A paradox in electromagnetic theory

[lugita15]

A "paradox" in electromagnetic theory

In Volume II Chapter 28 of his Lectures on Physics, Feynman describes a fundamental inconsistency in classical electromagnetic theory, concerning electromagnetic mass:
Consider an electron in which all of the charge q is uniformly distributed on the surface of a sphere of radius a. The magnitude E of the electric field at a distance r from the center of the electron is E=\frac{q}{4\pi\epsilon_{0}r^{2}}and the density u of its field energy is given by
u=\frac{\epsilon_{0}E^{2}}{2}=\frac{q^{2}}{32\pi^{2}\epsilon_{0}r^{4}}
In order to find the total energy in the electric field produced by the electron, we must integrate this density over all space:
U=\int^{\infty}_{a}\frac{q^{2}}{8\pi\epsilon_{0}r^{2}}dr=\frac{1}{2}\frac{q^{2}}{4\pi\epsilon_{0}}\frac{1}{a}=\frac{1}{2}\frac{e^{2}}{a}
where e^{2}=\frac{q^{2}}{4\pi\epsilon_{0}}
Now according to the theory of relativity, the energy U of a particle is always mc^{2}. So according to relativity, the mass of an electron due to its own electric field is
m=\frac{1}{2}\frac{e^{2}}{ac^{2}}
However, the mass of an electron due to its own field can also be calculated another way. Consider an electron moving at a speed v<<c. The momentum density \vec{g} of its electric field is given by
\vec{g}=\epsilon_{0}\vec{E}\times\vec{B} where \vec{B}=\vec{v}\times\vec{E}/c^{2}.
In order to find the total momentum in the electric field produced by the electron, we must integrate this density over all of space. This integration is quite messy, but the result is \vec{p}=\frac{2}{3}\frac{e^{2}}{ac^{2}}\vec{v}.
At low velocities, \vec{p}=m\vec{v}. So according to electromagnetic theory, the mass of an electron due to its own electric field is
m=\frac{2}{3}\frac{e^{2}}{ac^{2}}
So electromagnetic theory and special relativity give contradictory results for the mass of an electron due to its own electron field. What is the resolution to this apparent paradox?

Any help would be greatly appreciated.
Thank You in Advance.

 

[meopemuk]

There are lot of paradoxes in classical electromagnetic theory. You can find dozens of references (including those relevant to the "electromagnetic mass" controversy that you mentioned) in

G. Spavieri, G. T. Gillies, "Fundamental tests of electrodynamic theories: Conceptual investigations of the Trouton-Noble and hidden momentum effects", Nuovo Cim. B 118 (2003), 205.

In my personal opinion, many of these paradoxes simply cannot be resolved within Maxwell's theory. We need a better theory of electromagnetism.

Eugene.

 

[lugita15]

There are lot of paradoxes in classical electromagnetic theory. You can find dozens of references (including those relevant to the "electromagnetic mass" controversy that you mentioned) in

G. Spavieri, G. T. Gillies, "Fundamental tests of electrodynamic theories: Conceptual investigations of the Trouton-Noble and hidden momentum effects", Nuovo Cim. 118 (2003), 205.

In my personal opinion, many of these paradoxes simply cannot be resolved within Maxwell's theory. We need a better theory of electromagnetism.

Eugene.

Is that paper online? If so, what is the link?

 

[meopemuk]

Is that paper online? If so, what is the link?

If you are willing to pay for it:

http://cat.inist.fr/?aModele=afficheN&cpsidt=15187191

Otherwise, you should visit a library.

Eugene.

 

[cesiumfrog]

A large number of these "paradoxes" have turned out to be subtle coordinate transformation errors.

 

[lugita15]

A large number of these "paradoxes" have turned out to be subtle coordinate transformation errors.

Is there any error in coordinate transformations in the paradox discussed in my post? For that matter, were there any coordinate transformations in my post?

 

[lugita15]

If you are willing to pay for it:

http://cat.inist.fr/?aModele=afficheN&cpsidt=15187191

Otherwise, you should visit a library.

Eugene.

My library doesn't subscribe to the journal in question. Is it included in some book? How did you read the paper?

 

[cesiumfrog]

Is there any error in coordinate transformations in the paradox discussed in my post? For that matter, were there any coordinate transformations in my post?

I'm not claiming to resolve this specific paradox, just commenting on other examples I've studied (here, for example, is one that Griffiths published earlier work on - having obtained a remarkably similar difference in leading factor). In your post you needed to transform the electric and magnetic fields into a moving frame of coordinates.

 

[meopemuk]

My library doesn't subscribe to the journal in question. Is it included in some book? How did you read the paper?

So bad. You can try to search for other articles of these authors on Google Scholar. Perhaps, some of them are more readily available, e.g., as preprints on the arXiv. I read this paper in the physics library. I am lucky to have a good research university nearby.

I've seen this "electromagnetic mass" controversy mentioned in many places. However, I don't think anybody has a logical resolution of this paradox within classical electrodynamics.

My other point was that there are many more electromagnetic paradoxes not mentioned in most textbooks. There are also recent experiments, which seem to contradict Maxwell's electrodynamics:

A. L. Kholmetskii, et al. "Experimental evidence of non-applicability of the standard retardation condition to bound magnetic fields and on new generalized Biot-Savart law",
J. Appl. Phys. 101 (2007), 023532; http://www.arxiv.org/abs/physics/0601084

N. Graneau, T. Phipps Jr., D. Roscoe, "An experimental confirmation of longitudinal electromagnetic forces" Eur. Phys. J. D 15 (2001), 87

Eugene.

 

[lugita15]

I'm not claiming to resolve this specific paradox, just commenting on other examples I've studied (here, for example, is one that Griffiths published earlier work on - having obtained a remarkably similar difference in leading factor). In your post you needed to transform the electric and magnetic fields into a moving frame of coordinates.

Yes, but we're dealing with speed much smaller than the speed of light, so Lorentz transformations of E and B don't really matter.

 

[lugita15]

I just wanted to give Feynman's derivation of \vec{p} by integrating \vec{g} over all of space, since I didn't include it in my original post:

 

[meopemuk]

Suppose that the electron is first at rest. Suppose also that the electric field makes contribution U/c^2 to the electron's mass. Now, let the electron accelerate under the influence of some external force. If we accept that perturbations in the field propagate at the speed of light, then the shape of the field around the electron changes with time. Does it mean that electron's mass is changing? Does electron's mass depend on its acceleration? It doesn't make sense to me.

Eugene.

 

[lugita15]

Suppose that the electron is first at rest. Suppose also that the electric field makes contribution U/c^2 to the electron's mass. Now, let the electron accelerate under the influence of some external force. If we accept that perturbations in the field propagate at the speed of light, then the shape of the field around the electron changes with time. Does it mean that electron's mass is changing? Does electron's mass depend on its acceleration? It doesn't make sense to me.

Eugene.

Although the electromagnetic energy density u will change as the electron accelerates, when you integrate the density over all of space, the total energy U in the electric field stays the same. So the electron's mass doesn't depend on its acceleration.

 

[meopemuk]

Although the electromagnetic energy density u will change as the electron accelerates, when you integrate the density over all of space, the total energy U in the electric field stays the same. So the electron's mass doesn't depend on its acceleration.

Can you prove that?

Eugene.

 

[pervect]

I think the point probably is that it's inconsistent to view the electron as some sort of small, charged rigid sphere, which was right in the initial problem statement.

Consider an electron in which all of the charge q is uniformly distributed on the surface of a sphere of radius a.

Rigid spheres don't exist in relativity, as is well known.

One would like to argue that if the sphere is small enough, one could avoid this difficulty. If it is a small sphere, deformations of this sphere shouldn't matter much. But the self-energy of a small sphere approaches infinity as the sphere shrinks to a point. This means that one is attempting to justify small changes to an infinite quantity, an argument that is very suspect.

That's just my take, I haven't read on this in detail. This concept apparently has a lot of history. If you happen to be able to find "Concepts of mass in classical and Modern Physics" by Max Jammer, he has a chapter devoted to the idea of "electromagnetic mass" which is related to this problem. So there is quite a lot of literature on this to plow through, something I haven't done.

You'll probably need to get access to a university library to get to the better papers. Usually the biggest problem is getting parking on campus :-). Printing the papers also might get expensive (you can ask the librarian about alternatives available if any, I know of at least one university that attempts to put USB ports on the library computers so you can put a flash drive onto the library computers).

 

[meopemuk]

There is another well-known paradox associated with assigning such properties as energy and momentum to electromagnetic fields. Electrons have spin, therefore there is a magnetic field B even around a stationary electron. One can find the (Poynting) vector of the field's momentum by integrating the vector product \mathbf{E} \times \mathbf{B} over entire space. This integral is non-zero. This means that the linear momentum of the field is non-zero. Shall we then conclude that the electron at rest has a non-zero momentum?

Eugene.

 

[cesiumfrog]

I think the point probably is that it's inconsistent to view the electron as some sort of small, charged rigid sphere, which was right in the initial problem statement.

Rigid spheres don't exist in relativity, as is well known.

That seems a bit of a red herring.

Rigidity is irrelevant to comparing a static charge distribution between stationary and constant-velocity reference frames.

Whether or not it models an electron is irrelevant to whether or not different methods for determining the invariant mass of a given static classical charge distribution should be expected to give the same results.

 

[pervect]

That seems a bit of a red herring.

Rigidity is irrelevant to comparing a static charge distribution between stationary and constant-velocity reference frames.

Whether or not it models an electron is irrelevant to whether or not different methods for determining the invariant mass of a given static classical charge distribution should be expected to give the same results.

I don't think it's as much of a red herring as you might think, since the invariant mass of a system of nonzero volume is only invariant if the system is a closed system. If you take a small piece of a bigger system, that small piece does not in general have a 4-momentum that transforms as a 4-vector. The correct approach is to divide the system into pieces of infinitesimial volume, which will transform properly - this approach winds up giving you the stress-energy tensor (which however transforms as a rank 2 tensor, not a 4-vector).

We had a thread on the energy in the field not too long ago, in fact, where I pointed out the need to include the energy in the charges as well as the energy in the field:

https://www.physicsforums.com/showthread.php?t=180779

where I said:

Of course, \partial_{a}T^{ab} = 0 for the combined system of particles and fields (see Jackson, page 611 for instance). However, if one considers the field only, and ignores energy and momentum of particles or media with which the field is interacting, there can be a non-zero divergence of the stress energy tensor of the field alone.

So I think the missing piece here is that one needs to consider more than just the field, and one will "find" the missing energy and momentum. When one has an isolated system, the total energy and momentum does transform as a 4-vector. The problem here, I think, is that the invariant "mass" of the field isn't coming out right because it isn't a 4-vector, and it isn't a 4-vector because a field with a source isn't a closed system unless you consider the energy in the source.

 

[lugita15]

I don't think it's as much of a red herring as you might think, since the invariant mass of a system of nonzero volume is only invariant if the system is a closed system. If you take a small piece of a bigger system, that small piece does not in general have a 4-momentum that transforms as a 4-vector. The correct approach is to divide the system into pieces of infinitesimial volume, which will transform properly - this approach winds up giving you the stress-energy tensor (which however transforms as a rank 2 tensor, not a 4-vector).

We had a thread on the energy in the field not too long ago, in fact, where I pointed out the need to include the energy in the charges as well as the energy in the field:

https://www.physicsforums.com/showthread.php?t=180779

where I said:



So I think the missing piece here is that one needs to consider more than just the field, and one will "find" the missing energy and momentum. When one has an isolated system, the total energy and momentum does transform as a 4-vector. The problem here, I think, is that the invariant "mass" of the field isn't coming out right because it isn't a 4-vector, and it isn't a 4-vector because a field with a source isn't a closed system unless you consider the energy in the source.

Could you please explain, in simple terms, what the stress-energy tensor is?

 

[lugita15]

There is another well-known paradox associated with assigning such properties as energy and momentum to electromagnetic fields. Electrons have spin, therefore there is a magnetic field B even around a stationary electron. One can find the (Poynting) vector of the field's momentum by integrating the vector product \mathbf{E} \times \mathbf{B} over entire space. This integral is non-zero. This means that the linear momentum of the field is non-zero. Shall we then conclude that the electron at rest has a non-zero momentum?

Eugene.

Yes, a particle with spin has not only linear momentum, but angular momentum as well. But this isn't as counterintuitive as it may seem at first glance. In quantum mechanics, no particle is really "moving," in the sense that it is going on a continuous path from point A to point B. So momentum is no longer a property of "motion."

 

[meopemuk]

Yes, a particle with spin has not only linear momentum, but angular momentum as well. But this isn't as counterintuitive as it may seem at first glance. In quantum mechanics, no particle is really "moving," in the sense that it is going on a continuous path from point A to point B. So momentum is no longer a property of "motion."

The paradox that I mentioned refers to classical objects as well. For example, you can take a macroscopic charged magnet and see that even if the whole thing is at rest, there is a non-zero momentum in the field.

Eugene.

 

[cesiumfrog]

The paradox that I mentioned refers to classical objects as well. For example, you can take a macroscopic charged magnet and see that even if the whole thing is at rest, there is a non-zero momentum in the field.

You call that a paradox? I thought Griffiths used something like that to resolve a number of other apparent paradoxes (like non-obvious conservation of momentum)..

 

[lugita15]

The paradox that I mentioned refers to classical objects as well. For example, you can take a macroscopic charged magnet and see that even if the whole thing is at rest, there is a non-zero momentum in the field.

Eugene.

But in classical electromagnetism, all magnets are ultimately moving charges or currents.

 

[meopemuk]

For example, you can take a macroscopic charged magnet and see that even if the whole thing is at rest, there is a non-zero momentum in the field.

You call that a paradox?

Yes, this is rather disturbing. For example, according to rather general principles, the center-of-mass position \mathbf{R} of an isolated system should have time dependence (in the non-relativistic case)

\mathbf{R}(t) = \mathbf{R}(0) + \frac{\mathbf{P}t}{M}

How are you going to define the center-of-mass for the charged magnet (plus its field) to make sure that it is not moving even though its momentum is non-zero?

Moreover, it seems suspicious that the field momentum is an unmeasurable quantity. It doesn't add to theory's credibility, when its defects are hidden in something that can't be verified experimentally.

I thought Griffiths used something like that to resolve a number of other apparent paradoxes (like non-obvious conservation of momentum)..

Thanks for the reference. I am collecting electromagnetic paradoxes. It would be interesting to see what Griffiths has to say.

Eugene.

 

[cesiumfrog]

How are you going to define the center-of-mass for the charged magnet (plus its field) to make sure that it is not moving even though its momentum is non-zero?

Griffiths (from an example indexed under momentum): "a magnetic dipole in an electric field carries linear [..] hidden momentum[, that] precisely cancels the electromagnetic momentum stored in the fields". That one seems resolved to me.

 

[meopemuk]

Griffiths (from an example indexed under momentum): "a magnetic dipole in an electric field carries linear [..] hidden momentum[, that] precisely cancels the electromagnetic momentum stored in the fields". That one seems resolved to me.

So, Griffiths is explaining one unobservable quantity (field momentum) by introducing another unobservable quantity ("hidden" mechanical momentum). Brilliant!

Eugene.

 

[cesiumfrog]

Sorry Eugene, but I think I've given you an incorrect impression. You should read the entire example (which goes into explicit details) rather than just my brief excerpt.

 

[lugita15]

Thanks for the reference. I am collecting electromagnetic paradoxes. It would be interesting to see what Griffiths has to say.

If you want an electromagnetic paradox, here's one. It is an "exception to the Flux rule" pointed out by Feynman in his Lectures on Physics:
Consider a copper disk rotating with constant angular velocity, as shown in the figure. A bar magnet is directed normal to the surface of the disk, as shown in the following figure:

If a galvanometer is used to measure the induced current in the outer rim of the disk, a nonzero induced current is detected. Using the Lorentz force law, this is easily explainable. Consider a small charge dq on the outer rim of the disk. At any given time, it has a nonzero velocity because of the rotation of the disk. For this reason, the bar magnet exerts a magnetic force on the charge. This makes the charge move differently than the disk itself. Some of these charge will go through the galvanometer, and thus the galvanometer will indicate the existence of a current.

But wait a minute. Let's try applying the integral form of Faraday's Law: the emf along a closed loop is equal to the rate of change of the magnetic flux through the loop. Take the loop to be the outer rim of the copper disk. But the magnetic flux through surface of the copper disk is constant, and thus the rate of change of magnetic flux is zero. This is because both the magnetic field and the area are both constant. So we have a strange situation in which the induced emf is nonzero even though the rate of change of magnetic flux is zero.

 

[meopemuk]

Thank you, lugita15. That's a good one.

Eugene.

 

[rewebster]

Thank you, lugita15. That's a good one.

Eugene.

Could you send me that list that you're putting together, if it isn't an imposition? If you're putting it together for a paper, no problem--I could and would wait for that to be published--thanks