Talk:Lorentz force/Archive 2

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Archive 3

What is the principle that causes the magnetic part of Lorentz force?

As stated in the header, the article doesn't seem to explain the physics principle that causes the magnetic part of lorentz force. From what I can tell, a magnetic field can act on a moving electric charge and create a kinetic force perpendicular to the plane of magnetic force and movement of electric charge. And we call this force lorentz force, or at least the magnetic force part of it.(If I'm not wrong, this specific part of the lorentz force is also called laplace force(?)) There's also an electric field in this, but that's irrelevant as it's obvious an electric field can and will act directly on an electric charge, though I'm not sure if than an actual term for that. electrostatic force?

However, what is the explanation that allows the magnetic field to act on the moving charge? There doesn't seem to be an explanation for this in the article.

--218.186.9.246 (talk) 13:58, 21 July 2010 (UTC)

In my experience, when people imagine an electric force, they imagine the push-pull of a spring (or something like that). It's "obvious" meaning intuitive, and it's intuitive because it has everyday mechanical analogues. But there aren't other common everyday experiences of perpendicular velocity-dependent forces, so the magnetic force is somehow unintuitive and "not obvious". In reality (as opposed to in our intuitions), the magnetic force and electric force come from the exact same place (quantum electrodynamics) and stand on equal footing.

For this article, I don't think it's a good idea to try to "explain" magnetic forces: Magnetic forces are caused by magnetism. What is magnetism caused by? It's a law of physics, it's ultimate justification is that we can see it and measure it and experience it. Or you could say magnetism is caused by quantum electrodynamics, which is too complicated to discuss much in this article. :-)

Some teachers try to encourage students to believe in magnetism...a famous example is the argument that "If you believe in electricity and you believe in special relativity then to be consistent you have to believe in magnetism too". This is given in some textbooks including Purcell, and on wikipedia in Relativistic electromagnetism. I don't buy this pedagogical approach (personally), I think people should believe in magnetism because they can see it in front of their eyes. Maybe in a semester-long course it's worth doing this discussion...not in an encyclopedia article (at least not this one). Just my opinion... :-)

The "Laplace force" term is already defined in the article, just like you said. :-) --Steve (talk) 16:18, 21 July 2010 (UTC)

Ehh... so basically, just believe that it's there and it works because it's much too complicated for this level? Well...that seems somewhat wrong...--218.186.9.246 (talk) 10:27, 22 July 2010 (UTC)

Belief does not enter into this. The Lorentz force is a real experimental fact. (The easiest way to show this is to place a beam of electrons in a magnetic field.) If you are looking for a proof of the existence of the magnetic Lorentz force from first principles then you probably need to go to QED. You can start with some basic assumptions and go through some math to show that if one set of laws are true then the Lorentz force is true. The simplest form of this that I am aware of is in Griffith (522-525). There he shows that the magnetic Lorentz force on a charge moving parallel to a wire in a magnetic field due to the wire is observed as a pure electric force from the point of view of the particle (in its stationary reference frame) when relativistic contraction of the distance between the moving charges of the wire is accounted for. In other words, the magnetic part of the Lorentz force is a necessary to be consistent with relativity. It is not strictly true to say that the magnetic portion of the Lorentz force is caused by the electric portion due to relativity, though. In relativity there is no preferred reference frame; it is equally valid to say that the magnet portion of the force caused the electric.

What Steve is trying to say I think is that we could provide a 'proof' similar to Griffith's but that it:

a) Is more complicated then it is worth (there are no simple 'proofs')

b) Distracts from the fact that the Lorentz force is an experimental fact in its own right

c) Gives the impression that the electric force is more fundamental then the magnetic force just because we can understand it easier.

For the most part I agree with a) although it may have a role in the advanced sections. I agree strongly with b) which is a point I think you might have missed. As for c) I can see Steve's point, but it seems too subtle of a point to worry about. It is an important point to remember if you are prone to out-thinking a problem for instance by trying to find the best reference frame. But for the majority of people it isn't an issue, in my opinion. TStein (talk) 18:07, 4 August 2010 (UTC)

Anonymous 218.186.9.246 there is no explanation for the Lorentz force in modern physics. It is an unexplained fact. David Tombe (talk) 19:44, 21 October 2010 (UTC)

I want to hijack the magnetic force page which redirects here to use for force between magnets.

The magnetic force page redirects here. I would like to use that page for a new page covering the force between magnets. There are pros and cons to this. My main reason is that the force between magnets is now duplicated on two pages magnet and magnetic moment when it really should be its own page. I thought about using force between magnets but I prefer something simpler. To me, most people who would use magnetic force are those who are looking for the force between magnets. But I am looking for input, before I go through the work of creating a new page.

(This post was the main reason I came to this page. I almost forgot by responding to the other talk posts though ;) ). TStein (talk) 19:36, 4 August 2010 (UTC)

Based on google books search and WP links, "magnetic force" is used about equally for both force between magnets and Lorentz force. So I guess either way is OK.

In general I prefer very specific titles, like force between magnets, because if the article is called magnetic force then probably some editor will not look too carefully and waste their time writing a new article on Lorentz force and adding it on to the magnetic force entry. In other words, the scope and topic of an article tends to evolve over time to match the title, even if it's originally written and intended to have a narrower scope. :-P

Certainly the article should exist whatever it's called. Good luck! --Steve (talk) 20:33, 4 August 2010 (UTC)

Thanks for doing the grunt work on the popularity. I didn't expect that result. I suppose for now I could use force between magnets and include something in the { {for}} template of the Lorentz Force. "Force between magnets" just doesn't seem very discoverable. The ideal situation in my mind is that magnetic force becomes its own article with subsections having main templates to both Lorentz Force and Force between magnets and magnetic force on dipole perhaps. The nice thing about this new article is that I don't think I will have to write too much new material.TStein (talk) 22:14, 4 August 2010 (UTC)

Equation (77) in Maxwell's 1861 paper and equation (D) in his 1864/65 paper

Fyzixfighter, regarding your reverts, you are probably correct that Lord Kelvin (William Thomson) did a mathematical formulation of Faraday's lines of force. In fact, I do believe that he wrote a paper around about 1847 which is mentioned in the preamble of Maxwell's 1861 paper. But as I have never read it, I am unsure about the precise contents. If you are sure that Lord Kelvin did a full mathematical treatment of Faraday's lines of force, then I am happy to leave mention of that fact in the article. However, you are wrong about the other matter. Maxwell's original eight equations appeared as a concise list in his 1864/65 paper. Most of these equations had also appeared in his 1861 paper but not in a concise list. Equation (77) in his 1861 paper is the same as equation (D) in his 1864/65 paper. That equation is to all intents and purposes the same as the Lorentz force equation. But if you read the paper, you will see that Maxwell was talking about the force on an electric current, whereas the modern day Lorentz force equation applies more particulary to a discrete charged particle. I intend to restore the information about Maxwell's equation (D), but first I'd like to hear the views of some of the other participants.

And to save me going to the cross product article where you have also reverted my edits, I might as well explain that Heaviside's four equations are not taken exactly from Maxwell's original list of eight. In the Heaviside four, there is a partial differential equation which Heaviside always referred to as 'Faraday's law'. That equation does not appear in the list of Maxwell's original eight equations in the 1864/65 paper. In Maxwell's original eight equations, he caters for all aspects of electromagnetic induction using that equation (77)/equation (D) which is mathematically identical to the Lorentz force equation. As regards the idea that Maxwell had 20 original equations, this is based purely on the fact that in the first six, he wrote each of the three cartesian components out separately. So your reverts at cross product were wrong. David Tombe (talk) 11:09, 22 October 2010 (UTC)

From what I read in Darrigol, I wouldn't say that Thomson did a full mathematical treatment of Faraday's lines of force, but he does appear to be the first to have attempted to subject them to a mathematical treatment. The Verschuur reference touches a bit on this, but I cited Darrigol because his is by far the most comprehensive on the subject. The prize of course goes to Maxwell for doing the first complete and general mathematical treatment of Faraday's lines and fields.

If you can find a source that equates Maxwell's 1865 (D) with the Lorentz force equation then cite it. Just because you think they look alike is not sufficient for WP:V and WP:OR. When I re-wrote the history section, I was very particular to follow a few reliable secondary sources as a basis of the history section. None of the sources that I can find draw this comparison you want to make.

As to the cross-product/Heaviside subject, I would advise that you look at the reference that I cited which clearly states that Heaviside reduced Maxwell's 20 (scalar) equations with 20 variables, to 4 (vector) equations. Part of the problem is that what you call an equation, the 1865 (D) for example, is what the secondary sources would call groups or sets of equations. Thus (D) is three equations in Nahin's (the reference I gave) and others' counts. Additionally, your summary of what Heaviside did is drastically over-simplified and flawed, namely because you ignore the fact that Heaviside produced his duplex equations for the special case where the conductor was stationary. Heaviside was a telegraph operator and was initially more interested in the propagation along stationary conducting lines. He was well aware of the velocity dependent terms that arise when the duplex equations are written for moving conductors, in fact he was the first Maxwellian physicist to give an exhaustive list of all these terms. He even introduced additional velocity dependent terms that Maxwell had in fact missed (for example, a Dxv contribution to H). In 1888 and 1889 he produced duplex equations with all the proper terms, including the ones Maxwell missed, for all cases of motion. However, the modern convention is to define the electric and magnetic field in the inertial frame of the observer and not the possibly moving conductor (I believe Jackson said this, but I don't have my copy in front of me at the moment), hence the modern adaptation of Heaviside's first (1885) set of duplex equation. --FyzixFighter (talk) 03:19, 23 October 2010 (UTC)

FyzixFighter, First of all, I am happy enough to leave the mention of Lord Kelvin in the article. I know that he did what looks like a very interesting paper in 1847 which Maxwell talks about in the preamble of his 1861 paper. I have been intending to read it. It might however be a good idea to call him Lord Kelvin as opposed to his actual name William Thomson, purely for the reason that the other scientist JJ Thomson is mentioned in the next line. You know, and I know, that they are two different people, but we need to aim at not confusing the readers.

On the other points, I'm actually fully aware of all the minute details regarding what Maxwell did and what Heaviside did. In fact both Maxwell and Heaviside dropped the v×B term when they derived the EM wave equation/telegraphy equation. Maxwell makes it quite clear that he is considering a point which is stationary in the aether, and so he drops the v×B term. And I know all about Heaviside's 1889 paper in which he extrapolates the EM wave equation for a moving point. And I know that even Maxwell himself referred to his 8 equations as 20 equations. That was because, as you say, six of them were expressed in each of their x, y, and z, components. But the problem with your revert at cross product is twofold. First of all, Heaviside's 4 equations don't involve dot product and cross product, so much as they involve divergence and curl. I know that div and curl are simply the application of dot product and cross product to the differential space operator, but I wanted to make that point more clear, and besides, a pure mathematician may or may not have some issue with the idea that a div and a curl are simply dot product and cross product extended to the differential operator. Secondly, although Heaviside reduced Maxwell's equations to a group of 4 which are fundamental to deriving the EM wave equation at a fixed point in the aether, it is not technically true to say that he reduced Maxwell's 20 equations to 4. Heaviside's 4 equations did all appear in Maxwell's papers. But only 1 of the 4 in question actually appears on the list of 20. So it is somewhat misleading to state that Heaviside reduced a particular set of 20 equations to 4 equations. That is the inaccuracy which I was trying to address.

As regards the equation (77), I am awaiting more comments, but I have already had one editor at WT:PHYS who acknowledges that it is one and the same as the Lorentz force. Even Maxwell's description of the v×B term, which is pre-Gibbs goes into elaborate detail about the parallelogram associated with the cross product. Both equations deal with electromotive force, even though nowadays we use the term 'magnetic force' for the v×B term. David Tombe (talk) 11:17, 23 October 2010 (UTC)

You still need a secondary source that connects Maxwell's equations (77) or (D) to the Lorentz force law, advice which you were given on WP:PHYS. If you want to say that Maxwell came up with such-and-such a form for the electromotive force, then yes we can say that - but connecting it to the Lorentz force law requires a secondary source. Certainly there is a relationship between Maxwell's equations elsewise Thomson and Heaviside and others wouldn't have been able to derive it, and the article says as such, but to say that the Lorentz force law is blatantly one of Maxwell's equations needs a secondary source. The statements you've been adding are based on your own interpretation of those equations and therefore fails WP:V and WP:OR.

I've re-shortened the initial sentence for two reasons. First, the emphasis is on the mathematical foundation being established for the idea of electric and magnetic fields, which was done by Kelvin and Maxwell, and not necessarily on Faraday's lines of force. Second, Kelvin's work on this spans multiple years and papers. In his 1847 work he developed a mathematical strain analogy the was a "mechanical representation of electric forces that integrated Faraday's notion of stresses in the field." (Darrigol, p 127). In 1851, he used Faraday's law and gave its mathematical expression for a particular case, preempting Maxwell's 1854 breakthrough of being the first to give Faraday's law a completely general mathematical expression. Later, at an 1852 British Association meeting, he presented "pretty diagrams of lines of force" that he had used to calculate the force on paramagnetic and diamagnetic spheres. All of this is in Darrigol, which is why secondary sources are so much better than primary sources.

Darrigol also provides the most comprehensive history of the path and various missteps of getting to the Lorentz force that I can find. --FyzixFighter (talk) 16:27, 23 October 2010 (UTC)

I haven't fully digested Maxwell's paper yet (and I doubt I will finish it - it is hard to read nowadays) but I can hazard a guess as to the problem here. Maxwell was describing the effects of electric and magnetic fields on moving conductors whereas Lorentz was describing the effect on charges. So Maxwell had formulated the electromagnetic force correctly - but not in the modern form. Since the electron had not been discovered in Maxwell's day, how could he do otherwise? --Michael C. Price ^talk 18:02, 24 October 2010 (UTC)

Michael, That's exactly right. And that's exactly how I worded it in my edits. I clearly stated that Maxwell applied the force to electric currents as opposed to discrete charged particles. David Tombe (talk) 18:28, 24 October 2010 (UTC)

Check out E. Whittaker, A History of the Theories of Aether and Electricity, vol. I, 2nd edition, Nelson, London (1951). Reprinted by the American Institute of Physics, (1987). p. 310. It is of interest to note that James Clerk Maxwell gave the expression for the Lorentz force already in his historic memoir of 1865. --Michael C. Price ^talk 19:12, 24 October 2010 (UTC) PS I don't see the statement in the book. --Michael C. Price ^talk 19:41, 24 October 2010 (UTC)

@Michael - I too couldn't find where they find that in Whittaker's book. Pages 350-360 cover a bit of what Thomson did. I don't think we can say Maxwell got expressions for the forces within a moving conductor. I think we can say that he gave us the expressions (though slightly incomplete) expressions for the fields and the electromotive force in a moving conductor. But the electromotive force was not understood as a mechanical force as we think about it - it was the driving cause of currents, in this case due to induction. Why Maxwell and others didn't immediately make the leap to a general force law on charged objects was not that no one had discovered charged atomic/subatomic particles, but more generally that they had not yet shown that an electrical current was identical to moving charges. Maxwell, and even Faraday, had inklings of this, but it was Rowland that first showed this. Thomson's and Heaviside's theoretical work may have been motivated in part by the discovery of subatomic particles (eg cathode rays) but they did the calculations for a charge sphere with a finite radius. This is what I gather from my reading of the Nahin and Darrigol references. I would highly recommend the two books as reading material as they have been very enlightening.

When I rewrote the history section a few months ago, I did my best to summarize the secondary sources that I had found. I did not approach it with an agenda to diminish or to amplify Maxwell's role. It should also not be up to us to interpret primary sources except maybe in rare occasions. This is exactly why wikipedia emphasizes a preference for secondary sources. Secondary sources go through a peer review process, our (your, mine, David's) interpretations haven't and do not pass WP:V. If what David is saying is well known, than a secondary source should exists that says as much. Otherwise it's likely not a mainstream view. --FyzixFighter (talk) 22:15, 24 October 2010 (UTC)

I'm happy with the version now; I thought my mentioning of the conductors was sufficient, but you have further improved it. Good job all round.

PS Perhaps the article can explain how Heaviside's version - the first correct form - differs from Lorentz's?

--Michael C. Price ^talk 22:27, 24 October 2010 (UTC)

Michael, Thanks for your help. Off the top of the head, my guess is that the Lorentz version, being a product of the Lorentz transformations applied to Heaviside's 4 Maxwell equations, would have the additional Lorentz factor (gamma factor) as a coefficient. This web link here shows how the Heaviside 4 are brought to equation (19) using the Lorentz transformation. [1]. Or maybe that's the relativity method. I'm not so sure as my knowledge of physics stops at Heaviside. But note how the Heaviside 4 don't involve the v×B term. Heaviside uses Maxwell's equation (54) as opposed to Maxwell's equation (77) and he always refers to equation (54) as Faraday's law. So with the Heaviside 4 not including the v×B term, Lorentz re-introduces it by applying a Lorentz transformation to the Heaviside 4. I'm only guessing, but it will be something like that. I've just looked at what the article says about the matter. It says partially the same as what I have said except that it talks about Lagrangian mechanics being applied to the Heaviside 4 as opposed to the Lorentz transformation. I have no idea what that is all about. There are some editors here who are experts on Lorentz. Somebody like DH springs to mind. You need to call him in. David Tombe (talk) 23:27, 24 October 2010 (UTC)

The Lorentz factor is, IIRC, a red herring here. Lorentz noted the invariance of MEs under what we now call Lorentz transformations, but did not develop the idea further. --Michael C. Price ^talk 07:15, 25 October 2010 (UTC)

Michael, That's probably true, and I would assume we are specifically talking about the 4 Heaviside Maxwell equations. But like I said, I haven't really studied Lorentz in any detail and so I'll be keeping out of most post-Heaviside issues. But just to add my 2 cents on Lorentz, it seems that he noted how Heaviside had established a kind of Doppler shift on the Coulomb force around moving charges and he thought that this could explain length contraction and hence explain the null result in the Michelson-Morley experiment. David Tombe (talk) 09:22, 25 October 2010 (UTC)

FyzixFighter, Just as an additional piece of information, although it doesn't need to go into the article, Maxwell also produced the v×B term in his force equation at equation (5) in his 1861 paper. The difference between the v×B in equation (5) as compared to in equation (77) is that equation (5) applies v×B in the context of being the force on an electric current in a magnetic field, whereas equation (77) which came in part II is purely about EM induction, and so it is about the electromotive force that drives the current in a wire that is moving at right angles to a magnetic field. But as you are aware, Maxwell was not altogether clear about what his model for electric current actually constituted. He seems to have seen it in terms of some kind of velocity of a stuff which he referred to as 'free electricity'. It was clearly something not unlike the old vitreous fluid of DuFay and Franklin. Nevertheless, when Heaviside produced the v×B force in relation to discrete charges, he referred to it unambiguously as Maxwell's electromagnetic force and as being double JJ Thomson's result. David Tombe (talk) 23:41, 24 October 2010 (UTC)

Dual Lorentz force - does electric field affect movement of magnetic moment?

I couldn't find it in the article, but it seems that there is kind of dual Lorentz force for magnetic moment travelling in electric field (like Aharonov-Casher effect is dual to Aharonov-Bohm).

For example imagine classical electron traveling in proton's electron field - let's change reference frame such that for infinitesimal time electron stops and proton is moving in also magnetic field created by quite large electron's magnetic moment - because of 3rd Newton's law, resulting Lorentz force should also work on electron ...

(3) equation here is Lagrangian for such electron's movement: $\mathbf {L} ={\frac {1}{2}}m\mathbf {v} ^{2}+{\frac {Ze^{2}}{r}}+{\frac {Ze}{c}}\left[\mathbf {v} \cdot \left({\frac {\mu \times \mathbf {r} }{r^{3}}}\right)\right]$

the last term here corresponds to this electric field-magnetic moment interaction.

This force seems to be quite important (if true?) - maybe it should be somehow included in the article?--195.150.224.239 (talk) 11:30, 21 February 2011 (UTC)

Definitely should be discussed at magnetic moment. Here, it's more off-topic, maybe a few words and a link.

On the other hand, something that's very directly relevant to this article is the "Lorentz force equation" for a moving magnetic monopole in an electric field, see magnetic monopole. This is an exact analogue (or "dual") and the equation is the same form as the conventional one. :-) --Steve (talk) 18:58, 21 February 2011 (UTC)

Seeing Lorentz force in the article about monopoles was my final reason for this comment :-) - here we have magnetic dipoles so it's a bit more complicated. There are needed some better sources for this quite classical force ... --195.150.224.239 (talk) 21:32, 21 February 2011 (UTC)

Does anyone have the original reference?

The history section of the article says: "It was not until 1784 when Charles-Augustin de Coulomb, using a torsion balance, was able to definitively show through experiment that this was true.[4]". But reference #4 is a modern book. Does anyone have the original reference? Was it published? There are a lot of statements before this about early experiements and/or thoughts, but no references are given. Drkirkby (talk) 09:44, 29 November 2011 (UTC)

The citations need to be secondary sources -- WP:PRIMARY -- but the original publication could be incorporated into the text I suppose. Good luck in finding sources etc if you want to improve it. --Steve (talk) 13:11, 29 November 2011 (UTC)

Clean up

Main changes:

• add colour box to main equation
• add physics eqn navbox
• clean up nomenclature from inline LaTeX to html, remove the repetition of the same quantities such as E and B all the way through when their meaning is easily known
• clean up formulae
• re-section: the sections seem very random e.g. why is the force on a wire way down at the end when Faraday’s law and EM induction in moving wires half way? why is EMF discussed here and there?
• changed some of the headings, e.g. "Trajectories of particles in a Lorentz force" sounds a bit weird...

There is plenty of this to do.-- F = q(E + v × B) 19:13, 25 February 2012 (UTC)

If you write something like $dq/dV$, it looks like a derivative (where V is an independent variable, dV is the change in V, and q is the dependent variable). But of course, here, it's not a derivative, it's just a ratio of two infinitesimal quantities. I rewrote to avoid this confusion. --Steve (talk) 14:30, 26 February 2012 (UTC)

I know that but fair enough, the reader may not. Even with the statement of dq and dV why did you delete the equations

$\mathbf {f} ={\frac {d\mathbf {F} }{dV}},\quad \rho ={\frac {dq}{dV}}$

They were there to make the quantities more transparent and obvious that they are volume densities. But why do I ask - there is a reason for that too. Again I don't care, just thought I would raise the point.

Furthermore... while at it could you please point out anything else I have done wrong? Is the re-sectioning fine? -- F = q(E + v × B) 14:39, 26 February 2012 (UTC)

I deleted $\mathbf {f} ={\frac {d\mathbf {F} }{dV}},\quad \rho ={\frac {dq}{dV}}$ because first, they look like derivatives and therefore may cause confusion, and second, I thought it was an excessive level of detail for the situation. (If I was writing a textbook, or exam solutions, instead of an encyclopedia, I would have made that section even longer instead of making it shorter.) I guess I was thinking that the many readers will have no problem understanding that "force per unit volume" means "force divided by volume". The readers who can't understand what "per unit volume" means, even after being told to "divide both sides by volume", are readers who have no hope of understanding this section anyway. (They will have long ago been scared away by the "dF" and "infinitesimal element" etc. Adding more equations to the section won't help them.) Just my opinion. --Steve (talk) 23:38, 26 February 2012 (UTC)

Lorentz Factor

"where \scriptstyle \gamma is the Lorentz factor defined above."

...And perhaps it was in a previous version. This is in the Relativistic Force >> Field Tensor section of this article. While I was able to find a very well-stated syntax of the Lorentz factor in the article on Four-Velocity, there was no clear statement of it in this section (where it should be) or even in this article. It would not take up much more space to simply insert it, and that is what I advise be done. (Ahfretheim (talk) 00:24, 7 February 2015 (UTC))

You are correct that the Lorentz factor is undefined, so I will link to it and insert the explicit expression. Thanks for catching, M∧Ŝc²ħεИ_τlk 17:02, 7 February 2015 (UTC)

Lorentz force in terms of potentials

...and using an identity for the triple product simplifies to

$\mathbf {F} =q\left[-\nabla \phi -{\frac {\partial \mathbf {A} }{\partial t}}+\nabla (\mathbf {v} \cdot \mathbf {A} )-(\mathbf {v} \cdot \nabla )\mathbf {A} \right]$

Not sure this is correct. Should be $\nabla _{\mathbf {A} }(\mathbf {v} \cdot \mathbf {A} )$ (Feynmann subscript notation). Basically because operators like Nabla don't commute like vectors do, so you can't just lift the identity for the vector triple product and replace $\mathbf {B}$ with $\nabla$.

Joleneth (talk) 15:48, 28 April 2015 (UTC)

True you can't use the vector identities with the nabla operator (or at least one shouldn't). Going through the calculation:

$\mathbf {F} =q\left[\mathbf {E} +(\mathbf {v} \times \mathbf {B} )\right]$

$\mathbf {E} =-\nabla \phi -{\dfrac {\partial \mathbf {A} }{\partial t}}$

$\mathbf {B} =\nabla \times \mathbf {A}$

$\mathbf {F} =q\left[-\nabla \phi -{\dfrac {\partial \mathbf {A} }{\partial t}}+(\mathbf {v} \times (\nabla \times \mathbf {A} ))\right]$

using tensor index notation for the triple cross product (much better and more general than Feynman's notation):

$\left(\mathbf {v} \times (\nabla \times \mathbf {A} )\right)_{i}=\varepsilon _{ijk}v_{j}\varepsilon _{k\ell m}\partial _{\ell }A_{m}=\varepsilon _{ijk}\varepsilon _{\ell mk}v_{j}\partial _{\ell }A_{m}$

$=\left(\delta _{i\ell }\delta _{jm}-\delta _{im}\delta _{j\ell }\right)v_{j}\partial _{\ell }A_{m}$

$=\left(v_{j}\partial _{i}A_{j}-v_{j}\partial _{j}A_{i}\right)$

$=\left(\partial _{i}A_{j}v_{j}-v_{j}\partial _{j}A_{i}\right)$

$=\left((\nabla \mathbf {A} )\cdot \mathbf {v} -(\mathbf {v} \cdot \nabla )\mathbf {A} \right)_{i}$

where $(\nabla \mathbf {A} )_{ij}=\partial _{i}A_{j}$ is a (tensor) dyadic product and the dot is a tensor contraction $[(\nabla \mathbf {A} )\cdot \mathbf {v} ]_{i}=(\partial _{i}A_{j})v_{j}$, in all

$\mathbf {F} =q\left[-\nabla \phi -{\dfrac {\partial \mathbf {A} }{\partial t}}+(\nabla \mathbf {A} )\cdot \mathbf {v} -(\mathbf {v} \cdot \nabla )\mathbf {A} \right]$

so if you mean $\nabla _{\mathbf {A} }(\mathbf {v} \cdot \mathbf {A} )=\nabla _{\mathbf {A} }(\mathbf {A} \cdot \mathbf {v} )=(\nabla \mathbf {A} )\cdot \mathbf {v}$ then you are correct on that term, but the antisymmetry includes the other term. M∧Ŝc²ħεИ_τlk 13:22, 15 May 2015 (UTC)

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"Discussion of units out of place"

[2] No, it is not. The problem is that the article keeps saying "in SI units" without it becoming clear why the choice of units should matter, until the reader encounters a separate h2 section, tucked in after "History", which gives the explanation. It is indeed "out of place" to give this discussion in such a random place. The discussion, which was indeed in the article as it stood, should be provided when "units" are first mentioned, or people will not know what the units are about.

This is an important point, and a source of confusion, and anything but "out of place". To quote these lecture notes I just googled (emphasis mine),

"This is a very important difference! It makes comparing magnetic effects between SI and cgs units slightly nasty. Notice that, in cgs units, the magnetic field has the same overall dimension as the electric field: v and c are in the same units, so B must be force/charge. For historical reasons, this combination is given a special name: 1 dyne/esu equals 1 Gauss (1 G) when the force in question is magnetic. (There is no special name for this combination when the force is electric.) In SI units, the magnetic field does not have the same dimension as the electric field: B must be force/(velocity × charge). The SI unit of magnetic field is called the Tesla (T): the Tesla equals a Newton/(coulomb×meter/sec). To convert: 1 T = 10⁴G [not strictly true, they are talking about the numerical value only, dab].

--dab (𒁳) 12:01, 13 January 2017 (UTC)