Relativity and Electromagnetism

[RMJ]

Hello,
There is a common setup used when describing the intimate relationship between electricity and magnetism. I have a question about the setup.

Setup:
There is some long current-carrying wire. Outside of that wire, there is some test charge.

In the first situation, the test charge is stationary.
The wire (carrying non-zero current) is said to be electrically neutral (let's say with average charge velocity Vdrift). In this case the charged particle is said to be subject to no electric or magnetic forces, and thus does not accelerate.

In the second situation, the test charge begins to move with the same speed (and in the same direction, let's say) as the moving charges in the wire.
In the reference frame of the wire, the particle experiences a magnetic force (simply, F=qv x B). In the reference frame of the particle, however, there seems to be an electric force because the charges moving in the current (in the reference frame of the wire/lab) are not moving from the test charge's perspective and the charges not moving in the current (in the reference frame of the wire/lab) are now moving in the opposite direction of the test charge (with average charge velocity of -Vdrift) meaning that the the lengths between them are now contracted (in the reference frame of the particle). When deriving an expression for the coulomb force experienced by this test charge in its own reference frame we might construct an argument saying that the charges that are not moving in the particle's reference frame are more separated than the particles that are moving in the particle's reference frame, and so there is a non-zero linear charge density in the wire which produces an electric field.

My question is this:
In the second situation, we can think about it from the reference frame of the moving test charge and we know that because one kind of charge is stationary and the other kind of charge is moving with average charge velocity -Vdrift. The moving charges undergo length contraction and create a nonzero charge "on" the wire. (The wire is not electrically neutral.)
If we view the FIRST situation from the perspective of the stationary test charge we see that the roles of the charges from the second situation (from the reference frame of the moving test charge) are reversed. There is one kind of charge moving at velocity Vdrift and another kind of charge that is stationary. Why doesn't length contraction occur with the moving charges in this scenario now and create an electric field too (just in the other direction)?

 

[Dale]

Length contraction requires that you know the distance in one frame. Once you know the distance in one frame then you can use length contraction to change the distance to or from the rest frame. But you need to be given the length in one frame first.

Here, you are not given the length in the current’s rest frame. You are only given the length in the lab frame. Those lengths are fixed by the fact that the conductor is given to be neutral. Given that fact then you can transform to the current’s frame, but you cannot go the other way since you are not given that information.

 

[RMJ]

Length contraction requires that you know the distance in one frame. Once you know the distance in one frame then you can use length contraction to change the distance to or from the rest frame. But you need to be given the length in one frame first.

Here, you are not given the length in the current’s rest frame. You are only given the length in the lab frame. Those lengths are fixed by the fact that the conductor is given to be neutral. Given that fact then you can transform to the current’s frame, but you cannot go the other way since you are not given that information.

I don't think you must know the explicit distance between successive charges. This is a conceptual argument.

My assumption is that if the current were 0, then the average distance between positive charges would be the same as the average distance between negative charges. All that one must know to know that length contraction occurs between moving charges in the wire is that non-zero current is flowing.

May you shed some light on my original point? I'm really stuck on this.

 

[Dale]

I already did shed light on it and you completely shrugged it off.

My assumption is that if the current were 0, then the average distance between positive charges would be the same as the average distance between negative charges.

Since the wire is neutral the distance between positive charges is the same as the distance between negative charges, regardless of the current. That is what “neutral” means

 

[RMJ]

I already did shed light on it and you completely shrugged it off.

Since the wire is neutral the distance between positive charges is the same as the distance between negative charges, regardless of the current. That is what “neutral” means

Pardon me. I didn't define the wire to be neutral. I said it is often defined that way. I'm trying to understand how it can be truly neutral even if current flows. (Relativistically)

 

[Dale]

I'm trying to understand how it can be truly neutral even if current flows. (Relativistically)

It can be neutral by having the same charge density of electrons and protons. That can be arranged fairly easily with standard electrical components. All you have to do is keep the overall voltage low and minimize any self capacitance.

 

[RMJ]

It can be neutral by having the same charge density of electrons and protons. That can be arranged fairly easily with standard electrical components. All you have to do is keep the overall voltage low and minimize any self capacitance.

By keeping voltage low and minimizing self-capacitance does one ensure that electron density is approximately equal to proton density by keeping current low?

 

[Orodruin]

When you go to a boosted frame in the first situation the wire will also be charged. This will lead to an electric force on the test charge. However, this force will be exactly balanced by the magnetic force on the charge that will appear since the test charge is moving in that frame and the magnetic field is non-zero.

 

[Dale]

By keeping voltage low and minimizing self-capacitance does one ensure that electron density is approximately equal to proton density by keeping current low?

No, current can be arbitrarily high. For a superconductor the net charge can be 0 everywhere regardless of current. For a good conductor the charge can be approximately 0 even for reasonably large currents. For a poor conductor there will be some excess positive charge on one side and negative charge on the other side of a wire that is overall neutral and is conducting a current.

 

[RMJ]

This is my attempt to understand the behaviour of the wire from the perspective of the test charge:

In situation #1 the particle experiences the coulomb force from the wire, but it also experiences a magnetic force equal in magnitude but opposite in direction because the magnetic field is generated by particles that are moving in its reference frame (the current) at average velocity Vdrift..

In situation #2 the particles that were moving in the current from the perspective of the test charge in situation #1 are now stationary, and the particles that were stationary from the perspective of the test charge in situation #1 are now moving in the opposite direction as the current from the wire's perspective, with average drift velocity -Vdrift. Here the particle experiences a magnetic force equal to that of situation #1, but it also experiences an electric force equal in magnitude to the one in situation #1, but in the opposite direction. Now the magnetic and coulomb forces experienced by the particle are in the same direction and add constructively to accelerate the wire toward the particle (in the particle's reference frame, of course).

 

[jartsa]

This is my attempt to understand the behaviour of the wire from the perspective of the test charge:

In situation #1 the particle experiences the coulomb force from the wire, but it also experiences a magnetic force equal in magnitude but opposite in direction because the magnetic field is generated by particles that are moving in its reference frame (the current) at average velocity Vdrift..

Test charge's views about the situations are these: (The test charge stands still next to the wire)

With no current in the wire: No electric field, no force, distance between electrons 4 nm. (test charge has no idea about magnetic field, because test charge has no magnetometer)

With current in the wire: No electric field, no force, distance between electrons 4 nm. (test charge has no idea about magnetic field, because test charge has no magnetometer)

In short: nothing is different between situation when current flows and situation when current does not flow.

 

[Dale]

In situation #1 the particle experiences the coulomb force from the wire, but it also experiences a magnetic force equal in magnitude but opposite in direction because the magnetic field is generated by particles that are moving in its reference frame (the current) at average velocity Vdrift..

Neither of these statements are correct. In this frame the wire is neutral, so the E field is zero, so $qE=0$. There is a current so B is nonzero, but the velocity of the charge is zero so $qv\times B=0$.

In situation #2 ... Here the particle experiences a magnetic force equal to that of situation #1,

Yes, in situation 2 in the particle’s frame the particle is again at rest so again the force is 0.

but it also experiences an electric force equal in magnitude to the one in situation #1, but in the opposite direction. Now the magnetic and coulomb forces experienced by the particle are in the same direction and add constructively

None of this is correct in the particle’s frame.

 

[RMJ]

Neither of these statements are correct. In this frame the wire is neutral, so the E field is zero, so $qE=0$. There is a current so B is nonzero, but the velocity of the charge is zero so $qv\times B=0$.

Yes, in situation 2 in the particle’s frame the particle is again at rest so again the force is 0.

None of this is correct in the particle’s frame.

In this frame the wire is clearly not neutral because the density of the particles flowing in the current is higher than that of the stationary charges in the wire.

 

[Dale]

In this frame the wire is clearly not neutral because the density of the particles flowing in the current is higher than that of the stationary charges in the wire.

No:

In the first situation, the test charge is stationary.
The wire (carrying non-zero current) is said to be electrically neutral

 

[RMJ]

No:

Excuse me. I must not have been clear.
I posited that this "neutrality" is part of the usual setup of the problem. The point of this thread is roughly that I truly don't understand how the wire can be assumed to be neutral if it also carries current and therefore contains particles of one charge moving in one direction while the particles of the other charge are stationary. Does the distance between the moving particles decrease due to their movement at some average drift velocity, or does it not? If it does, mustn't there be more charges of the type that are moving than there are of the stationary type enclosed in any finite length of the wire (due to length contraction due to relative velocities)? If not, why not?
If there are more charges of the moving type (in conventional current these would be positive charges) than of the stationary type (in conventional current, negative charges) in any finite length of the wire, then gauss' law tells us that the electric field outside the wire (in the stationary test charge's frame) in non-zero. This is all based on the assumption that, if there were no current flowing, the average distance between positive charges in the wire would be equal to the average distance between negative charges in the wire.
Please explain what is wrong with each step in my logic of this post. Obviously I don't understand how a current-carrying wire can truly be neutral but I would like to. Are the effects that I'm talking about present but negligible?

Thanks,
RMJ

 

[Orodruin]

Your basic assumption is that the separation between charges is always the same in their rest frame. This is not necessarily true.

 

[RMJ]

Your basic assumption is that the separation between charges is always the same in their rest frame. This is not necessarily true.

Let's say that I set the current (conventional) in the wire to zero. Once I did that, I then measured the average distance between the positive charges in the wire in some long sample region of interest. Let's call this distance "d".

Now let's say that I set the (conventional) current in the wire to be nonzero, with each positive charge in the wire moving at an average velocity Vdrift. Once I did that, let's say I stood with one foot on each of two of the many positive charges in the wire and measured their separation distance, then repeated this process simultaneously for the many pairs of adjacent positive charges in the wire in the region of interest, and found the average separation distance between consecutive positive charges to be some number I call "L."

Are you saying that d, average positive charge separation distance when no current is flowing in the lab frame (which, when no current is flowing, is the same reference frame of the positive charges) would not necessarily be the same as L, the average positive charge separation distance as measured in the reference frame of the positive charges moving at average velocity Vdrift in relation to the lab frame? If so, why do you say this?

 

[Orodruin]

If so, why do you say this?

Why do you say it would happen that way? There is no reason for your assumption and it would lead to a net charge on the conductor.

 

[RMJ]

Why do you say it would happen that way? There is no reason for your assumption and it would lead to a net charge on the conductor.

I don't understand what you mean.
'Why do I say what would happen what way? Which assumption are you talking about? Are you saying it would lead to a net charge on the conductor?

 

[Orodruin]

Your assumption that the distance between the charges in their rest frame is the same before and after they are set in motion. This assumption has no physical basis.

Edit: In order to keep the conductor neutral, the distance between the charges in the lab frame must remain the same. Hence, since this distance is a length contraction of their rest frame distance, the rest frame distance must be larger.

 

[RMJ]

Your assumption that the distance between the charges in their rest frame is the same before and after they are set in motion. This assumption has no physical basis.

Okay. I accept that my assumption is likely incorrect. May you please prove to me why it is false?

 

[Orodruin]

You have already been told why this is repeatedly in this thread. I see no point in repeating it again.

Charges are not little balls separated by a fixed distance.

 

[RMJ]

Your assumption that the distance between the charges in their rest frame is the same before and after they are set in motion. This assumption has no physical basis.

Edit: In order to keep the conductor neutral, the distance between the charges in the lab frame must remain the same. Hence, since this distance is a length contraction of their rest frame distance, the rest frame distance must be larger.

I understand the post might be confusing and I apologise for that. I'm trying to understand why the conductor is neutral. I'm not defining it as neutral. My point is that many people assume it (or know it) to be neutral when current is flowing, but that I don't understand why.
Why is it neutral?

 

[RMJ]

I understand the post might be confusing and I apologise for that. I'm trying to understand why the conductor is neutral. I'm not defining it as neutral. My point is that many people assume it (or know it) to be neutral when current is flowing, but that I don't understand why.
Why is it neutral?

Or maybe a better question is: Why does the rest frame separation distance of the moving charges increase just enough to make the conductor electrically neutral?

 

[Ibix]

Somebody on here pointed out to me that a moving rubber band is length contracted. But if it is stretched as it moves, the length contracted stretched rubber band can be the same length as its unstretched unmoving length.

Something similar happens here. At rest (no current) the separation between protons must be the same as the separation between electrons. Starting a current does not add electrons, so the wire must remain overall electrically neutral in the lab frame. Thus the separation of electrons in the lab frame must remain the same. So, just as the rubber band was stretched, the spacing of electrons in the electron rest frame must have increased so that the lab separation remains constant.

 

[Ibix]

Or maybe a better question is: Why does the rest frame separation distance of the moving charges increase just enough to make the conductor electrically neutral?

Because we haven't changed the number of electrons.

 

[RMJ]

Because we haven't changed the number of electrons.

Thank you so much. That really ties it up for me. I think I have a way better grasp on the concepts discussed than I did before this chat.

Thanks to everyone who replied and helped me along the way. I hope this chat can be of help to other students in the future.

Regards,
RMJ

 

[Dale]

The point of this thread is roughly that I truly don't understand how the wire can be assumed to be neutral if it also carries current

Ok, that is fine. We can discuss how the facts are possible, but we have start with a correct set of facts. So first we must assert as an experimental fact that a current carrying wire may be neutral. If we start with false facts, ie the false assertion that a current carrying wire must be charged, then we will get nowhere.

If not, why not?

The distance between conduction electrons is not rigidly fixed. They can be nearer or further in order to adjust in response to the external fields. In this case the electric field in the wire simply adjusts their spacing as needed so that the wire is neutral.

Note, it is possible to make the wire non-neutral in the lab frame instead. The neutrality or charge of the wire is under experimental control. Basically, we can control the voltage on either end of the wire. The difference in voltage controls the current and the average voltage controls the charge.

Since we can make the wire charged or uncharged (without a current) that demonstrates that we can change the spacing between charges in their rest frame. Since we can change the spacing in the rest frame without a current it should not be terribly surprising that we can change the spacing with a current