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- This video states that a static E field and a static H field at right angles to each other, have momentum. Is this true?

Is it true, what he's saying from 04:28 to 04:56 ? I have my doubts, but I thought I'd better ask here.

- Thread starter Swamp Thing
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- Summary
- This video states that a static E field and a static H field at right angles to each other, have momentum. Is this true?

Is it true, what he's saying from 04:28 to 04:56 ? I have my doubts, but I thought I'd better ask here.

Mentor

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D. Babson, S. P. Reynolds, R. Bjorkquist and D. J. Griffiths

Hidden momentum, field momentum, and electromagnetic impulse

American Journal of Physics 77, 826 (2009); https://doi.org/10.1119/1.3152712

Abstract said:Electromagnetic fields carry energy, momentum, and angular momentum. The momentum density, ##\epsilon_0 (\mathbf{E} \times \mathbf{B})##, accounts (among other things) for the pressure of light. But evenstaticfields can carry momentum, and this would appear to contradict a general theorem that the total momentum of a closed system is zero if its center of energy is at rest. In such cases, there must be some other (nonelectromagnetic) momenta that cancel the field momentum. What is the nature of this “hidden momentum” and what happens to it when the electromagnetic fields are turned off?

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Thanks, I'll take a look.

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Another great free online resource are the essays by Kirk McDonald:

http://puhep1.princeton.edu/~mcdonald/examples/

The ones about "hidden momentum" are gems!

Science Advisor

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Perhaps that is why he doesn't publish them.

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Science Advisor

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"The cleanest example of hidden momentum is the following: Imagine a rectangular loop of wire carrying a steady current. Picture the current as a stream of noninteracting positive charges that move freely within the wire. When a uniform electric field E is applied (see Fig. 9), the charges will accelerate up the left segment and decelerate down the right one."

Figure 9 shows that the E field in the two vertical segments of the loop is different than the E field in the two horizontal segments of the "wire carrying a steady current." This violates Ohm's law, j=\sigma E.

List the site of one of McDonald's papers, and I will point out his error.

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$$\vec{j}=\gamma \sigma (\vec{E}+\vec{v}/c \times \vec{B})$$

or in covariant form

$$j^{\mu} = \sigma F^{\mu \nu} u_{\nu}?$$

Most mistakes about electrodynamics is using non-relativistic approximations for the motion of the charge carriers and then using field momenta, which are of a higher order in the relativistic expansion (formally in powers of ##|v|/c=|\beta|##. That's the case in the current-loop example shown in said Fig. 9 of the paper by Babson et al.

I don't see any obvious errors in the nice paper by Babson quoted in #2. I'm also not aware of any errors in McDonald's manuscript quoted in #4.

Science Advisor

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Are you saying that \gamma E is constant in their example?

Science Advisor

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Which of McDonald's papers should I look at?

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The here discussed example is also discussed by McDonald with some more very nice elaborations starting from it:

/home/arch/hees2/paper/textbooks/class-edyn/electrodynamics-of-moving-media-Penfield.djvu

A more puzzling one is

Another good explanation is found in

Science Advisor

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Science Advisor

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Its mistake is assuming the center of energy theorem.

The Griffith's Hnizdo paper just repeats Babson.

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Science Advisor

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A proof the "center of energy" theorem (quote from Grifffiths) in an EM field is illusive.

McD refers to a paper by Griffiths, but Griffith's just refers to a paper by Coleman and Van Vleck.

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Where is this quote from? How can this in a Poincare invariant theory be true? If a theory is Poincare covariant the center-of-**momentum** theorem automatically holds thanks to Emmy Noether. See, e.g.,

https://arxiv.org/abs/physics/0501134v1

https://arxiv.org/abs/physics/0501134v1

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Science Advisor

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What quote do you mean. My post only quoted you and two words from a Griffiths paper.

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Science Advisor

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D.J. Griffiths, Dipoles at rest, Am. J. Phys. 60, 979 (1992) .

"Center of Energy Theorem" is always used by Griffiths , but why does the name matter when \gamma E is not constant for Babson?

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I also don't see anything in the quoted paper by Griffiths which contradicts standard relativistic theory; to the contrary Griffiths discusses the resolution of apparent paradoxes, which always occur when using non-relativistic approximations under circumstances where a relativistic treatment or more careful systematic non-relativistic approximations are due.

He nicely discusses even the case of electrodynamics with magnetic monopoles and corresponding nonstandard magnetic dipoles. So far all empirical evidence rules out (elementary) magnetic monopoles and nonstandard magnetic dipoles (as is also explained at the end of the paper).

Science Advisor

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For now, I just want to repeat again that keeping ##{\bf j}## constant while ##{\bf\gamma E}## varies contradicts

##{\bf j=\gamma\sigma E.}##

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Well all you do is repeating your enigmatic claims that basicly "everyone is wrong", and give us no justification whatsoever. It does not look good.I just want to repeat again

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Obviously we have some misunderstanding here. The correct Ohm's Law is

For now, I just want to repeat again that keeping ##{\bf j}## constant while ##{\bf\gamma E}## varies contradicts

##{\bf j=\gamma\sigma E.}##

$$\vec{j}=\gamma \sigma (\vec{E}+\vec{v}/c \times \vec{E}),$$

where ##\vec{v}## is the velocity of the charge carriers making up the current. In covariant form it reads

$$j^{\mu}=\sigma F^{\mu \nu} u_{\nu},$$

where

$$u^{\mu}=\gamma \begin{pmatrix} 1 \\ \vec{v} \end{pmatrix},$$

is the four-velocity of the charge carriers making up the current.

BTW. All this is well-established physics, already understood by Minkowski in 1910 (posthumously published by Born, who completed the work). It's also all contained in the famous first textbook on SRT by Minkowski. I don't know, why you question this well-established physics all the time!

As I said, it's somewhat unfortunate to call the here addressed phenomenon "hidden momentum", because in fact it's simply momentum, which is not hidden at all, but somehow this terminology came up in the later literature of the mid 1960ies for some reason.

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Science Advisor

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What did I question there? I know textbook physics, and regret Minkowski's premature death.

I meant to write your Ohm's law equation with ##\bf v\times B=0##. B external is zero isn't it?

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By chance, I've just written an Insights article about the much simpler problem of a straight wire. Maybe that helps to understand the importance of the relativistically complete version of Ohm's Law when discussing relativistic covariant electrodynamics:

The direct-current-conducting infinitely long wire is often discussed in the context of relativistic electrodynamics. This article clarifies this debate.

www.physicsforums.com

Another example, where it is important, is the homopolar generator, which also usually gives rise to a lot of confusion, which is just due to the fact that Ohm's Law is not written down in its full relativistic form:

It is surprising that the homopolar generator, invented in one of Faraday's ingenious experiments in 1831, still seems to create confusion...

www.physicsforums.com

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