Explanation of EM-fields using SR

[universal_101]

Lets say the loop of wire consists of 2 parallel wires (A and B), each 600m long and ignore the length of the connecting sections at each end. Let's also say there are 200 electrons in the entire circuit so they are 6m apart from each other, when equally spaced out and at rest with the wire in the wire/lab rest frame. (Yes, I know that is unrealistic but stick with me ;)

When the current is moving relative to the wire/lab, there are still 200 electrons distributed around 1200 m of wire as measured in the lab so the gap between electrons is still 6m as measured in the lab.

Why do you need to measure it, I think it should be predicted(logically deduced) by the standard EM theory, that given a wire with NO current and NO excess charge should remain neutral NO matter how high the current through it.

 

[Dale]

Just switching the current on and off is in conflict with the standard EM theory(i.e. SR).

In what possible way could you think this statement is true? What possible conflict between Maxwells equations and an electrical switch could you think exists?

To make a claim like this you need a rigorous derivation showing exactly how closing a switch violates Maxwells equations. Otherwise it is just your personal speculation.

 

[A.T.]

There is no need to transform the constant flowing current for different inertial frames

You claimed that the situation was symmetrical from both frames, and I explained why this not the case. Of course you have to compare the frames to show that. And when you claim there is excess charge, you also have to specify in which frame. Otherwise it is meaningless.

Alright, let's consider a particle accelerator, and suppose there are highly energetic(fast moving) Muons in the tunnel, and there is a sample of them in a stationary lab w.r.t the accelerator. Now, my question is since the particles in the tunnel are length contracted, time dilated etc. compared to the samples in the stationary lab because of the relative motion(this is all in the lab's frame). Why does the same not follow for the case of switching the current on from off because of the relative motion between electrons and protons(this is all in the wire's frame).

The individual electrons are length contracted in the wire's frame. But their spacing doesn't decrease in the wire's frame, when the current starts flowing.

 

[Dale]

Why does the same not follow for the case of switching the current on from off because of the relative motion between electrons and protons(this is all in the wire's frame).

Length contraction does follow, always. If the current is on then different frames disagree about the distance between electrons. If the current is off then different frames disagree about the distance between electrons. Different frames always disagree about the distance between electrons. That is length contraction. It is always present.

The comparison of distances at different times in one frame is not length contraction. Length contraction is a disagreement between two frames, not a change over time in one frame. Because of that, your comment about length contraction all in one frame is a self-contradiction. There is no such thing as length contraction in one frame.

 

[harrylin]

[..] Length contraction is [..] not a change over time in one frame. [..] There is no such thing as length contraction in one frame.

On a side note: sorry, but No. Once more, please digest our earlier comments here:

https://www.physicsforums.com/showthread.php?p=4517568
https://www.physicsforums.com/showthread.php?p=4517685

Thus, length contraction is also the Lorentz contraction of a body that changed from rest to motion as determined in a single inertial reference system; as a matter of fact, historically it's even the first meaning of that expression.

 

[A.T.]

Thus, length contraction is also the Lorentz contraction of a rigid body that changed from rest to motion as determined in a single inertial reference system; as a matter of fact, historically it's even the first meaning of that expression.

Fixed it for you. This the key element that people often forget, when assuming "length contraction" in that historical sense. And it is a good reason to avoid that historical usage, because it is based on this often not explicitly stated assumption, that the proper length is constant which is not always true. This leads to confusion like we see here with universal_101, and generally in Bell-Spacehip-Paradox threads.

 

[yuiop]

In every inertial frame that is true. But when you accelerate from the inertial wire's frame to the inertial current's frame your line of simultaneity rotates. In this accelerating frame there are more electrons per second leaving the segment, than entering it. So when you arrive in the current's frame you end up with less electrons in the segment, than there were in the wire's frame.

So yes, the relativity of simultaneity can be used to explain why there a different counts of electrons in the wire segment, between the two frames: Two events (electron entering, electron leaving) that are simultaneous in the wire's frame, are not simultaneous in the current's frame (leaving comes first).

Get it now. Thanks A.T. Must have been tired last night. The initial loss of electrons from the section of wire is not recovered later in the steady state phase when the amount of electrons entering is the same as the amount of electrons leaving. As a side note, generally speaking the electrons do not all start moving simultaneously in the rest frame of the wire when the switch is closed. I am checking out the situation when the switch on event propagates down the wire at the speed of light which is probably faster than in reality. The end result is the same as far as charge density is concerned, but the explanation appears to be a little different.

 

[Dale]

Thus, length contraction is also the Lorentz contraction of a body that changed from rest to motion

This the key element that people often forget, when assuming "length contraction" in that historical sense. And it is a good reason to avoid that historical usage

As noted, this is a deprecated historical usage. Furthermore, even the deprecated historical usage is not relevant to the current discussion. For both reasons (historical and irrelevance) a further discussion of this usage would be off-topic.

Please do not hijack the thread.

 

[A.T.]

Furthermore, even the deprecated historical usage is not relevant to the current discussion.

The confusion of the two meanings of "length contraction", and the misapplication of the deprecated meaning might be the reason why there still a discussion. So maybe it's not bad to clarify it.

 

[Dale]

I think that your post clarifies why the historical usage is not relevant. The electrons are not a rigid body, so the only usage which is relevant to the current discussion is the standard modern usage. If universal_101 has questions about the modern usage then he can ask, but I don't want any historical apologists cluttering up the thread.

 

[harrylin]

Get it now. Thanks A.T. Must have been tired last night. The initial loss of electrons from the section of wire is not recovered later in the steady state phase when the amount of electrons entering is the same as the amount of electrons leaving. As a side note, generally speaking the electrons do not all start moving simultaneously in the rest frame of the wire when the switch is closed. I am checking out the situation when the switch on event propagates down the wire at the speed of light which is probably faster than in reality. The end result is the same as far as charge density is concerned, but the explanation appears to be a little different.

It appeared that you were not tired but lucid; see my earlier reply to you. If I'm not mistaken then RoS merely gives you in an indirect way the electron distance as transformed from the stationary frame, and which can be found directly. Isn't that what you meant? The transformation doesn't contain information about final force equilibrium according to the cat's reckoning, other than that according to PoR the cat should obtain that result.

 

[TrickyDicky]

I think the problem with the clip's explanation is that it doesn't clarify why(and I'm referring to the current switch on) for observers at rest wrt the protons(like the guy who explainss it) the electrons are not length contracted while for observers at rest wrt the electrons (like the cat moving along them) protons are length contracted.

 

[yuiop]

I think the problem with the clip's explanation is that it doesn't clarify why(and I'm referring to the current switch on) for observers at rest wrt the protons(like the guy who explainss it) the electrons are not length contracted while for observers at rest wrt the electrons (like the cat moving along them) protons are length contracted.

In DrGregs excellent diagram, the electrons are shown as length contracted in the rest frame of the wire when the current is on (bottom left drawing). It is just the gaps between the centres of the electrons that remains the same as when the current was off in the rest frame of the wire (top left drawing). I think it is also debatable whether electrons length contract if they are considered as quantum point particles and the length contraction of the electrons themselves is not important to the basic theme of this thread. It is only the inter electron gaps that has any significance.

I think a better question is why the battery (which is a store of excess electrons) does not take the opportunity to pack extra electron into the wire when the circuit is switched on. I am sure there is a good reason that does not happen, but it is just sort of assumed without explanation.On the other hand, it would be possible to construct a circuit with a coil at one end and induce a current, so that we are certain no additional electron are added to the circuit.

 

[Dale]

I think the problem with the clip's explanation is that it doesn't clarify why(and I'm referring to the current switch on) for observers at rest wrt the protons(like the guy who explainss it) the electrons are not length contracted while for observers at rest wrt the electrons (like the cat moving along them) protons are length contracted.

The video may not draw it correctly (not sure), but they are both length contracted in fact. The spacing between the electrons is greater in the electrons' frame than in the protons' frame. The spacing between the protons is greater in the protons' frame than in the electrons' frame. Length contraction occurs for both protons and electrons.

 

[Dale]

I think a better question is why the battery (which is a store of excess electrons) does not take the opportunity to pack extra electron into the wire when the circuit is switched on. I am sure there is a good reason that does not happen, but it is just sort of assumed without explanation.

It is not really an assumption, it is more like part of the specification of the problem. It is what I called earlier a boundary condition.

You could pack additional electrons on if you used a very high voltage battery (MV or more) and a big resistor in series with the wire so that very little voltage is dropped across the wire. But that would be a different problem.

If you were working a projectile problem and the problem description says that the projectile leaves the barrel at 45º angle and 1000 m/s then I wouldn't call the 1000 m/s an assumption, it is just part of the specification of the problem. Would you?

 

[yuiop]

It is not really an assumption, it is more like part of the specification of the problem. It is what I called earlier a boundary condition.

Fair comment, but I don't think the video took much time to define the boundary conditions clearly. I was just making sure that the door was firmly closed on allowing extra electrons into the circuit. With a fixed number of electrons I am not sure how anyone can claim the gap between the electrons should contract in the rest frame of the wire, unless they imagine the electrons going around in a small clump with a massive gaps between the ends of the clump.

 

[TrickyDicky]

In DrGregs excellent diagram, the electrons are shown as length contracted in the rest frame of the wire when the current is on (bottom left drawing). It is just the gaps between the centres of the electrons that remains the same as when the current was off in the rest frame of the wire (top left drawing). I think it is also debatable whether electrons length contract if they are considered as quantum point particles and the length contraction of the electrons themselves is not important to the basic theme of this thread. It is only the inter electron gaps that has any significance.

The video may not draw it correctly (not sure), but they are both length contracted in fact. The spacing between the electrons is greater in the electrons' frame than in the protons' frame. The spacing between the protons is greater in the protons' frame than in the electrons' frame. Length contraction occurs for both protons and electrons.

I know that, I said that the video din't clarify it well enough, it gave the impression that it was that asymmetry that explained what makes magnets work, but it isn't that asymmetry since both electrons and protons length contract from each other's frame, instead they should have emphasized the fact that the proton's rest frame and the wire's is the same.

 

[universal_101]

The individual electrons are length contracted in the wire's frame. But their spacing doesn't decrease in the wire's frame, when the current starts flowing.

Then why do you not apply this "individual electron contraction and no change in spacing" to the video this thread is based on and to the picture that DrGreg has posted, and see what comes out. Let me guess, it does not give correct results !?

 

[universal_101]

The comparison of distances at different times in one frame is not length contraction. Length contraction is a disagreement between two frames, not a change over time in one frame. Because of that, your comment about length contraction all in one frame is a self-contradiction. There is no such thing as length contraction in one frame.

You could have said it far too earlier in this thread and could have saved every one a lot of typing! And it is not over yet, although it is nice to have the modified definition of Length contraction yet again, but this definition makes me wonder, why do you then not apply the same definition to the Time Dilation, Mass Increment etc. in a simple scenario of twin paradox(where you watch everything from the frame of staying twin, and I hope that where there is time Dilation there is length Contraction i.e all the relativistic effects are present) or when you work with the results of accelerators!

 

[universal_101]

Fixed it for you. This the key element that people often forget, when assuming "length contraction" in that historical sense. And it is a good reason to avoid that historical usage, because it is based on this often not explicitly stated assumption, that the proper length is constant which is not always true. This leads to confusion like we see here with universal_101, and generally in Bell-Spaceship-Paradox threads.

First of all the word rigid has NO meaning here or anywhere where relativity is discussed(that is as long as EM forces are concerned), if it means anything then it is how particles were related to each other before there was any relative motion, and you are using it to define Length Contraction.

"Since electrons are not rigidly connected to each other, there is NO length Contraction", I don't know what does the underlined statement means, because as far as I know electrons are connected to each other by Coulomb forces(mostly) which according to relativity are liable to change if there is relative motion, which should result in electrons coming closer to each other. Just like Twin Paradox!

 

[universal_101]

In DrGregs excellent diagram, the electrons are shown as length contracted in the rest frame of the wire when the current is on (bottom left drawing). It is just the gaps between the centres of the electrons that remains the same as when the current was off in the rest frame of the wire (top left drawing). I think it is also debatable whether electrons length contract if they are considered as quantum point particles and the length contraction of the electrons themselves is not important to the basic theme of this thread. It is only the inter electron gaps that has any significance.

What is the meaning of Length contraction if instead you are contracting individual points(which by definition cannot be contracted). And why do you not apply it to the whole picture of DrGreg's and instead of contracting the inter electron gaps just contract the electrons. So you can't have Length Contraction and same spacing both in one case, and Length contraction and different spacing in the other case, just so it suits the observed facts!

 

[TrickyDicky]

Universal, instead of bringing further cofusion by questioning everyone's relativity knowledge thru your distorting misconceptions, why don't you tell us if you finally understood why in the presence of current length contraction in the wire's frame retains neutrality while in frames moving wrt the wire it doesn't.

 

[A.T.]

First of all the word rigid has NO meaning here or anywhere where relativity is discussed(that is as long as EM forces are concerned), if it means anything then it is how particles were related to each other before there was any relative motion, and you are using it to define Length Contraction.

By "rigid" I meant: Keeping a constant proper length (length measured in the object rest frame) over time. That is the premise, based on which you can assume that an accelerating object will shorten in the original rest frame.

as far as I know electrons are connected to each other by Coulomb forces(mostly)

No, they are not connected by Coulomb forces. They are repulsing each other by Coulomb forces

which according to relativity are liable to change if there is relative motion,

Yes, the E-fields of moving electrons are contracted in the wire's frame.

which should result in electrons coming closer to each other.

No, contracted repulsive fields are still repulsive. They don't become attractive. The electrons still try to distribute as far as possible from each other.

 

[harrylin]

Universal, instead of bringing further cofusion by questioning everyone's relativity knowledge thru your distorting misconceptions, why don't you tell us if you finally understood why in the presence of current length contraction in the wire's frame retains neutrality while in frames moving wrt the wire it doesn't.

All the counterproductive quibbling over words may have made you overlook post #74, #75 and #103. When we do find the answer, we may also be able to explain the steady state charge imbalance according to the physics from the cat's perspective.

 

[A.T.]

Then why do you not apply this "individual electron contraction and no change in spacing" to the video this thread is based on and to the picture that DrGreg has posted, and see what comes out.

This is exactly what DrGreg's diagram shows.

 

[Dale]

You could have said it far too earlier in this thread and could have saved every one a lot of typing!

So does that mean that you understand it now?

but this definition makes me wonder, why do you then not apply the same definition to the Time Dilation, Mass Increment etc.

I do apply the same defintion to time dilation (I don't use relativistic mass at all).

 

[TrickyDicky]

All the counterproductive quibbling over words may have made you overlook post #74, #75 and #103. When we do find the answer, we may also be able to explain the steady state charge imbalance according to the physics from the cat's perspective.

Find the answer to what question exactly, can you phrase it in specific terms?

 

[yuiop]

In this pape http://www.chip-architect.com/physics/Magnetism_from_ElectroStatics_and_SR.pdf the charge density measured in the rest frame of a test charge moving at v relative to the wire, is given as:

$g_L = I v/c^2$

I might be wrong, but I suspect that is just for the special case when the drift velocity of the electrons $(v_e)$ is equal to the velocity of the test charge. If that is the case, then the more general expression should be:

$g_L = I (1/v_e)* (v^2/c^2)$

which reduces to the previous expression when $v_e = v$

Expressed like this, it is easy to see that the slow drift velocity of the electrons (about 1mm per second) relative to the speed of light actually magnifies the effect and explains why the EM effect is one of the few relativistic effects readily observable at everyday velocities.

Any thoughts?

 

[Dale]

So, if you start with a four-current $(0,I,0,0)$ and boost to any arbitrary frame moving at velocity v you get a four-current $(\gamma I v/c, \gamma I,0,0)$ regardless of what the drift velocity of the charge carriers are. So I get the charge density is $\rho=\gamma I v/c^2$, which is the same as their expression to first order in v.

 

[yuiop]

So, if you start with a four-current $(0,I,0,0)$ and boost to any arbitrary frame moving at velocity v you get a four-current $(\gamma I v/c, \gamma I,0,0)$ regardless of what the drift velocity of the charge carriers are. So I get the charge density is $\rho=\gamma I v/c^2$, which is the same as their expression to first order in v.

In their derivation they equate the magnetic force with the electric force, but they do not allow for the fact that the forces are measured in different reference frames. If we allow for this using the Lorentz transformation of transverse force, then I get $F_{mag} = F_{elec}/\gamma$ and this gives the same result as yours, $Q_L=\gamma I v/c^2$. This of course implies that when the electric force is measured in the same reference frame as the magnetic force is measured in (two separate experiments alongside each other in the same lab) the equation for the electric force is gamma greater than the equation quoted in the paper. My argument is partly supported by equation 29 of this text if I understood it correctly. This can thought of as the difference between the transverse force on a particle that has relative motion, compared to the force on a static particle.

Equation (0) of this article uses an approximation, which when carried out accurately also indicates the charge density and electric force is greater by a factor of gamma than the result they obtain.

Trouble is I guess I am still a bit confused because this additional gamma factor is well hidden in this https://www.mtholyoke.edu/courses/tdray/phys310/electromag.pdf. Also, the implication is that the Lorentz Force law (Eq 8.4) should actually be $F = q(\gamma*E + v \times B)$ although that seems unlikely as it has been tested to high velocities, or the equation for the Electric force field is gamma greater than the usually quoted formula as in (Eq 8.1) of the text?

As for the relevance of the drift velocity, I will have to give that some more thought. It is just that to me, the derivations of electric force require that the electrons are stationary in the rest frame of the test charge and so the derivation is only valid for the case where the test charge is moving at the drift velocity.