I Electric & Magnetic Fields Inside Conducting Wires

[bob012345]

Suppose a straight current carrying wire is immersed in a uniform electric field along its axis. For this problem, please don't worry about complete loops or return paths, just consider a segment of straight wire carrying a current. Since external electric fields don't get inside conductors due to charges on the surface, I expect that field doesn't interfere with the current and voltage drop along the wire due to an independent source such as a battery. I'm assuming that's true.

Now consider that the wire moves through a region of space containing a uniform magnetic field at right angles such that there is a force on the wire. The magnetic field permeates the wire. As the wire picks up speed wrt the field, there should be an electric field seen in the reference frame moving with the wire. This electric field is part of the relativistic transformation of the magnetic field. The question I'm asking is if this electric field acts within the wire to oppose the current and thus acts substantially different that the electric field of the first case? Thanks.

 

[CWatters]

Are you describing Counter/back EMF??

https://en.m.wikipedia.org/wiki/Counter-electromotive_force

 

[pervect]

The electromagnetic field transformations, for instance the transformation of the perpendicular part of the field, $E_\bot' = \gamma \left( E_\bot + v \times B \right)$ are completely general. There are not any different "flavors" of E and B. So I don't see how it makes any sense to consider the two cases you describe as being fundamentally different. In both cases in the frame of the wire there is an E field, thus they are similar.

Ohm's law, however, is not as simple. It takes some work to come up with a covariant formulation of it - the covariant formulation would make it true in any reference frame, the non-covariant formulation requires that one consider a special frame, the frame of the conductor. Thus $j = \sigma E$ may not in general be correct. So in the problem at hand, the charges in the wire distribute themselves so that the electric field in the frame of the wire is small, i.e. zero if the wire is an ideal conductor. This doesn't mean that the E field in the interior of the conductor is zero in all frames, though. I believe is impossible if the B field is not also zero.

 

[bob012345]

Are you describing Counter/back EMF??

https://en.m.wikipedia.org/wiki/Counter-electromotive_force

No, I don't think so.

 

[jartsa]

So we consider just a short part of a long wire. We note that to get the external electric field canceled inside the wire, there should be an axial charge density gradient on the outside of the short wire segment.

And then in the other case there should be an axial charge density gradient on the outside of the wire segment, in order to create an electric field inside the wire, in order to get the electrons moving against the Lorentz-force. It's the battery's job to create that charge density gradient. (Total electric field inside the wire is zero, if we consider the Lorentz-force to be a force caused by an electric field)

 

[sweet springs]

Say we put a conductor, e.g. metal piece, into static electric field. Surface of the body is charged so that there exist no electric field inside.
In other IFRs we observe external electric and magnetic field and electromagnetic field of moving surface charges. I guess detailed analysis of such a case would give answer to your question.

Say we put a conductor into static magnetic field. No charge appear on surface.
Say we put a moving conductor into static magnetic field. Isn't the body charged?

 

[bob012345]

So we consider just a short part of a long wire. We note that to get the external electric field canceled inside the wire, there should be an axial charge density gradient on the outside of the short wire segment.

And then in the other case there should be an axial charge density gradient on the outside of the wire segment, in order to create an electric field inside the wire, in order to get the electrons moving against the Lorentz-force. It's the battery's job to create that charge density gradient. (Total electric field inside the wire is zero, if we consider the Lorentz-force to be a force caused by an electric field)

There is no reason the fields would in general cancel. They are caused by independent phenomenon.

 

[bob012345]

The electromagnetic field transformations, for instance the transformation of the perpendicular part of the field, $E_\bot' = \gamma \left( E_\bot + v \times B \right)$ are completely general. There are not any different "flavors" of E and B. So I don't see how it makes any sense to consider the two cases you describe as being fundamentally different. In both cases in the frame of the wire there is an E field, thus they are similar.

Ohm's law, however, is not as simple. It takes some work to come up with a covariant formulation of it - the covariant formulation would make it true in any reference frame, the non-covariant formulation requires that one consider a special frame, the frame of the conductor. Thus $j = \sigma E$ may not in general be correct. So in the problem at hand, the charges in the wire distribute themselves so that the electric field in the frame of the wire is small, i.e. zero if the wire is an ideal conductor. This doesn't mean that the E field in the interior of the conductor is zero in all frames, though. I believe is impossible if the B field is not also zero.

The difference I think, if there is one, is that in the first case, it's an external electric field that the wire can keep out but in the second case, it's an internal magnetic field transformed into an internal electric field by relativity. If one were inside the wire at rest wrt the magnetic field, one sees just that field. Once moving wrt to that magnetic field, one sees that internal field transformed and now has an electric field component opposing the current flow and that has nothing to due with back-emf's. The question is if that electric field is inside the wire or not. I think it is.

 

[sweet springs]

The magnetic field permeates the wire. As the wire picks up speed wrt the field, there should be an electric field seen in the reference frame moving with the wire.

Texts says E=0 within conductor so that no force works on electrons. It's about the case v=0 or B=0. In general force acting on electrons is F=E+vxB=0 for static case. Electric field can appear inside conductor.

 

[pervect]

The difference I think, if there is one, is that in the first case, it's an external electric field that the wire can keep out but in the second case, it's an internal magnetic field transformed into an internal electric field by relativity. If one were inside the wire at rest wrt the magnetic field, one sees just that field. Once moving wrt to that magnetic field, one sees that internal field transformed and now has an electric field component opposing the current flow and that has nothing to due with back-emf's. The question is if that electric field is inside the wire or not. I think it is.

I don't think I have a better answer than saying that there is no difference. Perhaps some background observations will help where we differ - I believe we do differ, as I'm not sensing agreement.

Electric and magnetic field components are, in and of themselves, frame dependent quantities. They also transform according to a specified laws, so that if you specify all the electric and all the magnetic field components in one frame, then you can compute what they are in another frame. You can't perform the computation unless you know all of the quantites, both electric and magnetic compoents, but when you do, and you specify the frames, there computation is well documented, though I didn't write out all the relevant laws, just some of them.

So, in the frame of the conductor/wire, in that frame, in both cases, there is an external electric field. You can pick another frame if you like, and analyze things in that frame, except that the description of a conductor is difficult in any frame other than the frame of the conductor. This relates to the remarks I made about covariance. Therefore we pick the frame of the conductor, and in that frame, in both cases, there is an external electric field.

Given this much, the rest follows. The wire then does what wires do, which I would describe as the charges flowing in the conductor to keep the field in the wire low - zero in the idealized case of an ideal conductor. This happens as fields in the wire causes charges to flow - they flow naturally in a direction to keep the field low. That's all there is to it.

You seem to be disagreeing with this, but I really can't quite figure out exactly where. It seems pretty straightforwards to me, once we know the key facts, which are the ones I outlined.

I'll give the propositions numbers for easy reference so that you can give the number of the proposition that you may disagree with.

#1 In any frame, there are components of the electric and magnetic field unique to that frame
#2 There are laws that tell us how to transform fields from one frame to another
#3 Conductors have the property that, in the frame of the conductor, current flows proportionally to the electric field in the conductor. $j = \sigma E$. Unlike #1 and #2, statement #3 is a frame dependent statement, it's true only in the frame of the conductor
#4 As a consequence of #3, charges in the conductor move in such a way to make the field low (zero in the case of an idealized conductor) inside the conductor in the frame of the conductor. We need to specify which frame the field is zero in because of remarks #1 and #3 - the field components depend on the frame, and ohm's law is a frame dependent law.

 

[jartsa]

There is no reason the fields would in general cancel. They are caused by independent phenomenon.

Well I think they have to always cancel, because:

The electrons inside a good conductor never feel any net force, they would accelerate if they did. If they accelerated, there would be a current of billions of amperes very quickly.

So if the electrons inside a good conductor feel a Lorentz-force, then there must be also a charge distribution that cancels that force.
But maybe you are right. If we very slowly accelerate a conductor in a magnetic field, we can have current increasing and electrons inside the conductor being accelerated by an electric field as long time as we want.

 

[bob012345]

Well I think they have to always cancel, because:

The electrons inside a good conductor never feel any net force, they would accelerate if they did. If they accelerated, there would be a current of billions of amperes very quickly.

So if the electrons inside a good conductor feel a Lorentz-force, then there must be also a charge distribution that cancels that force.

You can't have a current without a potential difference and a current was assumed for this problem. I didn't say it was an ideal conductor, but it is a real one.

 

[bob012345]

I sense this simple problem is becoming greatly over complicated. I'm simply asking if the current, driven by a fixed source like a battery moving with the wire, is opposed and thus reduces as the wire goes faster. Or, does the current stay the same as the wire goes faster.

 

[Dale]

I'm simply asking if the current, driven by a fixed source like a battery moving with the wire, is opposed and thus reduces as the wire goes faster. Or, does the current stay the same as the wire goes faster.

The current density is the spacelike part of a four vector $(c\rho,\mathbf J)$. So it transforms like a length, ie it contracts.

 

[bob012345]

Ok, I've been thinking and I now think you all are correct in saying there is no difference. Fields have the same effect on matter regardless of how they are generated. My original assumption may have been wrong though. I assumed an electric field parallel to a current carrying wire would have no effect on the circuit but I'm not sure about that. If there is a completed circuit, shouldn't the electric field act as an opposing EMF in my circuit unless the entire circuit is within the field and the net effect cancels around the loop which wasn't my original assumption.

Then, in my original magnetic field problem, the electric field as seen in the moving frame of the wire acts as a counter EMF to the battery which generates current that makes the wire move. That counter EMF can be equivalently seen by an observer at rest wrt the magnetic field as a motive EMF generated by the Lorentz force.

I'm getting this thinking from Feynman lectures V2 and from the way electrodynamic tethers work in space;

https://en.m.wikipedia.org/wiki/Electrodynamic_tether

I'm now thinking the answer to my original question "if this electric field acts within the wire to oppose the current and thus acts substantially different that the electric field of the first case?" Is yes but not because it's different, but because it's the same.