# Special Relativity problem

1. May 25, 2015

### arpon

Consider an infinite wire that has no electric current initially.

Then current starts to flow in the wire, i.e. the free electron drifts at speed $v$ (and the positive charges are fixed)
Applying special relativity, it appears that the distance between the electrons shrinks, i.e, density of electron in the wire increases.

So, it seems to me that the net charge is not conserved in this case, as the negative charge density has increased.

2. May 25, 2015

### Orodruin

Staff Emeritus
What makes you think the distance between the electrons in their rest frame is the same?

3. May 25, 2015

### arpon

Thanks for your reply. At first, it seemed very obvious to me. Later on, when I thought whether the distance should be same or not and why, I couldn't reason it out. Would you please give me some clue?

4. May 25, 2015

### Orodruin

Staff Emeritus
Your argumentation in the first post should already tell you that the distance between the electrons in their rest frame is different in the two scenarios if you want an uncharged conductor carrying a current.

However, there will be inertial frames where the conductor is negatively or positively charged. This follows from the Lorentz transformation of the 4-current.

5. May 25, 2015

### arpon

And in these inertial frames, the wire will always be negatively or positively charged (when any extra charge is not added to the system). The net charge will be conserved in a particular reference frame, but may vary from one frame to another.
As, there is no absolute reference frame, we cannot determine the 'absolute' net charge of the universe.
Do you agree with me?

6. May 25, 2015

### Orodruin

Staff Emeritus
Your argumentation assumes that this wire is infinite. This leads to a distribution of matter not going to zero at infinity and it is unclear how you want to start a current in an infinite wire (this would actually take an infinite amount of time).

7. May 25, 2015

### arpon

Does it matter?

8. May 25, 2015

### Orodruin

Staff Emeritus
Yes. The way to determine the total charge would be to integrate the charge density over all of space. If the charge density is not a function which decays sufficiently fast, this integral becomes nonsense.

9. May 25, 2015

### Staff: Mentor

To follow on with what Orodruin said, for a realistic scenario without an infinite wire (i.e. a circuit with a current loop), you do wind up with charge conservation. The charge density increase on one side of the circuit will be offset by a charge density decrease on the other side.

10. May 25, 2015

### pervect

Staff Emeritus
You didn't really specify what made the charges flow. If we assume that the charges flow because you applied a battery, and we additionally apply Maxwell's equations, we know that charge will be conserved, because a battery can't generate or destroy charges, it can only make them flow, and Maxwell's equations require charge conservation.

Applying Maxwell's equations to a current loop in free space is rather complex, it's much simpler if you have the wire over a ground plane, or some other system such as a pair of wires that acts as a transmission line. Charge will still be conserved either way, but it's a lot harder to analyze analytically what happens by applying Maxwell's equations - I don't think I've ever seen it done in the literature.

If you do do the transmission line analysis, you basically find that charge conservation does apply as expected, and that the assumption that the charges kept a constant proper distance when they started moving is wrong, at least when the charges are made to move by a battery.

It doesn't really matter whether you analyze a finite transmission line or an infinite one, but you do need to clarify the details of what makes the charges flow to analyze sensibly what happens. And you also need to consider the time element, how and when the charges start to move in a physically realilstic manner, if you want physically realistic results. Having all the charges "magicaly" start moving at the same time isn't physicall, having the charges flow as a pulse through a transmission line is.

AFAIK, if you have a current loop generated by a battery, you won't see any redistribution of charge from the viewpoint of a stationary observer, but you may and will see charge redistribution from the frame of reference of a moving observe. This is basically Purcell's analysis of the magnetic field.

11. May 25, 2015

### Fantasist

How can that be if the current loop is at rest (which the OP seemed to be implying)? The length contraction factor should be the same around the loop.

12. May 25, 2015

### Staff: Mentor

It is not at rest in the frames where the charge density changes.

Remember, length contraction is only occurs in the direction parallel to the motion, not in all directions.

13. May 25, 2015

### Ibix

The electrons are moving with respect to the loop - one direction on one side of the loop and the other direction on the other side. That means that there is no frame in which they are all at rest, and in any frame except the rest frame of the loop they have different speeds and hence different gammas.

14. May 25, 2015

### Fantasist

We are considering the rest frame of the loop. Consider a circular super-conducting loop where initially all electrons are rest. The OP was asking what happens if all electrons are set somehow in motion. Would a resulting increased electron density throughout the loop not violate charge conservation?

15. May 25, 2015

### Ibix

Why would there be an increased electron density in that frame? That there is not is the point Orodruin made in posts #2 and #4.

16. May 25, 2015

### Staff: Mentor

Post 1 was considering the rest frame of the wire. Post 4, 5, and on were considering other frames.

17. May 25, 2015

### A.T.

It would, that's why it doesn't happen in the rest frame of the loop.

18. May 26, 2015

### Fantasist

Maybe because of the length contraction effect? You can find it explained in many educational resources, for instance in http://physics.weber.edu/schroeder/mrr/MRRtalk.html where they say

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.

OK, in the picture posted by the OP the electrons are moving to the right, but this should not make any difference.

19. May 26, 2015

### Ibix

May I take it that you didn't re-read posts #2 and #4 as I recommended in the bit of my post that you didn't quote?

The point is that when no current flows, the electrons are at rest with respect to the wire, and are 0.1nm (for the sake of argument) apart in the rest frame of the wire. When the current flows, the electrons are moving. Unless electrons have appeared from nowhere they must still be 0.1nm apart in the rest frame of the wire. All that means is that they aren't 0.1nm apart in their own rest frame. But why should they be? They don't form a solid body held together by internal forces. They can be whatever distance apart they like and accelerated bodies don't necessarily end up the same distance apart in their final rest frame as they were in their original frame.

20. May 26, 2015

### Staff: Mentor

Read the reference you cited. Nowhere does it use length contraction to claim that the wire is charged in the lab frame. It uses length contraction, plus the fact that the wire is not charged in its rest frame, to show that it is charged in other frames.