# Talk:Relativistic electromagnetism

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## Regrets (I have a few!)

I really wish Id not started this page now as it is too complex a subject to illustrate without diagrams that I cant draw! Suggestions on how to get out of this mess will be most welcome!--Light current 17:51, 30 April 2006 (UTC)

Diagrams will be inserted to illustrate the text ASAP--Light current 16:03, 21 October 2005 (UTC)

Having trouble getting the equations right. Please bear with me (or help!)--Light current 17:26, 21 October 2005 (UTC)

Lots more work needed on article especially with diagrams & equations. Please bear with me.--Light current 01:46, 22 October 2005 (UTC)

Started work on diagrams. Any help from persons with drawing packages gratefully recieved.--Light current 21:45, 22 October 2005 (UTC)

## Request for help

I really need someone to help on improving my very basic (and not ver satisfactory) text based diagrams in this article. Any offers? You will be rewarded with thanks!!--Light current 00:30, 23 October 2005 (UTC)

## Cut from article to be worked on

Using angle brackets to denote an average over all points on the shell,

�sin2 θ� = �1 − cos2 θ� = 1− �x2� R2 .

Now since the origin is at the center of the sphere, the average value of x2 is the same as the average value of y2 or z2: �x2� = �y2� = �z2�. (4.9)

But this implies that �x2� = 1 3�x2 + y2 + z2� = 1 3�R2� = R2 3

since x2 +y2 +z2 = R2 and R is constant over the whole shell. Combining equations (4.8) and (4.10) gives

�sin2 θ� = 1− R2 3R2 = 2 3. (4.11) 4.3

So the average energy per unit volume stored in the transverse electric field is

q2a2 48π2�0c4R2.


## The physical basis of electromagnetic radiation

                \       wall
\     |
\    |
\   |
\  |
\ |
\|B
/|\
/ | \
/  |  \
A/   |   \C


Consider a positively charged particle, initially traveling to the right at 1/4 the speed of light. It bounces off a wall at point B. The particle is now at point A, but if there had been no bounce it would now be at C. An imaginary circle (actually a cross section of a sphere) encloses the region of space where news of the bounce has already arrived; inside this circle the electric field points directly away from A. Outside the circle the news has not yet arrived, so the field points directly away from C. As time passes, the circle expands outward at the speed of light, and points A and C move away from B at 1/4 the speed of light. The field at your location points away from where the particle would be now if there had been no bounce. We know from special relativity that no information can travel faster than the speed of light

Assume that the information travels at precisely the speed of light, but no faster. This assumption, together with Gauss’s law, is enough to determine the electric field everywhere around the accelerated charge. The complete map of the electric field of an accelerated charge turns out to be quite complicated. An abstract representation in terms of field lines will be used instead. Field lines are continuous lines through space that run parallel to the direction of the electric field. A drawing of the field lines in a region therefore indicates immediately the direction of the electric field.

A map of the field lines for the situation of figure is shown in figure . No field lines through the gray spherical shell in figure are shown, since this is the region that is just in the midst of receiving the news of the particle’s acceleration. To determine the direction of the field here, imagine a curved Gaussianpillbox”, indicated by the dashed line in the figure, which straddles the gray shell. This surface is symmetrical about the line along which the particle is moving; but viewed from along this line, it would be circular.

The Gaussian surface encloses no electric charge, so Gauss’s law tells us that the total flux of �E through it must be zero.

The direction of the field within the gray spherical shell can be found be considering the flux through the curved Gaussian “pillbox” indicated by the dashed line. Now consider the flux through various parts of the surface. On the outside (right-hand) portion there is a positive flux, while on the inside (left-hand) portion there is a negative flux. But these two contributions to the flux do not cancel each other, since the field is significantly stronger on the outside than on the inside. This is because the field on the outside is that of a point charge located at C, while the field on the inside is that of a point charge located at A, and C is significantly closer than A. The net flux through the inside and outside portions of the surface is therefore positive. To cancel this positive flux, the remaining edges of the pillbox must contribute a negative flux. Thus the electric field within the gray shell must have a nonzero component along the shell, in toward the center of the Gaussian surface. This component is the transverse field, since it points transverse (i.e., perpendicular) to the purely radial direction of the field on either side. To be more precise about the direction of the field within the gray shell, consider the modified Gaussian surface shown in figure.

Since the flux along segment cd must be zero, the electric field within the gray shell must be parallel to this segment. until it subtends the same angle, as viewed from C, that the inner surface ab subtends as viewed from A. Now the fluxes through ab and ef do indeed cancel. Segments bc and de are chosen to be precisely parallel to the field lines in their locations, so there is no flux through these portions of the surface. In order for the total flux to be zero, therefore, the flux must be zero through segment cd as well. This implies that the electric field within the gray shell must be parallel to cd. If you start at A and follow any field line outward, you will turn a sharp corner at the gray shell’s inner edge, then make your way along the shell and slowly outward, turning another sharp corner at the outer edge. The thickness of the gray shell is determined by the duration of the acceleration of the charge.

A complete drawing of the field lines for this particular situation is shown in figure. The transverse portion of the electric field of an accelerated charge is also called the radiation field, because as time passes it “radiates” outward in a sphere expanding at the speed of light. If the acceleration of the charged particle is sufficiently great, the radiation field can be quite strong, affecting faraway charges much more than the ordinary radial field of a charge moving at constant velocity. The radiation field can also store a relatively large amount of energy, which is carried away from the charge that created it.

### Strength of the radiation field.

A complete sketch of the electric field lines for the situation shown in the preceding figures, including the transverse radiation field created by the acceleration of the charge.

Consider a somewhat simpler situation, in which a positively charged particle, initially moving to the right, suddenly stops and then remains at rest. Let v0 be the initial speed of the particle, and let the deceleration begin at time t = 0 and end at time t = t0. Assume that the acceleration is constant during this time interval; the magnitude of the acceleration is then a = |�a| = v0 t0. Assume that v0 is much less than the speed of light, so that the relativistic compression and stretching of the electric field discussed above is negligible. Figure shows the situation at some time T, much later than t0. The “pulse” of radiation is contained in a spherical shell of thickness ct0 and radius cT. Outside of this shell, the electric field points away from where the particle would have been if it had kept going; that point is a distance v0T to the right of its actual location. (The distance that

R = cT v0T sin θ ct0 θ v0T θ

For clarity, only a single field line is shown here. it traveled during the deceleration is negligible on this scale. A single field line is shown in the figure, coming out at an angle θ from the direction of the particle’s motion. There is a sharp kink in this line where it passes through the shell, as discussed in the previous section.

The radial component Er of the kinked field can be found by applying Gauss’s law to the pillbox shown. How strong is the electric field within the shell? Break the kinked field up into two components: a radial component Er that points away from the location of the particle, and a transverse component Et that points in the perpendicular direction. The ratio of these components is determined by the direction of the kink; See that Et Er = v0T sin θ ct0 = aT sin θ c

We can find the radial component Er by applying Gauss’s law to a tiny pillbox that straddles the inner surface of the shell. Let the sides of the pillbox be infinitesimally short so that the flux through them is negligible. Then since the net flux through the pillbox is zero, the radial component of �E (that is, the component perpendicular to the top and bottom of the pillbox) must be the same on each side of the shell’s inner surface. But inside the sphere of radiation the electric field is given by Coulomb’s law. Thus the radial component of the kinked field is Er = 1 4π�0 q R2 , where q is the charge of the particle. Combining equations and using the fact that R = cT , it can be shown that Et = qa sin θ 4π�0c2R.

Looking back at the figure, we see that the size of the kink in the field is a qualitative indication of the field strength. Also, the strength of the transverse field is proportional to a, the magnitude of the particle’s acceleration. The greater the acceleration, the stronger the pulse of radiation. This pulse of radiation carries energy. The energy per unit volume stored in any electric field is proportional to the square of the field strength. This implies Energy per unit volume ∝ a2 R2.

Since the volume of the spherical shell (the shell itself, not the region it encloses) is proportional to R2, the total energy it contains does not change as time passes and R increases. Thus when a charged particle accelerates, it loses energy to its surroundings, in an amount proportional to the square of its acceleration. This process is the basic mechanism behind all electromagnetic radiation: visible light and radio waves to gamma rays.

### The Larmor Formula

#### Energy radiated by a charged particle

In any electric field, the energy store per unit volume is epsilon_0/2 E^2

The total energy in the pulse is:

${\displaystyle Ep={\frac {e^{2}a^{2}t}{6\pi \epsilon _{0}c^{3}}}}$ where t is the duration of the acceleration

If we divide both sides by the duration t of the particles acceleration, we obtain the power: P.

${\displaystyle P={\frac {e^{2}a^{2}}{6\pi \epsilon _{0}c^{3}}}}$

where a is the acceleration, e is the electronic charge, epsilon0 is the permittivity of free sapce and c is the speed of electromagnetic radiation This is the Larmor formula

If the direction in which the energy goes is not important, we can average equation (4.7) over all directions. Using a mathematical device, introduce a coordinate system with the origin at the center of the sphere and the x axis along the particle’s original direction of motion. Then for any point (x, y, z) on the spherical shell, cos θ = x/R.

To obtain the total energy stored in the transverse electric field, we must multiply equation by the volume of the spherical shell. The surface area of the shell is 4πR2 and its thickness is ct0, so its volume is the product of these factors. Therefore the total energy is Total energy in electric field = q2a2t0 12π�0c3 . The total energy is independent of R; that is, the shell carries away a fixed amount of energy that is not diminished as it expands. There is also a magnetic field, which carries away an equal amount of energy. Many details about magnetic fields have been omitted. A factor of 2 would need putting in. Thus the total energy carried away by the pulse of radiation is twice that of the previous equation or

When a charged particle accelerates, part of its electric field breaks free and travels away at the speed of light, forming a pulse of electromagnetic radiation. Often, in practice, charged particles oscillate back and forth continuously, sending off one pulse after another in a periodic pattern.

An example of the electric field around an oscillating charge is shown in figure. A map of the electric field lines around a positively charged particle oscillating sinusoidally, up and down, between the two gray regions near the center. Points A and B are one wavelength apart. If you follow a straight line out from the charge at the center of the figure, you will find that the field oscillates back and forth in direction. The distance over which the direction of the field repeats is called the wavelength. For instance, points A and B in the figure are exactly one wavelength apart. The time that it takes the pattern to repeat once is called the period of the wave, and is equal to the time that the source charge takes to repeat one cycle of its motion. The period is also equal to the time that the wave takes to travel a distance of one wavelength. Since it moves at the speed of light, we can infer that the wavelength and the period are related by

speed = wavelength period or c = λ T ,

where λ (“lambda”) is the standard symbol for wavelength, T is the standard symbol for period, and c is the speed of light.

### Electromagnetic Waves

When a charged particle accelerates, part of its electric field breaks free and travels away at the speed of light, forming a pulse of electromagnetic radiation. Often, in practice, charged particles oscillate back and forth continuously, sending off one pulse after another in a periodic pattern. An example of the electric field around an oscillating charge is shown in figure. A B

If you follow a straight line out from the charge at the center of figure you will find that the field oscillates back and forth in direction. The distance over which the direction of the field repeats is called the wavelength. For instance, points A and B in the figure are exactly one wavelength apart.

If you sit at a fixed point and watch the electric field as it passes by, you will again find that its direction oscillates. The time that it takes the pattern to repeat once is called the period of the wave, and is equal to the time that the source charge takes to repeat one cycle of its motion. The period is also equal to the time that the wave takes to travel a distance of one wavelength. Since it moves at the speed of light, we can infer that the wavelength and the period are related by

speed =wavelength.period

or c = λT

where λ (“lambda”) is the standard symbol for wavelength, T is the standard symbol for period, and c is the speed of light. The frequency of an oscillation or a wave is the reciprocal of the period.

extracted/modified from Daniel V Schroeder paper

## Off the beaten track in Wikipedia

As the person who started it, I tend to agree with you. It was a mistake. But maybe some bits of it might be saved for use in other articles.--Light current 22:59, 21 November 2006 (UTC)
OTOH, maybe it could be pruned down to the bare min to explain the concept only. Other more knowledgable editors may then wish to expand it properly in the fullness of time!--Light current 23:18, 21 November 2006 (UTC)

## How do we know that Purcell was the first to take this approach to electromagnetism & relativity ?

Are there any sources that show conclusively that this had never been done before Purcell ?

The reason I ask this is that I attended a brilliant lecture course in 1968 (yes - that's after Purcell) on electromagnetism & relativity in which the lecturer derived all of Maxwell's equations from just two postulates: the invarance of c (velocity of light) and e (charge of an electron) through all the standard transformations of special relativity. I don't recall any reference having been made to Purcell's work, though it's feasible that the lecturer had made use of some of his material. My suspicion is that the ideas go back a lot earlier than 1963. (Incidentally, I'm currently reading Subtle is the Lord by Abraham Pais – it was that which led me to browse Wikipedia for articles related to electromagnetism & relativity and hence find this one.) DFH 20:27, 23 November 2006 (UTC)

Thats what Daniel V Schroeder said in his paper on this subject.--Light current 23:41, 23 November 2006 (UTC)
In my 1956 edition of the "Encyclopaedia Britannica" (before Purcell) article on "electricty" v8, p159, it is shown how the "magnetic force" between two parallel wires of current is really an electrostatic attraction due to special relativity. However, the math only went as far to show qualitatively that special relativity demands: A/R = (i/c)^2 where A=attraction due to i, R = repulsion between free electrons. Understanding the connection is basic to a good theoretical understanding of special relativity. Feynman in his famous 1964 physics course derives this relationship in v2 13-6. Ywaz (talk) 17:45, 17 July 2010 (UTC)

## Source material

Here [1] is the source material for this article. See if YOU can rewrite it--Light current 03:29, 1 January 2007 (UTC)

## Another explanation on "The origin of magnetic forces"

I have a more simple explanation on "why" magnetic forces appear. However, I'm not sure whether it's correct or not:

Consider two identical charged objects with each with mass m moving at the same velocity v at a distance 2x from each other perpendicular to the velocity. Since their time slows down when they move, they repel each other less (slower) than we expect them to. Therefore, we say there's a magnetic force counteracting the repulsive electric force.

Mathematically: The objects' masses, charges and distance are the same in their reference frame as in the observer's. Due to time dilation, however, the acceleration - and therefore also the electric forces - observed in the observer's frame (primed) will be different:

${\displaystyle F'=m\cdot a'=m\cdot {\frac {d^{2}x}{d(t')^{2}}}=m\cdot {\frac {d^{2}x}{dt^{2}}}\cdot (t/t')^{2}=m\cdot a\cdot {\sqrt {1-(v/c)^{2}}}^{2}=F\left(1-(v/c)^{2}\right)}$

The observer concludes that there must be a magnetic force accounting for the difference:

${\displaystyle F_{m}=F'-F=F\left(1-(v/c)^{2}\right)-F=-(v/c)^{2}F}$

Pellishau (talk) 13:04, 18 January 2008 (UTC)

## Electricity from magnetism?

This article describes how magnetic fields originate from electrostatics and special relativity. However, it is not apparent to me how the same principles can be used to explain how magnetic fields create electrical currents. This seems like an important subject that needs to be addressed to complete this article.

Just because when you have an electric field from magnetic one (relation can be reverse), you then have a voltage difference, or electromotive force. — Preceding unsigned comment added by Klinfran (talkcontribs) 08:00, 1 September 2011 (UTC)

192.104.67.122 (talk) 18:17, 4 August 2009 (UTC)db —Preceding unsigned comment added by 192.104.67.122 (talk) 17:05, 4 August 2009 (UTC)

Collapsed conversation—not really on topic on artile talk page anymore

I think its best to severely down-grade the term magnetism; what this is all about is that electric charge in motion and relativity explains everything called magnetism. Magnetism is just the name given to a force that wasn't seen as electric force because relativity hadn't been understood.

So the question 'how magnetic fields create electrical currents' is sort of, how an electric charge causes a force on another charge (which is just how it is), and is also then exactly the same physics as the other way round, ie 'how electrical currents create magnetic fields' they both always exist in unison.

(note.1 changing the viewers velocity changes the interpretation of what is and what isn't a magnetic field, because its a poor non-fundamental conceptualisation of the physics.) (note.2 the term electromagnetism is fundamentally replaced by electro-weak, because the weak nuclear force has the same basis as electric force, magnetism is just a relativistic side-effect of a sub-category of a 'real' force.)

also the 'causes' part of the question isn't quite clear, 'cause' can really only refer to the way energy moves though the system, because the physics of the currents and magnetic fields do not contain causality, better to think of them as a single thing, and that energy (driving changes in the system) can be added, (the 'cause'), too and/or taken from, (the 'effect'), many different places in this thing, one or more interpreted as a magnetic effect, or not, depending of the viewers relative speed.

so basically, the use of the term Magnetism, only for historical reference and possibly an example of an evolution in understanding, would have great benefits to education.—Preceding unsigned comment added by 86.175.21.115 (talk) 23:24, 30 January 2011 (UTC)

Are you arguing that the electricity is a more fundamental force of nature than magnetism?
absolutely, i guess you didn't really understand this then.
If so, could you explain why, and what reliable sources support this?
and you ignore Einstein's quote, at the top of the article, why?
All you're saying above is, "starting with electricity and special relativity, you can derive that there has to be magnetism". Yes, OK. It is also true that "starting with magnetism and special relativity, you can derive that there has to be electricity". Actually we know that neither electricity nor magnetism is a fundamental force of nature: They are emergent phenomena. The true law of nature is quantum electrodynamics (QED), and in the classical limit QED gives rise to both electricity and magnetism simultaneously. Again, the derivation is "QED --> Electromagnetism", not "QED --> Electric force, then Electric force --> Magnetic force".
well apart from QM having nothing to do with it, this is all classical, i remember this from Feynmans lectures, and didn't he basically invent QED.
as a visual aid try:- (Q + relativity + QM)== QED
and see magnetism is not needed at all
The stuff in this article, Purcell's textbook, etc., is good pedagogy, because people find the electric force intuitive and special relativity sort-of-intuitive-eventually, but somehow the magnetic force is much more unintuitive. So it's helpful to show students that if you "believe in" electricity and "believe in" special relativity than you have to also "believe in" magnetism. But you shouldn't get carried away and think that as a matter of fundamental physics (not physics pedagogy) the electric force is a more basic and fundamental force in the universe than the magnetic force.
the 'unintuitiveness' of magnetism IS the 'unintuitiveness' of relativity.
try picturing the Magnetic force like the Coriolis force, each a lower-dimensional 'fictitious' force of a more fundamental higher-dimensional reality.
By the way, if you're more interested in practical matters, please feel free to try to write out all the laws of electromagnetism (at least including Maxwell's equations and the Lorentz force) without mentioning magnetism or the magnetic field, and see whether this way they become simpler or more complicated. I think you will find it is much more complicated! If you finish that exercise, a more challenging exercise is to try to explain spin–orbit coupling without mentioning magnetism. :-) --Steve (talk) 01:44, 31 January 2011 (UTC)
again forget QM, not relevant at all, really do yourself a favor and try harder to get this, its a really good one to understand. Feyman lectures are good on this.
this has a lot of parallels with; Energy == Mass, and for pretty much the same reasons, and that, i'd have to say is MUCH harder to get intuitively.
and so educationally, 'getting' this concept is immensely powerful.
In the future could you please write blocks of text instead of interspersing your comments? It makes it easier for people to read the whole conversation later on. Thank you.
i disagree
Here is Richard Feynman answering the question "Why do bar magnets attract each other?" His answer is basically, Because electrical forces and magnetic forces are fundamental forces of nature. He does not mention special relativity at all, and he talks on and on about what is the ultimate cause of magnetic forces but never says that the answer is electrical forces. In fact he clearly puts electrical forces and magnetic forces on the same level, just saying they're intimately related (which I certainly agree with). I don't think I've seen Feynman say anywhere that electricity is more fundamental and magnetism is less fundamental, and I read his whole "Lectures" textbook series. What page number of what book does he say this?
from memory, a very long time back, middle of book 3, but you have to understand it not just read it.
You are making the argument that magnetism is a less fundamental force of nature than electricity, and that magnetism is ultimately caused by electricity. It's shocking that you would think QED, which is our most fundamental understanding of electromagnetism, has nothing to do with whether this is true or false. I'm saying "Most basic and fundamental laws of physics ==> Electromagnetism" and you're saying "Most basic and fundamental laws of physics ==> Electricity ==> Magnetism". We can't decide on this question if you're unwilling to discuss the "Most basic and fundamental laws of physics", namely the standard model of particle physics including QED (or maybe string theory from which you derive the standard model).
Could you please explain "(Q + relativity + QM)== QED"? What does "Q" mean?
charge.
The Einstein quote is emphasizing that a magnetic force can also be viewed as an electric force. I don't see how he's saying that the first view is wrong (or narrow-minded or whatever) and the second view is right. But it's hard to say for sure in a quote from a 3-paragraph commemorative letter. Didn't Einstein write other things too? For example, if Einstein that thought electricity was fundamental and magnetism was unnecessary, then he sure didn't say so in his lecture notes on electromagnetism. His memorable moving magnet and conductor example is notably symmetric in its treatment of electricity and magnetism: You can view it as an electric force or a magnetic force, but he didn't say that one view was more correct than the other. In any case, it doesn't really matter because the fundamental nature of classical electromagnetism (namely, that it is fundamentally nothing more than an approximation to QED) was not understood in Einstein's day.
The Coriolis effect is indeed a "fictitious force", and not coincidentally if you use an inertial frame of reference the Coriolis effect disappears from the laws of physics, which become much simpler as a consequence. On the other hand, there is no inertial frame of reference where magnetism disappears from the laws of physics,
you need to think 4-dimensionally.
and as I said above, if you try to write Maxwell's equations (let alone the spin-orbit coupling equation)
Maxwells equations are formulated in terms of magnetism, they now just disappear, or rather ARE an expression of relativity, so this is the ultimate simplification. This is often referred too as a unification.
without explicitly mentioning magnetism, it becomes not simpler but massively more complicated and very contrived. That's part of the reason I reject your analogy between the Coriolis force and the magnetic force. --Steve (talk) 20:56, 5 February 2011 (UTC)
there must be plenty of resources on the internet that explain this better than i can.
http://en.wikibooks.org/wiki/Electrodynamics/Relativistic_Electromagnetism
http://easther.physics.yale.edu/Richard_Easther/Relativistic_E_and_M.html
http://www.mtholyoke.edu/courses/tdray/phys310/electromag.pdf
of course the article itself lists many resources that use this as the basis for teaching, so read any of those.
Maybe I should be more specific. I AGREE that electricity and magnetism are intimately linked together by special relativity. I AGREE that an electric force in one frame of reference may be a magnetic force in another frame of reference or vice-versa. These things are basic and uncontroversial. I DISAGREE that electricity is more fundamental and magnetism is less fundamental. I believe that the vast majority of physicists would also disagree. I see electricity and magnetism as being like Siamese twins: Equally important and linked together. You are saying they are "Parent and child", where electricity is the cause and magnetism is the effect. When I say "Siamese twins", I mean something similar to space and time. No one would say "According to special relativity, the reason that time exists is because of space", and no one would say "According to special relativity, the reason that space exists is because of time". Instead they say "According to special relativity, space and time are linked together into spacetime and neither is possible without the other". My objection is to the asymmetry in your point of view.
In special relativity, the electric field and magnetic field are put together into a single object, the electromagnetic tensor F. The whole beauty of special relativity is in illuminating how the different components of a covariant tensor like F are all linked together. It goes against everything in special relativity to say that six of the nonzero components of F are the "fundamental cause" and the other six nonzero components of F are the "non-fundamental effect". But that's exactly what you're saying. You're saying that ${\displaystyle F^{01},F^{02},F^{03}}$ (the electric field) is a fundamental and important part of the universe, whereas ${\displaystyle F^{12},F^{23},F^{13}}$ are non-fundamental effects which only exist because of the components ${\displaystyle F^{01},F^{02},F^{03}}$. The insights of special relativity show that it makes no sense to think along those lines: All the tensor components are interconnected into a single whole that should not be split up conceptually any more than it can be split up physically.
Why are you posting links like this one? This does not say anywhere that electricity is more fundamental and magnetism is less fundamental. It says that the two are intimately linked together in special relativity, something that, again, I understand very well. Feynman's textbook says the same thing. Again, I am saying electricity and magnetism are "Siamese twins", and you are saying they are "Parent and child". You are welcome to try to find a page number in Feynman that says that electricity is the parent and magnetism is the child. I cannot find any. I find lots of discussion that implies they are Siamese twins.
When you say, "(Q + relativity + QM)== QED", I guess you're saying that combining the laws of classical electrostatics with the laws of special relativity and the laws of non-relativistic quantum mechanics, you get the laws of QED.
Q(quantum==QM) E(electro==Q) D(dynamics==relativity)
Are you really a QED expert? Because I have studied QED, and I believe your statement is completely false. QED contains much much more than just a concatenation of the previously-known laws of physics. QED does not come from the previously-known laws of physics: That's backwards. The previously-known laws of physics come from QED (plus the rest of the standard model). I hope you agree that this is fundamentally the correct way to think about the laws of physics: The most exact laws are always the most fundamental laws. --Steve (talk) 23:56, 5 February 2011 (UTC)
maybe i didn't make this clear enough; the question you originally asked, at the top, is basically a product of the misunderstanding of magnetism, until you can allow for that, there will never be an answer to the question, because the question has no meaning.
BTW QED is consistent with relativity, does not 'produce' it, if it did it would predict gravity.
anyway if you've done such a course, recently?, (because they were really rare things when i did one.) wouldn't you be able to get hold of a real physicist (not one of those self styled quantum-mecahanicists) to go through this with you, because (and you're going to hate this, but its really true.) its more fundamental than QED. i really appreciated the day, and can actually remember the circumstances when i suddenly could 'see' this.
How's this: The professor who taught my QED/QFT course has recently posted a long rant on this exact topic: [2]. He was a professor of theoretical physics at Harvard University, I hope that counts as a "real physicist". --Steve (talk) 21:39, 7 February 2011 (UTC)
Yes thats a rant, and a completely useless way to answer a question.
and he doesn't ever appear to have been a professor.
interesting link further down thought, should be included here; http://physics-quest.org/Magnetism_from_ElectroStatics_and_SR.pdf
it claims to derive magnetism from charge+relativity, in a way that covers the current being produced by charges with a general velocity distribution, something that i have worried about, i'll check through it sometime. —Preceding unsigned comment added by 86.175.122.183 (talk) 16:28, 8 February 2011 (UTC)
Lubos was indeed an assistant professor at Harvard for a number of years, including when he taught my QFT class. I don't know why you're arguing against facts that can be easily checked. OK, here is proof.
i really dont want to disparage qualifications but just look up what an "assistant professor" and a "professor" are, particularly look up the AMERICAN meaning of the terms.
Can you please clarify a few points for me:
• In inertial reference frame A, a particle feels a magnetic force. In inertial reference frame B, a particle feels an electric force. Do you agree or disagree with this statement: "The physical analysis in frame B is fundamentally correct, and the physical analysis in frame A is fundamentally incorrect."
There're both right, that's because Magnetism is really only a 'name' mistakenly given the electric force, when viewed from a moving frame and not analysing it relativistically. again compare with 'Coriolis force' its 'real' in an accelerating 2-dimension frame, because that's how its defined, it has no other meaning.
• I am holding two bar magnets stationary and near each other on my desk. They are attracting each other. Do you agree or disagree: "In the correct inertial reference frame, the force between the magnets can be entirely explained as electrical attraction." If you agree, what is the velocity of this inertial frame? (Relative to the rest frame of the desk and the magnets.)
NO, in a magnet with 'poles' the charges are moving in various directions, (think of a solenoid instead of a bar magnet, its exactly the same fields. ) its only in the special case when the charges are all moving in the same direction is it possible to find a 3D reference frame where the force is interpreted as purely electric force.
The general solution is always 4-dimensional, try to get past 'reference frames', they are just a way to reduce the real world of 4-dimensions, to the 3-dimensions people feel happier with, and its that, physically meaningless process, that 'creates' the illusion of the magnetic field.
• Do you agree or disagree with this statement: "It is a conceivable possibility that magnetic monopoles exist in the universe."
thats a bit more interesting and involved, yes AND no, it hinges on a very clear definition of a monopole.
I generally assume its like 'spin' where the name is used because of a significant number of similarities with actual rotation, but it could have had, and might have been better, if it a new made-up name, to stop people extrapolating properties outside the QM world. So i wouldn't be surprised if a 'monopole' were found one day, but i don't think it would have significance here.
Obviously i think finding 'half a magnet' would be impossible.
I hope you answer these, it will help me understand your point of view. Thanks! --Steve (talk) 04:58, 10 February 2011 (UTC)
from my side, having seen and having myself struggled to get an intuitive grasp of relativity, your 'issues' with this are quite familiar.
if you can understand the problem below, then your relativistic understanding should be very well up to seeing this.
if two twins travel apart so that for each the other ages at a slower rate, and the same for a return journey, since this is all symmetrical, why don't their ages stay synchronised?

Steve and anon BT CENTRAL PLUS, would you please continue this on your talk page (for instance at User_talk:Sbyrnes321/Electricity from magnetism—feel free to move the conversation), as this is obviously never going to end? This back and forth between you seems no longer on topic here, since this talk page is not meant for discussing the subject or someone's understanding or misunderstanding thereof—see WP:TPG. Thanks. - DVdm (talk) 23:01, 10 February 2011 (UTC)

Ummmm, "talk page is not meant for discussing the subject or someone's understanding or misunderstanding thereof—" ?
this is a discussion page unless i'm doing this wrong, and this article is currently focused toward 'Relativistic electromagnetism' as a teaching scenario, the physics is contained elsewhere,(although the title should emphasise that really.) so is also about comprehending, in that context i think this discussion has a great deal of relevance, if only to display some of the problems. if the page had no focus on education i would agree. —Preceding unsigned comment added by 86.175.125.189 (talk) 00:35, 11 February 2011 (UTC)

Well I agree with DVdm, I started User talk:Sbyrnes321/Electricity from magnetism. :-) --Steve (talk) 01:30, 11 February 2011 (UTC)

Ok, I have collapsed the section again. As anon seems not to be prepared to adhere to talk page guidelines (no signatures and improper indentation), the above has become as good as unreadable anyway. Enjoy over there. Cheers - DVdm (talk) 12:20, 11 February 2011 (UTC)
I'm thrilled to see my original comment started such a discussion. My poor wording drove the discussion in a different direction than I intended. I agree the electricity and magnetism are the same thing looked at from two different viewpoints. But from a practical aspect, to make this point clear you need to start with one and derive the other. This article does that by starting with ampere's law and lorentz contraction combined with some vector mechanics to show that magnetism is the same as electricity. Instead of stating that the article should address how "magnetic fields create electrical currents", I should have said that the article should include a vector analysis using the principles already presented that shows why an accelerating or decelerating electrical charge field induces a neighboring electrical charge field to accelerate or decelerate. (or that an accelerating or decelerating electrical charge field resists its own acceleration or deceleration) — Preceding unsigned comment added by 192.104.67.222 (talk) 14:41, 30 November 2011 (UTC)

## Education and training

This article relates to the education and training of electrical and electronics engineers. The references show that in the 1960s there was a move to introduce special relativity into the teaching of electromagnetic fields. The textbook by Corson & Lorrain was perhaps the apex of this movement. Since the symmetries of relativity are exactly the symmetries of the EM field, there is good reason to treat the subjects together. However, at most, this program of study provides some intuition on how magnetic and electric fields are observer-dependent. Full derivation of the Maxwell equations is an overstatement. Some investigation is necessary to find why this initiative did not take hold. Edits to the page from today could expand from Purcell to the other sources, and explore engineering and physics education for traces of the movement.Rgdboer (talk) 01:53, 19 November 2009 (UTC)

The additional sources are now mentioned in the lead. Further, A. Einstein's comment on magnetism being electrical has been quoted and sourced. Scanning the sources for appropriate lesson plans may produce material to improve the article.Rgdboer (talk) 03:19, 20 November 2009 (UTC)

## Origin of magnetic force

I very doubt about the contraction of lenght that is exposed to talk about the origin of magnetic force, as it is described in most book. Why? First because the case is the one of an infinite wire, thus with an open finite wire, because of conservation of charges, if the whole negative lign, for example, contracts, then the tips will be positive, if you apply it to a closed loop, as there is speed of electron everywhere, and no motion of protons, you will have charge accumulations on some parts, very strange that charges are uniformly reparted in one frame and concentrated in another, this could be because of simultaneity, but simultaneity and time dilation could work in the opposite because charges have a certain pace, regular, to cross the wire section, and accelerate at each angle of the wire. Secondly does this contration is in addition to the transformation of electric field of each charge, that is also an outcome of lenght contration, then does both effects accumulate? This needs a simulation, because books have been written a long time ago.Klinfran (talk) 07:57, 1 September 2011 (UTC)

Do you say that accelecation can decrease the density? Why? Relativity says only about speed and only about contraction. Furthermore, it does not matter in which direction the charges move. So, acceleration, which only changes the direction, should not change anything. The neutral wire should become negatively charged when current is created in it. --Javalenok (talk) 12:43, 7 May 2013 (UTC)

## Something is wrong with the magnetic induction B

Extended content

When we analize the interaction between two current carying conductors we first represent the magnetic induction "B" following the right hand rule. But, if you look back at the origine of this representation you can find that it comes from Faraday and the orientation of the magnetic induction "B" was based on the pattern of iron filings formed around a magnet bar which resemble to " field of lines" conneting the two poles. If you look closer the iron piling particles also can be considered tini magnets and their orientation will always be that their internal atomic currents that creates their magnetic field will be aligned with the internal atomic currents of the large permanent magnet bar so they will tend to have the same direction ! The independent "lines" are formed because the adjacent particles of iron will align their internal currents with the stronger current of the large permanent magnet bar while between the "lines" there will be repulsion, also particle of iron from a "line" will form chains with the top and the bottom particle so that their internal currents will be aligned too. So now from the two parallel electric current situation we arrive to another two parallel electric current situation! There is no such a thing like magnetic induction "B" that is tangent to the conductors or circle the conductor...its just an illusion ! Can you see the error? — Preceding unsigned comment added by 71.185.132.57 (talk) 01:00, 30 June 2013 (UTC)

This is where we discuss format and content of the article, not (parts of) the subject — see wp:talk page guidelines. Questions like this might be welcome at the wp:reference desk/science. Good luck. - DVdm (talk) 11:23, 1 July 2013 (UTC)

## Electromagnetic theory of radiation is still valid?

If the magnetic field is really an electric filed doesn't this mean that the electromagnetic propagation of radiation is invalid? — Preceding unsigned comment added by 71.185.132.57 (talkcontribs) 03:01, 7 July 2013 (UTC)

DVdm has already indicated to you in the previous thread that this is not the forum for this kind of discussion; this question does not relate to editing the article. Also do not remove talk page content as you did, and rmeber to sign your posts on talk ages with for tildes: (~~~~). Please follow the suggestions. — Quondum 03:14, 7 July 2013 (UTC)

## Introductory sentence

"Relativistic electromagnetism is a modern teaching strategy for developing electromagnetic field theory from Coulomb’s law and Lorentz transformations." This statement would be more rigorously complete if it included conservation (invariance) of charge as a premise. Should it be changed? — Quondum 20:15, 29 July 2013 (UTC)

Yes, it is a good idea to get away from Coulomb’s law when doing relativity, even if the references use the electrostatic principle as a backdrop. Suggestion to focus on charge conservation is worthy. However, the topic is fraught with difficulties though well motivated. Caution should be taken regarding electro-mechanics as a deductive system.Rgdboer (talk) 22:36, 29 July 2013 (UTC)

I'm not sure we understand each other. I was not trying to change the concept, only saying that in the deductive process, there is an unstated assumption of charge conservation. Charge conservation + Coulomb's law + Lorentz transformations => Relativistic electromagnetism. If there is no charge conservation, as in relativistic mass + Newton's law of gravitation + Lorentz transformations, we get a different set of laws (being something like gravitoelectromagnetism). I was simply suggesting something like "Relativistic electromagnetism is a modern teaching strategy based on developing electromagnetic field theory from charge conservation, Coulomb’s law and Lorentz transformations." — Quondum 22:56, 29 July 2013 (UTC)

Conservation of charge is not so much the point as relative charge density. On the topic of relativistic electromagnetism, I have suggested that the eddy current brake is instructive. On March 5, someone logged in just to delete that suggestion, making no other contribution. At its best, this article may help students begin to appreciate EM field theory.Rgdboer (talk) 22:33, 30 July 2013 (UTC)

## Opposite charges with opposite motions and matching length contraction

In a frame of reference where length contractions of positive and negative charges in a neutral wire are non-zero and identical, it would seem that any charge whose frame deviates from this frame would receive a magnetic force. If the positive charges of the metal ions in a typical conductor are stationary in the lab frame (or "lab-stationary"), this would mean that in the lab frame the magnetic field "loops" propagate (or "drift") at half the drift velocity as the electron current (or approximately half as per the relativistic velocity addition formula). Therefore, even for a current flowing at a constant rate in a lab-stationary wire, the magnetic field produced is non-lab-stationary!

If, in the lab frame, you apply magnetic Lorentz force q(v x B) on a lab-stationary charge q via a magnetic field B traveling at this "midway" velocity v between that of the metal ions and that of the electron current, what you will find is that the otherwise neutral wire in the lab frame will have electric fields lines pointing to it (due to the negativity of conducting charge) which are proportional to (v x B) which, ceteris paribus, is proportional to v^2, or alternatively B^2.

Therefore, the distance between electrons in a wire does not have to decrease due to their relative motion with respect to the observer in order to explain this electric field. The increase of the electric field can be explained just as well by saying that the magnetic field of the wire is stationary only in the frame of reference between that of the metal ions and that of the electrons, and therefore an electric field is produced from an otherwise neutral wire via the magnetic Lorentz force on lab-stationary charges.

In the case of completely negative electron beam, the magnetic field of the beam travels at the velocity of the beam, and any lab-stationary charge would be subject to a magnetic Lorentz force based on an electric field that points toward the beam. The induced electric field complements the electric field of the charged particles of the beam, which in this special case are electrons.siNkarma86—Expert Sectioneer of Wikipedia
86 = 19+9+14 + karma = 19+9+14 + talk
08:55, 15 March 2014 (UTC)

## Old content

After almost 5 years with no support with references, the following sections have been removed today:

### Uniform electric field — simple analysis

Figure 1: Two oppositely charged plates produce uniform electric field even when moving. The electric field is shown as 'flowing' from top to bottom plate. The Gaussian pill box (at rest) can be used to find the strength of the field.

Consider the very simple situation of a charged parallel-plate capacitor, whose electric field (in its rest frame) is uniform (neglecting edge effects) between the plates and zero outside.

To calculate the electric field of this charge distribution in a reference frame where it is in motion, suppose that the motion is in a direction parallel to the plates as shown in figure 1. The plates will then be shorter by a factor of:

${\displaystyle {\sqrt {1-v^{2}/c^{2}}}}$

than they are in their rest frame, but the distance between them will be the same. Since charge is independent of the frame in which it is measured, the total charge on each plate is also the same. So the charge per unit area on the plates is therefore larger than in the rest frame by a factor of:

${\displaystyle 1 \over {\sqrt {1-v^{2}/c^{2}}}}$

The field between the plates is therefore stronger by this factor.

### More rigorous analysis

Figure 2a: The electric field lines are shown flowing outward from the positive plate
Figure 2b: The electric field lines flow inward toward the negative plate

Consider the electric field of a single, infinite plate of positive charge, moving parallel to itself. The field must be uniform both above and below the plate, since it is uniform in its rest frame. We also assume that knowing the field in one frame is sufficient for calculating it in the other frame.

The plate however could have a non zero component of electric field in the direction of motion as in Fig 2a. Even in this case, the field of the infinite plane of negative charge must be equal and opposite to that of the positive plate (as in Fig 2b), since the combination of plates is neutral and cannot therefore produce any net fields. When the plates are separated, the horizontal components still cancel, and the resultant is a uniform vertical field as shown in Fig 1.

If Gauss's law is applied to pillbox as shown in Fig 1, it can be shown that the magnitude of the electric field between the plates is given by:

${\displaystyle |E'|={\sigma ' \over \epsilon _{0}}\ }$

where the prime (') indicates the value measured in the frame in which the plates are moving. ${\displaystyle \sigma }$ represents the surface charge density of the positive plate. Since the plates are contracted in length by the factor

${\displaystyle {\sqrt {1-v^{2}/c^{2}}}}$

then the surface charge density in the primed frame is related to the value in the rest frame of the plates by:

${\displaystyle \sigma '\ ={\sigma \over {\sqrt {1-v^{2}/c^{2}}}}}$

But the electric field in the rest frame has value σ / ε0 and the field points in the same direction on both of the frames, so

${\displaystyle E'={E \over {\sqrt {1-v^{2}/c^{2}}}}\ }$

The E field in the primed frame is therefore stronger than in the unprimed frame. If the direction of motion is perpendicular to the plates, length contraction of the plates does not occur, but the distance between them is reduced. This closer spacing however does not affect the strength of the electric field. So for motion parallel to the electric field E,

${\displaystyle E'=E\ }$

In the general case where motion is in a diagonal direction relative to the field the field is merely a superposition of the perpendicular and parallel fields., each generated by a set of plates at right angles to each other as shown in Fig 3. Since both sets of plates are length contracted, the two components of the E field are

${\displaystyle E'_{y}={E_{y} \over {\sqrt {1-v^{2}/c^{2}}}}}$

and

${\displaystyle E'_{x}=E_{x}\ }$

where the y subscript denotes perpendicular, and the x subscript, parallel.

These transformation equations only apply if the source of the field is at rest in the unprimed frame.

### The field of a moving point charge

Figure 3: A point charge at rest, surrounded by an imaginary sphere.
Figure 4: A view of the electric field of a point charge moving at constant velocity.

A very important application of the electric field transformation equations is to the field of a single point charge moving with constant velocity. In its rest frame, the electric field of a positive point charge has the same strength in all directions and points directly away from the charge. In some other reference frame the field will appear differently.

In applying the transformation equations to a nonuniform electric field, it is important to record not only the value of the field, but also at what point in space it has this value.

In the rest frame of the particle, the point charge can be imagined to be surrounded by a spherical shell which is also at rest. In our reference frame, however, both the particle and its sphere are moving. Length contraction therefore states that the sphere is deformed into an oblate spheroid, as shown in cross section in Fig 4.

Consider the value of the electric field at any point on the surface of the sphere. Let x and y be the components of the displacement (in the rest frame of the charge), from the charge to a point on the sphere, measured parallel and perpendicular to the direction of motion as shown in the figure. Because the field in the rest frame of the charge points directly away from the charge, its components are in the same ratio as the components of the displacement:

${\displaystyle {E_{y} \over E_{x}}={y \over x}}$

In our reference frame, where the charge is moving, the displacement x' in the direction of motion is length-contracted:

${\displaystyle x'=x{\sqrt {1-v^{2}/c^{2}}}}$

The electric field at any point on the sphere points directly away from the charge. (b) In a reference frame where the charge and the sphere are moving to the right, the sphere is length-contracted but the vertical component of the field is stronger. These two effects combine to make the field again point directly away from the current location of the charge. (While the y component of the displacement is the same in both frames).

However, according to the above results, the y component of the field is enhanced by a similar factor:

${\displaystyle E'_{y}={E_{y} \over {\sqrt {1-v^{2}/c^{2}}}}}$

whilst the x component of the field is the same in both frames. The ratio of the field components is therefore

${\displaystyle {E'_{y} \over E'_{x}}={E_{y} \over E_{x}{\sqrt {1-v^{2}/c^{2}}}}={y' \over x'}}$

So, the field in the primed frame points directly away from the charge, just as in the unprimed frame. A view of the electric field of a point charge moving at constant velocity is shown in figure 4. The faster the charge is moving, the more noticeable the enhancement of the perpendicular component of the field becomes. If the speed of the charge is much less than the speed of light, this enhancement is often negligible. But under certain circumstances, it is crucially important even at low velocities.

### The origin of magnetic forces

Figure 5, lab frame: A horizontal wire carrying a current, represented by evenly spaced positive charges moving to the right whilst an equal number of negative charges remain at rest, with a positively charged particle outside the wire and traveling in a direction parallel to the current.

In the simple model of events in a wire stretched out horizontally, a current can be represented by the evenly spaced positive charges, moving to the right, whilst an equal number of negative charges remain at rest. If the wire is electrostatically neutral, the distance between adjacent positive charges must be the same as the distance between adjacent negative charges.

Assume that in our 'lab frame' (Figure 5), we have a positive test charge, Q, outside the wire, traveling parallel to the current, at the speed, v, which is equal to the speed of the moving charges in the wire. It should experience a magnetic force, which can be easily confirmed by experiment.

Figure 6, test charge frame: The same situation as in fig. 5, but viewed from the reference frame in which positive charges are at rest. The negative charges flow to the left. The distance between the negative charges is length-contracted relative to the lab frame, while the distance between the positive charges is expanded, so the wire carries a net negative charge.

Inside 'test charge frame'(Fig. 6), the only possible force is the electrostatic force Fe = Q · E because, although the magnetic field is the same, the test charge is at rest and, therefore, cannot feel it. In this frame, the negative charge density has Lorentz-contracted with respect to what we had in lab frame because of the increased speed. This means that spacing between charges has reduced by the Lorentz factor with respect to the lab frame spacing, l:

${\displaystyle l_{-}={l{\sqrt {1-v^{2}/c^{2}}}}}$

Thus, positive charges have Lorentz-expanded (because their speed has dropped):

${\displaystyle l_{+}=l/{\sqrt {1-v^{2}/c^{2}}}}$

Both of these effects combine to give the wire a net negative charge in the test charge frame. Since the negatively charged wire exerts an attractive force on a positively charged particle, the test charge will therefore be attracted and will move toward the wire.

For ${\displaystyle v\ll c}$, we can concretely compute both,

the magnetic force sensed in the lab frame

${\displaystyle F_{m}={QvI \over 2\pi \epsilon _{0}c^{2}R}}$

and electrostatic force, sensed in the test charge frame, where we first compute the charge density with respect to the lab frame length, l:

${\displaystyle \lambda ={q \over l}_{+}-{q \over l}_{-}={q \over l}{\bigl (}{\sqrt {1-v^{2}/c^{2}}}-1/{\sqrt {1-v^{2}/c^{2}}}{\bigr )}\approx {q \over l}{\Bigl (}1-{\frac {1}{2}}{v^{2} \over c^{2}}\,-\,1-{\frac {1}{2}}{v^{2} \over c^{2}}{\Bigr )}=-{q \over l}{v^{2} \over c^{2}}}$

and, keeping in mind that current ${\displaystyle I={q \over t}=q{v \over l}}$, resulting electrostatic force

${\displaystyle F_{e}=QE=Q{\lambda \over 2\pi \epsilon _{0}R}={Qqv^{2} \over 2\pi \epsilon _{0}c^{2}Rl}={QvI \over 2\pi \epsilon _{0}c^{2}R}}$

which comes out exactly equal to the magnetic force sensed in the lab frame, ${\displaystyle F_{e}=F_{m}}$.

The lesson is that observers in different frames of reference see the same phenomena but disagree on their reasons.

If the currents are in opposite directions, consider the charge moving to the left. No charges are now at rest in the reference frame of the test charge. The negative charges are moving with speed v in the test charge frame so their spacing is again:

${\displaystyle l_{(-)}={l{\sqrt {1-v^{2}/c^{2}}}}}$

The distance between positive charges is more difficult to calculate. The relative velocity should be less than 2v due to special relativity. For simplicity, assume it is 2v. The positive charge spacing contraction is then:

${\displaystyle {\sqrt {1-(2v/c)^{2}}}}$

relative to its value in their rest frame. Now its value in their rest frame was found to be

${\displaystyle l_{(+)}={l \over {\sqrt {1-v^{2}/c^{2}}}}}$

So the final spacing of positive charges is:

${\displaystyle l_{(+)}={l \over {\sqrt {1-v^{2}/c^{2}}}}{\sqrt {1-(2v/c)^{2}}}}$

To determine whether l(+) or l(-) is larger we assume that v << c and use the binomial approximation that

${\displaystyle (1+x)^{p}\approxeq 1+px\;{\text{ when }}\;x\ll 1}$

After some algebraic calculation it is found that l(+) < l(-), and so the wire is positively charged in the frame of the test charge.

citation: A. French (1968), no page given

One may think that the picture, presented here, is artificial because electrons, which accelerated in fact, must condense in the lab frame, making the wire charged. Naturally, however, all electrons feel the same accelerating force and, therefore, identically to the Bell's spaceships, the distance between them does not change in the lab frame (i.e. expands in their proper moving frame). Rigid bodies, like trains, don't expand however in their proper frame, and, therefore, really contract, when observed from the stationary frame.

### Calculating the magnetic field

#### The Lorentz force law

A moving test charge near a wire carrying current will experience a magnetic force dependent on the velocity of the moving charges in the wire. If the current is flowing to the right, and a positive test charge is moving below the wire, then there is a force in a direction 90° counterclockwise from the direction of motion.

#### The magnetic field of a wire

Calculation of the magnitude of the force exerted by a current-carrying wire on a moving charge is equivalent to calculating the magnetic field produced by the wire. Consider again the situation shown in figures. The latter figure, showing the situation in the reference frame of the test charge, is reproduced in the figure. The positive charges in the wire, each with charge q, are at rest in this frame, while the negative charges, each with charge −q, are moving to the left with speed v. The average distance between the negative charges in this frame is length-contracted to:

${\displaystyle {\sqrt {1-v^{2}/c^{2}}}}$

where is the distance between them in the lab frame. Similarly, the distance between the positive charges is not length-contracted:

${\displaystyle {\sqrt {1-v^{2}/c^{2}}}}$

Both of these effects give the wire a net negative charge in the test charge frame, so that it exerts an attractive force on the test charge.

Electromechanics and relativistic electromagnetism has several sources now listed. — Rgdboer (talk) 01:42, 28 November 2015 (UTC)

Well, as I recall, the Feynmann lectures in physics has a marvelous derivation of the magnetic field as the Lorentz-contracted electric field. However, it is far more subtle and nuanced than the above deleted section (e.g. you need to work out the cross-products, to get B pointing in the correct direction). Thus, yes, deleting the above seems appropriate: its somehow oversimplifying some very important details. 67.198.37.16 (talk) 19:20, 21 February 2016 (UTC)