone is able to say. I am not sure that principles of quantum mechanics allows one to know that.

But my point is that the proton is very different than the electron. The electron does not appear to be held together by forces.

The field of an electron is a mathematical abstraction that we find useful in calculating the forces between the electron and other charged particles. I don't see why it is necessary to conclude that the field of a single electron (as opposed to a field between two charged particles) should contain energy. I don't think it does.

AM

 

[Antiphon]

The field of an electron is a mathematical abstraction that we find useful in calculating the forces between the electron and other charged particles. I don't see why it is necessary to conclude that the field of a single electron (as opposed to a field between two charged particles) should contain energy. I don't think it does.

AM

That there is energy there (and how much) can be seen in the following experiment.

Bring an electron and positron together slowly from infinity. When they annihilate the rest masses get converted to energy. But on the way together a certain amount of energy can be extracted from the field. This energy is twice the field energy of the electron alone because you began with the isolated field of the two particles and ended with zero field as the charges almost coincided.

 

[Andrew Mason]

That there is energy there (and how much) can be seen in the following experiment.

Bring an electron and positron together slowly from infinity. When they annihilate the rest masses get converted to energy. But on the way together a certain amount of energy can be extracted from the field. This energy is twice the field energy of the electron alone because you began with the isolated field of the two particles and ended with zero field as the charges almost coincided.

A positron has potential energy relative to the distant electron. That potential energy is converted into other forms of energy, such as kinetic energy, as the particles get closer together (thereby reducing their potential energy). How does this prove that the field of a single electron or positron contains energy?

AM

 

[Antiphon]

A positron has potential energy relative to the distant electron. That potential energy is converted into other forms of energy, such as kinetic energy, as the particles get closer together (thereby reducing their potential energy). How does this prove that the field of a single electron or positron contains energy?

AM

In my experiment they are lowered slowly together, not freely accelerating toward one another. This way none of their energy becomes kinetic, it is all transferred to the agent lowering the charges in.

If this doesn't seem plausible, you can do the same thing with a full coulomb and ropes. As you slowly bring the charges together they do work on the ropes.

The only difference between the widely separated charge configuration and the close-in configuration is that 1) work was extracted and 2) the fields got smaller. Therefore the energy came from the electric field.

If you want to say the energy in the field is only there with two charges that are arbitrarily far apart but not with either one, I'd need to hear your thinking on why.

 

[Andrew Mason]

In my experiment they are lowered slowly together, not freely accelerating toward one another. This way none of their energy becomes kinetic, it is all transferred to the agent lowering the charges in.

If this doesn't seem plausible, you can do the same thing with a full coulomb and ropes. As you slowly bring the charges together they do work on the ropes.

The only difference between the widely separated charge configuration and the close-in configuration is that 1) work was extracted and 2) the fields got smaller. Therefore the energy came from the electric field.

Did it not come from the change in potential energy? If an object is lowered to the earth, does the work extracted come from the gravitational field of one of the Earth or of the object? Or does it come from the change in position and, hence, the change in potential energy?

If you want to say the energy in the field is only there with two charges that are arbitrarily far apart but not with either one, I'd need to hear your thinking on why.

Because potential energy, U = -kQq/r. There is no potential energy if you have only one charge.

AM

 

[Antiphon]

Did it not come from the change in potential energy? If an object is lowered to the earth, does the work extracted come from the gravitational field of one of the Earth or of the object? Or does it come from the change in position and, hence, the change in potential energy?

In the gravitational case the potential energy goes into/comes from the gravitational field. The net gravity of the two bodies is altered as well by the change.

Because potential energy, U = -kQq/r. There is no potential energy if you have only one charge.

The little-q charge is the mechanical locus of the energy extraction. But the energy comes from (or goes into) the field itself.

In relation to the original post the mass of an electron does not change at all in any electric field. The field itself acquires an inertia (and gravitates as well).

A charged capacitor is heavier and the center of mass of the added mass is in the space between the plates, not in the charges on either plate.

 

[Drakkith]

In my experiment they are lowered slowly together, not freely accelerating toward one another. This way none of their energy becomes kinetic, it is all transferred to the agent lowering the charges in.

If this doesn't seem plausible, you can do the same thing with a full coulomb and ropes. As you slowly bring the charges together they do work on the ropes.

I don't see how this applies to a single field. There is nothing for it to interact with, nothing that it can influence. By the definition of energy, a single field cannot have it, right?
Or is this whole point moot since in reality you can never have a 100% isolated field?

 

[Antiphon]

Then radiation by a single charge is either impossible or carries no energy?

 

[Drakkith]

Then radiation by a single charge is either impossible or carries no energy?

Charges aren't particles. They are a property of particles. Particles can radiate if the conditions are right. Saying that, could you elaborate and give a specific example? As your question doesn't really make sense right now. (IE what kind of radiation, what is producing it, ETC)

 

[Andrew Mason]

Then radiation by a single charge is either impossible or carries no energy?

This is a very good question. I would begin my answer with another question: does a charge ever radiate by itself? A similar related question would be: can a charge accelerate by itself?

I think the answer to both of my questions is that it cannot. It can accelerate only by interacting with another particle. Whether the acceleration is the result of the interaction that causes radiation or the cause of the radiation is another good question. Remarkably, this is still an unresolved question in science.

So in answer to your question, I would say that a single electron cannot produce radiation by itself. An electron can produce radiation only by interacting with other particles (eg. changing energy levels in an atom), in which case the energy comes from this interaction.

As a follow-up: if an electron could radiate by itself using the energy contained in its field, would the strength of the field (ie. the magnitude of the charge) not have to decrease?

AM

 

[Andrew Mason]

In the gravitational case the potential energy goes into/comes from the gravitational field. The net gravity of the two bodies is altered as well by the change.

I am not aware of any experimental evidence that the potential energy comes from or goes into the gravitational field. Are you?

In relation to the original post the mass of an electron does not change at all in any electric field. The field itself acquires an inertia (and gravitates as well).

That is not the question. The question is whether there is energy, hence inertia, in the field of a single electron.

If an electron is placed in an electric field, it has energy by virtue of its position. That means the mass of the electron and the charge/charges that create the field together are greater than their masses individually. But that does not mean that there is energy in the e-field of a single electron.

A charged capacitor is heavier and the center of mass of the added mass is in the space between the plates, not in the charges on either plate.

I am not aware of any experimental evidence that establishes where the added mass is physically located. Can you point me to such evidence?

AM

 

[Drakkith]

I think this is a result of a misunderstanding of what energy is.

 

[kof9595995]

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. .

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.

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"

AM

I think these are quite relevant to an old question I asked :https://www.physicsforums.com/showthread.php?t=397520
Now I tend to think in the framework of EM itself it is indeed impossible to test whether energy is localized by field, but perhaps general relativity provides us a way to test it?

 

[Antiphon]

If you assume that field energy is localized, it makes sense to assume field energy is localized with some functional relation to the field intensity. There's no experimental proof for this but GR says it should be so.

I went over the derivation of electrostatic potential energy E=1/2 epsilon E^2 from first principles beginning with two, then three then N charges.

You get a nested summation where indeed you don't count any of the self-terms. This means the field of a single electron is not part of the field energy calculation.

One is of course left wondering now what is to be concluded about the validity of taking the summation over to an integral of continuous charge density. It would seem there is a defect in the assumption of charges being point objects. These infinite self energies are rendered finite if we assume the electron is not a point particle. But I'm afraid we're lost in the woods beyond here. Or at least I am.