Electrostatic effect of a current carrying conductor

[ashokanand_n]

Consider an infinitely long stationary conductor which carries a steady uniform current.

1. At an arbitrary Test Point outside the conductor the Electric Field should be ZERO according to Maxwell's Equation.
2. But if Special theory of Relativity is considered, the moving charges (electrons) inside the conductor is an inertial frame moving with a fixed velocity with respect to the Test Point. Hence there will be a Lorentz Contraction for the moving frame resulting in the change of (negative) charge density. This will give rise to an imbalance in the +ve and -ve charges in the conductor and hence an Electric Field will be experienced at the Test Point.

How is this discrepancy resolved?? And what does happen in reality??

Ashok

 

[atyy]

Dan Schroeder, Purcell Simplified
http://physics.weber.edu/schroeder/mrr/MRR.html

 

[Dale]

Consider an infinitely long stationary conductor which carries a steady uniform current.

1. At an arbitrary Test Point outside the conductor the Electric Field should be ZERO according to Maxwell's Equation.
2. But if Special theory of Relativity is considered, the moving charges (electrons) inside the conductor is an inertial frame moving with a fixed velocity with respect to the Test Point. Hence there will be a Lorentz Contraction for the moving frame resulting in the change of (negative) charge density. This will give rise to an imbalance in the +ve and -ve charges in the conductor and hence an Electric Field will be experienced at the Test Point.

How is this discrepancy resolved?? And what does happen in reality??

Actually, it turns out that there is no discrepancy. Let's say in situation 1 that the test charge is at rest. There is no electrostatic force because the conductor is uncharged in this frame and there is no magnetic force because the velocity of the test charge is zero. In situation 2 there is, as you noticed, an electrostatic force, but here the test charge is moving so there is also a magnetic force. These forces always cancel each other out so that if one reference frame detects no force the same is true in all reference frames. Thus there is no experiment you can perform in different frames that will disagree on the measurable results, they will only disagree in what they call E and B.

 

[ashokanand_n]

Thanks for the reply.

Actually, it turns out that there is no discrepancy.

But I think I have not put the question very clearly. Actually there are no different situations discussed in the question. There is only one situation. i.e., There is a TEST CHARGE at a fixed point away from a Steady Current Carrying Infinite Wire. Since the TEST CHARGE is not moving, there is no magnetic force on it. At the same time if the moving electrons (inside the wire) are considered with respect to the TEST CHARGE (The reference frame is still the Stationary TEST CHARGE) they will experience what is called Lorentz Contraction resulting in an increased -ve charge density and hence the TEST CHARGE will experience an Electrostatic Force. If this is true then it means that there is always an Electric Field around a current carrying wire which is against Maxwell's Laws. And that is what I was referring to as "discrepancy".

 

[Dale]

If you are only interested in a single reference frame (rest frame of wire and test charge) then I don't understand where you think there is a discrepancy. There is no Lorentz contraction unless you consider at least two frames. In the lab frame the wire is uncharged, so the distance between charge carriers (in the lab frame) is fixed by that condition.

 

[ashokanand_n]

If you are only interested in a single reference frame (rest frame of wire and test charge) then I don't understand where you think there is a discrepancy. There is no Lorentz contraction unless you consider at least two frames. In the lab frame the wire is uncharged, so the distance between charge carriers (in the lab frame) is fixed by that condition.

In the Lab Frame the -ve charge carriers are moving whereas the +ve charge carriers are at rest (Supposing that the current is due to flow of electrons). So the -ve charge carriers can be treated as a second frame which will experience Lorentz Contraction because of which the wire should appear to be negatively charged.

Please correct me if i am wrong.

 

[Ich]

the wire should appear to be negatively charged.

Please correct me if i am wrong.

If it were negatively charged, the electrons would repel each other and drive away surplus electrons until the net charge is zero again.

 

[Dale]

Hi ashokanand,

I think you are making a mistaken assumption that the proper distance between the electrons is the same when they are at rest wrt the wire as when they are moving wrt the wire. This would be true if the free electrons in the wire formed a single very rigid object, but that is not the case at all. The distance between electrons is very flexible, particularly in a conductor.

The fact that the wire is uncharged in the lab frame is an experimental observation. Therefore that observed fact determines the distance between the flowing electrons. You are making an incorrect assumption about the distance between the electrons, recognizing that it implies a charged wire, and realizing that a charged wire goes against observed facts, but then you simply missed the conclusion that your assumption must therefore be incorrect.

 

[ashokanand_n]

Thank you DaleSpam,

I think i got your point.