I Electromagnetism question about moving a current-carrying wire instead of the test charge

OK so a wire with a current creates a b field in accordance with the right hand rule. Now moving charge will feel a force (if its moving in the correct direction) while stationary charges will not feel a magnetic force.

What I am curious about is if we move the wire itself, will a stationary charge possibly feel a magnetic force? From the perspective of the wire, the charge is moving, so I assume there would be a magnetic force even though the charge is stationary.
 
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A.T.

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What I am curious about is if we move the wire itself, will a stationary charge possibly feel a magnetic force? From the perspective of the wire, the charge is moving, so I assume there would be a magnetic force even though the charge is stationary.
The stationary charge cannot experience a magnetic force since its velocity is zero and the magnetic force is equal to ##q(\mathbf v \times \mathbf B)##.

However, for the symmetry reason that you mention it must experience a force. And since it is not moving this force must be an electric force. This implies that a purely magnetic field in one frame will transform to have at least some electric field in another frame.
 
The stationary charge cannot experience a magnetic force since its velocity is zero and the magnetic force is equal to ##q(\mathbf v \times \mathbf B)##.

However, for the symmetry reason that you mention it must experience a force. And since it is not moving this force must be an electric force. This implies that a purely magnetic field in one frame will transform to have at least some electric field in another frame.
So if I have a wire with AC current and then I physically move the wire back and forth along its length at the same frequency of the AC current, I would perceive my wire emitting an electric field?
 

tech99

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The stationary charge cannot experience a magnetic force since its velocity is zero and the magnetic force is equal to ##q(\mathbf v \times \mathbf B)##.

However, for the symmetry reason that you mention it must experience a force. And since it is not moving this force must be an electric force. This implies that a purely magnetic field in one frame will transform to have at least some electric field in another frame.
I think Maxwell was aware of this symmetry problem and was on the edge of understanding Relativity.
 
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So if I have a wire with AC current and then I physically move the wire back and forth along its length at the same frequency of the AC current, I would perceive my wire emitting an electric field?
Yes. Also yes if the current were DC or if the movement were at a different frequency or if the direction were different or if the motion were not sinusoidal.
 

vanhees71

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OK so a wire with a current creates a b field in accordance with the right hand rule. Now moving charge will feel a force (if its moving in the correct direction) while stationary charges will not feel a magnetic force.

What I am curious about is if we move the wire itself, will a stationary charge possibly feel a magnetic force? From the perspective of the wire, the charge is moving, so I assume there would be a magnetic force even though the charge is stationary.
There is only an electromagnetic field. The split into electric and magnetic components is frame dependent.

In this connection it is important to note that the wire is uncharged in the rest frame of the conduction electrons rather than the restframe of the wire. This is overlooked in almost all treatments of this problem, because a non-relativistic approximation for Ohm's Law is used and thus the Hall effect is neglected. For a nice treatment, see

P. C. Peters, In what frame is a current-conducting wire
neutral, Am. J. Phys. 53 (1985) 1156.
 
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P. C. Peters, In what frame is a current-conducting wire
neutral, Am. J. Phys. 53 (1985) 1156.
https://doi.org/10.1119/1.14075
Do you have a non-paywalled version of this paper? I am particularly interested in what assumptions they make regarding the grounding of the current-conducting wire. I have thought about this issue before but never pursued it in detail. When I thought about it I suspected that the bulk of the wire would be uncharged in the lab frame only if the center of the wire is grounded, which is indeed not the usual arrangement for a lab.
 

vanhees71

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Unfortunately, I guess there's no (legal) non-paywalled version of the paper. The argument is simple: In the frame with the wire at rest the (quasi-)free conduction electrons are deflected by the magnetic field, which builds up a negative charge density within the wire and an electric field such that the corresponding force compensates the magnetic force, such that in the steady state there's no radial current (Hall effect). In the frame where the conduction electrons are at rest there's also a magnetic field due to the moving positive wire, but no force acts on the freely moving conduction electrons since these are at rest. The bulk of the wire (i.e., the wire without the freely moving conduction electrons) doesn't contain freely moving charges and thus no electric field is built up. In this frame thus the charge density must vanish.

I once derived this result without knowing about this paper in a different way, simply using the correct relativistic version of Ohm's Law,
$$\vec{j}=\sigma (\vec{E}+\vec{v} \times \vec{B}/c),$$
assuming that ##\vec{j}## is homogenous over the cross section of the wire and directed along the wire (as in the usual textbook ansatz, using the incomplete Ohm's law neglecting the magnetic force). The resulting charge density is ##\mathcal{O}(v^2/c^2)##, where ##v## is the speed of the conduction electrons, i.e., totally academic in view of the fact that house-hold currents imply speeds in the range of mm/s.

Here's a writeup of my treatment of the DC-carrying coax cable (though it's in German). The relativistic part is Sect. 6 starting at p. 11:

 

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