Help with SRT and Maxwell

It doesn't look like I'll ever get access to any physics professors. I am too old and far away to enroll in any of their PhD programs. So I'll ask this question here – maybe someone knows the answer, or can steer me towards a good textbook on this subject.

One of my many textbooks states that all electromagnetic problems can be explained with Special Relativity Theory. The example given is of Ampere's law about parallel currents in parallel wires attracting each other. From the perspective of the moving electrons in wire #2, the apparent Lorentz contraction of the stationary protons in wire #1 gives wire #1 an apparent concentration of protons, and thus a plus charge which then attracts the electrons in wire #2. (And vice versa of course, but for discussion purposes, my questions will focus on the supposed attraction of the electrons in wire #2 to the protons in wire #1.)

I have some problems with that explanation. The first problem is that it seems to conveniently ignore the influence and behavior of the moving electrons in wire #1. They too would or should experience the increased concentration and charge of their own protons. So wouldn't they just pack in more electrons from the battery, and thus totally shield any net positive charge influence reaching out through space over to wire #2?

My second problem with that explanation is more serious – and much more difficult to describe. It concerns the behavior of a charged particle, say an electron, in the vicinity of a current carrying wire. The force on the electron is perpendicular to both its own velocity and the B field, i.e., q (v x B), etc. That makes some sense to me, as long as the electron's trajectory is inside the poles of the coil of wire causing the B field. The electron executes a little circle – in the opposite direction from the flow of electrons in the coil. No mater what direction it is traveling, it is attracted to that part of the coil where the wire electrons are moving parallel to it, and repelled from the opposite side. Except for my first problem with that mentioned above, this behavior can be explained with SRT.

But what about the behavior of the electron outside the coil of wire? The behavior seems consistent with the SRT explanation above when the trajectory is tangent to the coil – the electron is attracted to the coil if the current in the very local vicinity is parallel, etc. But when the trajectory is in the plane of the coil and (approaching) perpendicular to the local wire and current direction, the force has to be upstream with respect to the velocity of the electrons in the wire according to q (v x B).

I cannot understand that from the perspective of SRT. From symmetry, the upstream force on the electron cannot arise from the plus proton charges – the electron's approaching velocity is perpendicular to the local wire. Thus the force on our test electron must arise from the relative contraction of the wire electrons upstream and downstream from its trajectory. Yet the upstream electrons obviously have a higher relative (approaching) velocity than the downstream (departing) ones, so they should be contracted more than the downstream ones. Thus the apparent excess negative charge should be upstream from the test electron, tending to repel the test electron. Which ain't right, so how does one explain this with SRT?

Any help appreciated.

The drift speed of electrons is so very slow (maybe a mm/sec in a copper wire) it's hard to imagine that relativity has anything practical to do with attraction of opposite charges.

Here's an excerpt from THE RIDDLE OF GRAVITATION by Peter Bergmann, Page 60, a student of Einsteins:

Coulombs law of electric forces and Newton's law of gravitational forces differ from one another in that in Coulomb's law the electric charges play the role reserved in Newton's law for the masses. In electricity the apparent contradiction between relativity and the law of electric forces is resolved by the discovery that Coulomb's law holds rigorously only if the charged bodies do not move with respect to each other...large velocities affect masses differently from electric charges...a body's electric charge has the same value for all observers....it's mass depends on it's speed relative to the observer...Because the magnitudes the sources of gravitation depend so much on the frame of reference in which they are measured, the resulting field is bound to be more complex than the electromagnetic field....

Relativistic transforms for both E and B fields, and for particles (electrons) can be found in the LBL Particle Data Group book at

http://pdg.lbl.gov/2008/reviews/contents_sports.html

Click on the Kinematics category for the Lorentz transforms (see paragraph 38.1), and on the Constants, units etc. catgory and then Electromagnetic relations (see bottom of the table) for the E and B transforms.

JesseM
One of my many textbooks states that all electromagnetic problems can be explained with Special Relativity Theory. The example given is of Ampere's law about parallel currents in parallel wires attracting each other. From the perspective of the moving electrons in wire #2, the apparent Lorentz contraction of the stationary protons in wire #1 gives wire #1 an apparent concentration of protons, and thus a plus charge which then attracts the electrons in wire #2. (And vice versa of course, but for discussion purposes, my questions will focus on the supposed attraction of the electrons in wire #2 to the protons in wire #1.)
Here's a page that has a bunch of good explanations and graphics about how electromagnetic predictions are consistent between different frames in relativity (which is a little different from saying they 'can be explained' with SR):

http://physics.weber.edu/schroeder/mrr/MRRtalk.html

(edit: atyy beat me to it!)

Here's the text and graphic discussing why a charge moving parallel to the wire will experience an attractive force when viewed in the charge's rest frame (where it experiences no magnetic force since its velocity is zero), if you just want a conceptual explanation you can skip the equations:
Shown below is a model of a wire with a current flowing to the right. To avoid minus signs I'm taking the current to consist of a flow of positive charges, separated by an average distance of l. The wire has to be electrically neutral in the lab frame, so there must be a bunch of negative charges, at rest, separated by the same average distance. Therefore there's no electrostatic force on a test charge Q outside the wire. What happens, though, if the test charge is moving to the right? For simplicity, let's say its velocity is the same as that of the moving charges in the wire. Now consider how all this looks in the reference frame of the test charge, where it's at rest. Here it's the negative charges in the wire that are moving to the left. Because they're moving, the average distance between them is length-contracted by the Lorentz factor. Meanwhile the positive charges are now at rest, so the average distance between them is un-length-contracted by the Lorentz factor. Both of these effects give the wire a net negative charge, so it exerts an attractive electrostatic force on the test charge. Back in the lab frame, we call this force a magnetic force.

By the way, it's remarkable that we can measure magnetic forces at all, since the average drift velocity in a household wire is only a snail's pace: v/c is typically only 10^-13, so the Lorentz factor differs from 1 only by about one part in 10^26. We can still measure this effect because the total charge of all the conduction electrons in a meter-long wire is tens of thousands of coulombs; two such charges separated by only a few millimeters would exert enormous electrostatic forces on each other.

You can repeat the preceding calculation for the more complicated case where the test charge's velocity differs from the average drift velocity of the charge carriers in the wire. Their velocities can also be opposite in direction. In each case, you still get the correct expression for the magnetic force on the test charge. The case where the test charge moves perpendicular to the wire is still more complicated, but we can understand it qualitatively after digressing to consider how electrostatic fields transform from one reference frame to another.
exmarine said:
I have some problems with that explanation. The first problem is that it seems to conveniently ignore the influence and behavior of the moving electrons in wire #1. They too would or should experience the increased concentration and charge of their own protons. So wouldn't they just pack in more electrons from the battery, and thus totally shield any net positive charge influence reaching out through space over to wire #2?
From the above explanation, I think the idea is that the protons are at rest in the wire's rest frame while the electrons in the wire are moving in this frame (the author just talks about 'charges' rather than protons and electrons, but since real flow of charge in wires is in the form of electrons, I assume when he says 'To avoid minus signs I'm taking the current to consist of a flow of positive charges', that means he's labeling the electrons as positive charges and protons as negative ones), the Lorentz contraction between them will be different in the rest frame of the charged particle moving relative to the wire--the page makes the simplifying assumption that the charged particle outside the wire has the same velocity as the electrons in the wire, but then adds "You can repeat the preceding calculation for the more complicated case where the test charge's velocity differs from the average drift velocity of the charge carriers in the wire. Their velocities can also be opposite in direction. In each case, you still get the correct expression for the magnetic force on the test charge."
exmarine said:
My second problem with that explanation is more serious – and much more difficult to describe. It concerns the behavior of a charged particle, say an electron, in the vicinity of a current carrying wire. The force on the electron is perpendicular to both its own velocity and the B field, i.e., q (v x B), etc. That makes some sense to me, as long as the electron's trajectory is inside the poles of the coil of wire causing the B field.
It doesn't need to be inside a coil, it's simpler to show that the direction of the magnetic force on an electron moving relative to a straight wire always matches what you'd predict from the electric force in the rest frame of the same electron. Again, from the page:
If we put this system in motion to the right (shown at right), two things happen. The first is that the sphere gets length-contracted, flattened in the direction of motion. The second is that the components of the field perpendicular to the motion get stretched by the very same Lorentz factor. Therefore the field still points directly away from the point charge, but it's not the same in all directions: it's weaker in front of and behind the particle, and stronger to the sides, as shown below.

A Charge Moving Perpendicular to a Wire

With all this in mind, let's now consider our wire again, but with the test charge moving directly toward it.

In the frame of the test charge the wire is moving downward. The negative charges in the wire, which are moving straight down, have their electric fields distorted as shown previously, but these fields are symmetrical from left to right so they exert no net horizontal force on the test charge. The positive charges in the wire, however, are now moving diagonally, so their fields are distorted as shown above. At the location of the test charge, the field of the positive charge to the right is stronger than that of the positive charge to the left, so the test charge feels a net force pushing it to the left.

In summary, we can account for the direction of the magnetic force on the test charge no matter which way it's moving, and this motivates us to introduce a magnetic field vector that points into the page, with the force given by a cross-product of v and B.

Does that help settle the question, or do you still have questions about a coiled wire even if you accept the above explanation for a straight wire?

atyy
BTW, in the second part, when you talk about a coil of wire, are you talking about an infinitely long solenoid? I'm not sure I'm remembering right, but isn't the B field outside an infinitely long solenoid zero?

atyy
Regarding the first part, the question about why excess positive charge doesn't affect the electrons in the wire is stated in the coordinate system (moving frame) in which the electron outside the straight infinitely long wire is stationary, and seems difficult in that frame. That difficulty disappears in the coordinate system (lab frame) in which the electron outside the wire is moving, and the wire is electrically neutral, in that case the effect is explained as due to a B field. So perhaps an equivalent question in the lab frame would be: if the wire is electrically neutral, what is causing the electrons to move? That would be the battery, which is kinda just "given". I wonder what the battery transforms into in the moving frame ....

JesseM
So perhaps an equivalent question in the lab frame would be: if the wire is electrically neutral, what is causing the electrons to move? That would be the battery, which is kinda just "given". I wonder what the battery transforms into in the moving frame ....
I imagine there'd be some sort of sort of statistical mechanics aspect to why charge flows away from a battery when you connect it to a wire, sort of like how if you have a box filled with gas and then open up a route into a lower-pressure area, the gas will flow out until the pressures equalize. Of course it's probably more complicated with a battery since concentrated regions of charge are actually exerting a force on other charges whereas in an ideal gas you assume that the gas particles don't interact with one another.

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Thanks guys! The apparent DIAGONAL approach of the moving charges in the wire was the missing clue for problem #2. I knew that, but hadn't connected it up with this problem. I will have to go back and review - I assume that it tilts in the right (correct) direction to cause the correct force. Among all the amazing things about this is that the magnitude of the force on the test charge, i.e., q |(v x B)| is the same for both parallel and perpendicular trajectories w.r.t. the current flow in the wire. Yet the SRT effects causing the forces seems quite different. I wonder if that is an amazing coincidence or some more subtle connection...
I still don't see a good answer to problem #1 - why don't the electrons in wire #1 shield the apparent increased positive charge from wire #2?

JesseM
I still don't see a good answer to problem #1 - why don't the electrons in wire #1 shield the apparent increased positive charge from wire #2?
Ah, I missed that you were talking about two wires as opposed to just a single moving charge traveling next to a wire. But in this case, as long as the electrons in both wires are moving at the same speed and in the same direction in the wire rest frame, then in the rest frame of a given electron the distance between electrons (in either wire) is not contracted like with the distance between the protons (in fact the distance between electrons in this frame is larger than the distance between electrons in the wire rest frame). If you want to look at the situation with the currents flowing in opposite directions at equal speed in the wire rest frame, then in the rest frame of an electron in wire #2 the electrons in wire #1 will be even more contracted than the protons in wire #1 since their velocity is higher in this frame, so there will be a net repulsive force. And indeed, as Ampere discovered, if you place two wires next to each other with current flowing in the same direction they'll be attracted, but if their current is flowing in opposite directions they'll repel.

edit: I see you are also asking about how electrons in a given wire would be affected by the increased concentration of protons in their own wire (or maybe that's all you were asking and I misunderstood the question). Well, in this case wouldn't the forces from the protons on a given electron cancel out, since along the length of the wire the density of protons is the same in both directions from any given electron?

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atyy
I still don't see a good answer to problem #1 - why don't the electrons in wire #1 shield the apparent increased positive charge from wire #2?

How about something like this? In the moving frame in which the test charge outside the wire is stationary, in order to prevent more electrons coming in to neutralise the increased positive charge, we need to provide a constant source of energy, ie. the situation is steady state, but not equilibrium. If we transform this situation to the lab frame in which the test charge is moving, the wire is electrically neutral and a constant current flows, then we see that the situation is also steady state, but not equilibrium, with the constant energy needed to maintain the current being provided by the battery. Thus in both cases there is a source of energy which maintains the steady non-equilibrium state, which in the lab frame is the battery, and in the moving frame is the "Lorentz transform of the batter", whatever that may be. For the Purcell argument, it isn't necessary to know exactly how the source of energy produces this situation (although I would like to understand it); one only needs to know that the situation in the moving frame is exactly the same situation as the lab frame, just described in different space-time coordinates.

Yet the SRT effects causing the forces seems quite different. I wonder if that is an amazing coincidence or some more subtle connection...

Both! Maxwell's equations and the principle of relativity (that a cup of coffee in an aeroplane won't tip over just because it's moving 700 km/h relative to the ground) lead inexorably to the special theory of relativity.

I tried to describe 2 different situations.

#1 was 2 parallel wires carrying 2 parallel currents, and how could their attraction be explained with SRT. Your explanation makes sense if you imagine an INFINITE wire maybe? I had pictured a FINITE (real) length of wire, and the attractions at the ends would pack on more electrons from the source. So maybe the battery does come into the explanation with the end effects?

#2 was an electron test charge, approaching a current carrying wire. Your explanation of the rotation of bodies in transverse motion due to the relativity of simultaneity made sense – except it rotates the wrong way! Crap.

Check me here. And let's set the stage very carefully. First, the wire is carrying a flow of negative electrons (None of this flow of protons in the lab frame that some authors like for some unknown reason – it doesn't simplify any explanation for me!) And the test charge is one negative electron, approaching the wire exactly perpendicular. Now the upstream electron “clocks” appear “earlier” than the downstream ones? Isn't that correct? So don't the upstream electrons appear closer to the approaching test electron, and the downstream ones appear further away? So that would again, tend to push the approaching test electron downstream, rather than upstream as actually happens according to q (v x B)?

This is frustrating.

JesseM
#1 was 2 parallel wires carrying 2 parallel currents, and how could their attraction be explained with SRT. Your explanation makes sense if you imagine an INFINITE wire maybe? I had pictured a FINITE (real) length of wire, and the attractions at the ends would pack on more electrons from the source. So maybe the battery does come into the explanation with the end effects?
I would guess that if the distance between a charge in one wire and the nearest point in the other wire is small compared to the length of the wire, then treating the wire as infinite will be a good approximation to the actual answer where you take into account the length of the wire. I don't know where to find a source that discusses the accuracy of approximations in E&M though.
exmarine said:
#2 was an electron test charge, approaching a current carrying wire. Your explanation of the rotation of bodies in transverse motion due to the relativity of simultaneity made sense – except it rotates the wrong way! Crap.
Well, this was exactly the case that's dealt with in the last two diagrams I posted above--do you disagree with how the force lines point in that diagram?
exmarine said:
Check me here. And let's set the stage very carefully. First, the wire is carrying a flow of negative electrons (None of this flow of protons in the lab frame that some authors like for some unknown reason – it doesn't simplify any explanation for me!)
I think assuming a flow of positive charges just makes it so you don't have to deal with as many negative numbers in your calculations (and which charge we label 'positive' and 'negative' is arbitrary, so you can interpret a flow of positive charges as a flow of positively-charged electrons, you don't need to interpret it as protons). In any case, if you just switch all the +'s for -'s in the diagrams and vice versa it won't change the direction of the arrows.
exmarine said:
And the test charge is one negative electron, approaching the wire exactly perpendicular. Now the upstream electron “clocks” appear “earlier” than the downstream ones? Isn't that correct?
You mean "upstream" relative to the horizontal direction the electrons in the wire are flowing (as seen in either the lab frame or the test charge frame)? In this case no, if the electrons in the wire had clocks that were synchronized in their rest frame, then in a frame where they're moving, the "upstream" clocks show later times than the "downstream" ones.
exmarine said:
So don't the upstream electrons appear closer to the approaching test electron, and the downstream ones appear further away?
If I'm understanding the question right then no, it doesn't work that way, if the electrons in the wire have constant spacing in their own rest frame, then they'll have constant spacing in every frame (though in other frames the distance at a given moment will be shorter due to length contraction), the relativity of simultaneity doesn't create variable distance in one frame where you had constant distance in another. You might take a look at the diagram I made here of a row of clocks that are synchronized in their own frame as they look in a frame where they're all in motion.

Your attempted explanation here doesn't seem to have anything to do with the one on the page that atyy and I linked to, did you look over the explanation there for why the test particle experiences a horizontal force, having to do with the way the force vectors from the positive charges in the wire are squashed (weaker force) in the direction of motion and expanded (stronger force) in the direction perpendicular to their motion, and how in the test charge frame these positive charges are moving diagonally rather than straight down like the negative charges?

Thanks to everyone who has tried to 'splain this stuff to me. I've reviewed those figures again. All the velocity and force directions seem to conform to q (v x B), so they must be correct. My problem is understanding that from the perspective of SRT. The apparent ROTATION of a passing object due to varying clocks seems to be the missing clue. And I must have that rotation backwards. I'll go study that some more. (I thought the trailing end of the passing object had to indicate a later time than the leading edge. And thus the trailing end would appear closer to the approaching test charge. Is that backwards?)

Check out section 4 of this link – is it not correct about the apparent SRT rotation of a passing object?

http://www.pitt.edu/~jdnorton/Goodies/rel_of_sim/index.html

Thanks again everybody.
BB

JesseM
Check out section 4 of this link – is it not correct about the apparent SRT rotation of a passing object?

http://www.pitt.edu/~jdnorton/Goodies/rel_of_sim/index.html
OK, I see what you mean now. But section 4 would only match this example if there was a frame where the test charge was moving in a purely horizontal direction while the wire was moving straight down--in that case, in the test charge's own frame the wire would appear rotated like in the animation there. But I thought in this example we were talking about a situation where in the wire's rest frame the test charge is moving straight up towards the wire, in which case the wire would be coming straight down in the test charge rest frame, it won't be rotated in the same way.

? I thought the RELATIVE velocities in that annimation and in our case were identical??

His little observer is moving left, our stream of electrons in the wire is moving right - isn't that the same?

His rod is moving down towards the little observer, our little test electron charge is moving up approaching the wire and the line of electrons - isn't that also the same?

The left end of his rod - the trailing end appears closer to the observer, so wouldn't the trailing or the upstream end of our stream of electrons apprear closer to the negative electron test charge?

So how can the electron want to turn upstream, i.e., towards the "closest" end of the stream of electrons? (IF all this is correct - it must not be...)

JesseM
? I thought the RELATIVE velocities in that annimation and in our case were identical??

His little observer is moving left, our stream of electrons in the wire is moving right - isn't that the same?
No, in the wire rest frame the electrons are arranged horizontally and moving right and the test particle is moving straight up, which is not equivalent to saying there's a frame where the electrons are arranged horizontally and moving straight down and the test particle is moving left, which would be like what the diagram shows (of course you could find a frame where the test particle was moving left, but although I haven't done the calculations I would guess that in this frame the angle of the electrons would be skewed from the horizontal). Think of it this way, in the wire rest frame the protons are at rest and arranged horizontally while the test particle is moving straight up, so it should make sense that in the frame where the test particle is at rest, the protons are still arranged horizontally and moving straight down, right? Well, it's impossible that in the test particle's rest frame, the protons could be arranged horizontally while the electrons could be skewed--that would prevent each electron from passing each proton in turn, but those are local events which all frames must agree on. So, if the protons are arranged horizontally in the test particle rest frame, the electrons must be too.

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atyy
So, if the protons are arranged horizontally in the test particle rest frame, the electrons must be too.

Reading http://physics.weber.edu/schroeder/mrr/MRR.html, quickly, I did think Schroder meant the positive charges in the wire move diagonally (using Schroder's convention of horizontally moving positive charges and stationary negative charges in the lab frame). In the frame where the test positive charge is stationary, the negative charges move straight down, while the positive charges move straight down and horizontally, so the vector sum would be diagonal?

JesseM
Reading http://physics.weber.edu/schroeder/mrr/MRR.html, quickly, I did think Schroder meant the positive charges in the wire move diagonally (using Schroder's convention of horizontally moving positive charges and stationary negative charges in the lab frame). In the frame where the test positive charge is stationary, the negative charges move straight down, while the positive charges move straight down and horizontally, so the vector sum would be diagonal?
They all move diagonally in the test charge's frame, but I was talking about their arrangement at any single instant in the test charge's frame. Schroeder does show them arranged horizontally at a single instant, unlike with the rod that becomes skewed from the horizontal in section 4 of the relativity of simultaneity page which exmarine was referring to.

atyy
They all move diagonally in the test charge's frame, but I was talking about their arrangement at any single instant in the test charge's frame.

OK, got it.

Uh, are you guys confusing diagonal velocities with rotation of a passing object due to SRT?

...Well, it's impossible that in the test particle's rest frame, the protons could be arranged horizontally while the electrons could be skewed--that would prevent each electron from passing each proton in turn, but those are local events which all frames must agree on. So, if the protons are arranged horizontally in the test particle rest frame, the electrons must be too.

Look back at one of the figures in your own post #5, where the protons are rotated and the electrons are not. That's because of SRT I think?

JesseM