# Relativistic effects of a current carrying wire

1. Jan 6, 2014

### Jimmy87

Hi, I recently came across the idea that relativity can play a role in the repulsive force of a charge outside of a current-carrying wire. The situation described was when a positive charge (q) moves at the same speed and direction as the drift velocity of the electrons in the wire (assume non-conventional current flow). From our frame of reference the magnetic force causes repulsion. In the charge q's frame it is not moving with respect to the electrons but instead the positive charges appear to be moving in the opposite direction. It was argued that this cannot be explained classically as the charge is stationary in its frame so cannot interact with the magnetic field created by the moving positive charges. It was then argued that length contraction of the positive charges caused a higher positive charge density which causes the charge q to be repelled by an electric field from its reference frame.

The bit I really don't understand (because it wasn't even mentioned) was if the charge q is at rest but there is still a current flowing in the wire. From the charges frame it is not moving and neither are the positive charges but the electrons ARE moving as there is a current in the wire. Therefore surely you can equally argue that the electrons will length contract from the charge q's reference frame? But this means the wire would be negatively charged from the charge q's reference frame and be attracted to it but this doesn't happen when you put a stationary positive charge outside a current carrying wire! What am I not understanding? Thank you to anyone who can help me!

Last edited: Jan 6, 2014
2. Jan 6, 2014

### jartsa

Some moving electrons' electric fields will contract and concentrate in to the place where the test charge is.

Some moving electrons' electric fields will contract and concentrate in to the place where the test charge is not.

So therefore the test charge will say that every moving electron seems to have a decreased charge, except the nearest ones, which seem to have an increased charge.

EDIT: I see you mean increased electron density. Well, electron density does not increase if some electrons are "falling" in a homogeneous electric field, because - why would it increase?

On the other hand, when some electrons fall in a homogeneous gravity field, electrons get closer to each other. This is relevant, because from the equivalence principle we can see that if observer accelerates, he will see an electron formation to change shape.

Last edited: Jan 6, 2014
3. Jan 6, 2014

### Staff: Mentor

In the frame of the charge, the electrons don't give a current as they do not move (on average). The positive nuclei, moving in the opposite direction, give a current. There is a magnetic field in both frames.
For the same reason, there is no contraction for the electrons. They are at rest relative to the charge outside.

4. Jan 6, 2014

### Jimmy87

Thanks for the reply guys but I still can't see how that would work. The positive test charge outside the wire is stationary. If a current flows in the wire then electrons have a net movement whilst the positive charges (nuclei) are fixed. From the reference frame of the test charge outside, the electrons must surely be length contracted as they are moving relative to it. But this would imply that any STATIONARY positive charge outside a current-carrying wire is attracted as the wire has a net negative charge from its reference frame. My point is that basically every wire should be negatively charged because it has electrons flowing relative to a stationary observer (and the positive charges).

5. Jan 6, 2014

6. Jan 6, 2014

### Staff: Mentor

If I understand your question correctly, in frame A we have a neutral wire which is carrying current and a stationary charge, q. You would like to analyze it in frame A and some frame B where q is moving.

In A there is no electric field, so there is no electrostatic force on q. There is a magnetic field due to the current, but since q is stationary the magnetic force is also zero. The total force on q in A is therefore 0.

In B the wire is still carrying a current, but it also becomes charged. So in B there is an electric field and therefore an electrostatic force on q. There is also a magnetic field, and since q is moving there is therefore a magnetic force on q. Not coincidentally, those forces are equal and opposite, so the total force on q in B is also 0.

7. Jan 6, 2014

8. Jan 6, 2014

### Jimmy87

Thanks for your answer. I think the link to other thread along with your answer has clarified it. My thinking was that if there is a current in the wire then there will be electrons in the wire moving relative to the stationary charge outside the wire. This charge outside the wire would therefore see length contraction of the electrons meaning that an electric field would exist (i.e. wire becomes charged). But I think my error is that if the charge outside the wire is stationary then its in the lab frame which ALWAYS has a neutral wire therefore regardless of the relative motion of the charge carries in the wire, there will be no relativistic length contraction. Could someone confirm if I have correctly interpreted this? Many thanks to all!