Voltage due to relative motion of a charge and conductive loop

[particlezoo]

Consider an electric charge Q with rest frame R and a closed conductive loop L with rest frame R'. Q is moving relative to L, and vice versa. In R, the loop is moving and receives no magnetic flux from Q thus no voltage according the integral form of Faraday's law. In R', the loop receives changing magnetic flux from Q and thus voltage is induced. So, what am I missing here??

 

[Orodruin]

The usual thing that people miss when considering EM in different inertial frames is that they do not take into account that what appears as a magnetic effect in one inertial frame may appear as an electric effect in a different inertial frame. This should be the case here as well.

 

[particlezoo]

The usual thing that people miss when considering EM in different inertial frames is that they do not take into account that what appears as a magnetic effect in one inertial frame may appear as an electric effect in a different inertial frame. This should be the case here as well.

In the rest frame R of the charge Q, the loop L is moving, but initially no current is established anywhere. So in R, what electric field from Q would act on L to generate a circulating electric field which may produce a current? None it appears.

Although it is moving, loop L is without a magnetic dipole to begin with, so it does not by virtue of its motions in R produce an electric dipole exerting a linear force on Q.

So which is true you think:

1) There is no voltage induced into L by moving L in frame R, where Q is stationary.

2) There is voltage induced into L by moving Q in frame R', where L is stationary.

The problem for me is that they both seem to be true. I suspect that you must think 1 as false, but I have trouble seeing where you would get the closed loop electric field from, since the charge Q produces no magnetic field in frame R.

 

[ealbers]

In the rest frame R of the charge Q, the loop L is moving, but initially no current is established anywhere. So in R, what electric field from Q would act on L to generate a circulating electric field which may produce a current? None it appears.

No net linear force at all either because the loop L is neutral, and although it is moving, it is without a magnetic dipole to begin with, so it does not by virtue of its motions in R produce electric dipole exerting a linear force on Q.

So which is true you think:

1) There is no voltage induced into L by moving L in frame R, where Q is stationary.

2) There is voltage induced into L by moving Q in frame R', where L is stationary.

The problem for me is that they both seem to be true. I suspect that you must think 1 as false, but I have trouble seeing where you would get the closed loop electric field from, since the charge Q produces no magnetic field in frame R.

Any time there are two objects which are moving relative to each other, there is a space time curvature stress between them, created when one or both of them was accelerated to created the delta V between them. This curvature manifests itself from various perspectives as either a magnetic field (think of them as stress lines representing the curvature between the moving frames), or electric fields.
Remember, the spacetime curvature between moving frames (not accelerating), results in many effects, length delta, time delta as well, which create the appearance of magnetic/electric fields/effects, but are really just the bending of our space. Think what would happen if a 2d creature on a piece of paper encountered a 'bend' in the paper, it would 'feel a force', the force of having to change direction, it would be invisible to that creature, but obvious to us.

 

[particlezoo]

Any time there are two objects which are moving relative to each other, there is a space time curvature stress between them, created when one or both of them was accelerated to created the delta V between them. This curvature manifests itself from various perspectives as either a magnetic field (think of them as stress lines representing the curvature between the moving frames), or electric fields.
Remember, the spacetime curvature between moving frames (not accelerating), results in many effects, length delta, time delta as well, which create the appearance of magnetic/electric fields/effects, but are really just the bending of our space. Think what would happen if a 2d creature on a piece of paper encountered a 'bend' in the paper, it would 'feel a force', the force of having to change direction, it would be invisible to that creature, but obvious to us.

What I need to know is how would a stationary electric charge induce a voltage into a moving conductive ring, which should be possible since it is easy to see how the voltage would be induced in the rest frame of the ring. I am looking for an answer at the level of Maxwell's equations and the Lorentz force equation. Surely that must exist... they are the correct physical laws after all.

 

[Ibix]

Any time there are two objects which are moving relative to each other, there is a space time curvature stress between them, created when one or both of them was accelerated to created the delta V between them. This curvature manifests itself from various perspectives as either a magnetic field (think of them as stress lines representing the curvature between the moving frames), or electric fields.
Remember, the spacetime curvature between moving frames (not accelerating), results in many effects, length delta, time delta as well, which create the appearance of magnetic/electric fields/effects, but are really just the bending of our space. Think what would happen if a 2d creature on a piece of paper encountered a 'bend' in the paper, it would 'feel a force', the force of having to change direction, it would be invisible to that creature, but obvious to us.

This is not correct at all. There is no spacetime curvature in this problem or, more precisely, it is utterly negligible. You seem to be mistaking the change in coordinates used by a moving frame for a change in spacetime.

What I need to know is how would a stationary electric charge induce a voltage into a moving conductive ring, which should be possible since it is easy to see how the voltage would be induced in the rest frame of the ring. I am looking for an answer at the level of Maxwell's equations and the Lorentz force equation. Surely that must exist... they are the correct physical laws after all.

My electromagnetism is shakier than it probably should be, but I feel bad posting without offering any suggestions. It occurs to me that the E and D fields inside the conductor will not be the same as they would be in free space. That means you've got time-varying fields as the conductor enters a given volume. That might be a fruitful area for further thought.

 

[Dale]

what electric field from Q would act on L to generate a circulating electric field which may produce a current?

The usual Coulomb field.

What I need to know is how would a stationary electric charge induce a voltage into a moving conductive ring,

This should be very clear from Coulomb's law. Usually this is the first thing taught in an EM class

 

[particlezoo]

The usual Coulomb field.

This should be very clear from Coulomb's law. Usually this is the first thing taught in an EM class

The usual Coulomb field.

This should be very clear from Coulomb's law. Usually this is the first thing taught in an EM class

There is voltage across the ring in frame R to be sure, but if it is anything like an electrostatic field, I don't see it how it would produce a circulating current. There will be some rearrangement of the charge distribution of course with a corresponding current, that I know, but that current would need to depend on the velocity of the ring in order to match what is seen in the ring's rest frame R', where the voltage depends on the velocity of the charge Q (which would actually be a squared dependence on the velocity of Q in frame R').

 

[Dale]

but that current would need to depend on the velocity of the ring

Which it clearly does. Suppose you have two charge distributions A and B. Clearly there is some current necessary to change from A to B, and clearly the faster the change the greater the current.

You need to just work an example or two. That will be a pain but it will build your confidence.

 

[ealbers]

Remember that when Maxwell/Gauss created the laws, the concept of relative frame of reference was not understood like it

This is not correct at all. There is no spacetime curvature in this problem or, more precisely, it is utterly negligible. You seem to be mistaking the change in coordinates used by a moving frame for a change in spacetime.
My electromagnetism is shakier than it probably should be, but I feel bad posting without offering any suggestions. It occurs to me that the E and D fields inside the conductor will not be the same as they would be in free space. That means you've got time-varying fields as the conductor enters a given volume. That might be a fruitful area for further thought.

Actually you'd be surprised how much length contraction is the cause of a magnetic field when you are considering a moving charge.
There is a good youtube video showing the effect, while the effect is tiny per particle, the sum given the sheer number of particles involved results in what you see as a magnetic field around moving charges...its really just a electric field.
See
For a good explanation.

 

[Ibix]

Actually you'd be surprised how much length contraction is the cause of a magnetic field when you are considering a moving charge.

Not surprised at all. But this has nothing to do with spacetime curvature. It's purely a coordinate change.

 

[ealbers]

Ummm...If time dilation and length contraction due to velocity are not expressed as a curvature of spacetime...I mean then how d you express them? What are they? Come on its a 4d coordinate system, you can exchange the time dimension for any of the others x1,x2 or x3

 

[Orodruin]

Ummm...If time dilation and length contraction due to velocity are not expressed as a curvature of spacetime...I mean then how d you express them? What are they? Come on its a 4d coordinate system, you can exchange the time dimension for any of the others x1,x2 or x3

Again, this has absolutely nothing to do with space-time curvature. Length contraction and time dilation is an effect of coordinate choices in flat Minkowski space for which the curvature is zero.

 

[Orodruin]

Remember that when Maxwell/Gauss created the laws, the concept of relative frame of reference was not understood like itActually you'd be surprised how much length contraction is the cause of a magnetic field when you are considering a moving charge.
There is a good youtube video showing the effect, while the effect is tiny per particle, the sum given the sheer number of particles involved results in what you see as a magnetic field around moving charges...its really just a electric field.
See
For a good explanation.

While that might be a good heuristic and pop-sci explanation, it clearly does not tell you anything about how a general EM field transforms between frames, which has very little to do with length contraction and everything to do with how an anti-symmetric tensor field transforms. Please be aware that people posting here often have extensive subject knowledge well beyond the popular level. This includes @Ibix when it comes to SR.

 

[Dale]

Ummm...If time dilation and length contraction due to velocity are not expressed as a curvature of spacetime...I mean then how d you express them? What are they? Come on its a 4d coordinate system, you can exchange the time dimension for any of the others x1,x2 or x3

Please do not hijack another person's thread. If you have questions about the difference between length contraction and curvature, please open a new thread. The previous respondents are 100% correct that this problem has nothing to do with spacetime curvature.

 

[particlezoo]

Which it clearly does. Suppose you have two charge distributions A and B. Clearly there is some current necessary to change from A to B, and clearly the faster the change the greater the current.

You need to just work an example or two. That will be a pain but it will build your confidence.

I can see how how doubling the velocity of the loop L in frame R, where the charge Q is at rest, can double the rate of change of the electric field experienced by the loop in frame R. However, I do not see how that translates to to increasing the induced electric field in frame R at each point in space in this frame, which would be the electrostatic field of Q, independent of the velocity of loop L.

Also, in the loop's rest frame R', the magnetic field from the charge Q would increase with Q's velocity in R', and the charge Q would pass by with reduced time, generating a voltage varying with the square of the velocity of charge Q in frame R'.

 

[Dale]

I do not see

I would recommend that you simply work a few problems. Start with two point charges moving at the same velocity in two different frames. Then do two point charges moving at different speed. Then do an infinite line of charge and a point charge in two different frames. Then do an infinite line of current and a point charge in two different frames. Do a loop last.

 

[particlezoo]

I would recommend that you simply work a few problems. Start with two point charges moving at the same velocity in two different frames. Then do two point charges moving at different speed. Then do an infinite line of charge and a point charge in two different frames. Then do an infinite line of current and a point charge in two different frames. Do a loop last.

I know that if I have a loop carrying a current, there will be an electric dipole produced by the loop depending on the motion of the loop, which can act upon a stationary charge. Conversely, if I chose the rest frame of the loop, that interaction between the current loop (now stationary) and the charge (now moving) would be magnetic instead of electric. But even when the loop is moving, it's current would be uniform all around, and if there is not rotational E-field component from the stationary charge, there is no net total E-dot-J between the charge and the current loop (except when the loop accelerates), and the only B would be from the current loop.

 

[particlezoo]

I know that if I have a loop carrying a current, there will be an electric dipole produced by the loop depending on the motion of the loop, which can act upon a stationary charge. Conversely, if I chose the rest frame of the loop, that interaction between the current loop (now stationary) and the charge (now moving) would be magnetic instead of electric. But even when the loop is moving, it's current would be uniform all around, and if there is not rotational E-field component from the stationary charge, there is no net total E-dot-J between the charge and the current loop (except when the loop accelerates), and the only B would be from the current loop.

Upon futher inspection, I realize that J would have sources and sinks due to the charge density of the moving current loop. So the E-dot-J between the stationary charge and the moving current loop would not be zero, with exceptions depending on the loop's position, orientation, and direction of motion. This would address the question of the induced power in the case that there is a current in the loop. That was my underlying concern.

However, the induced voltage in the case that there is (initially) no loop current as the loop moves relative to a stationary charge in frame R remains an unanswered question to me.

 

[Dale]

But even when the loop is moving, it's current would be uniform all around

Be careful here. The source term in Maxwell's equations is current density, not current. How does current density transform?

Again, I highly recommend you work these problems for yourself. I have not worked your specific problem, so I don't have direct answers to it. But having worked many other related problems I can tell you that it all works out consistently if you do it correctly.

 

[vanhees71]

BTW, when working the problem yourself, it is important to use the fully relstivistic constituent equations, particularly in your case Ohm's Law, which reads
$$\vec{j}=\sigma(\vec{E}+\vec{v} \times \vec{B}/c).$$
It's also a good exercise to think, why this is the correct law by thinking about the current (density) as the flow of charged particles (usually electrons for metals) in the wire.

 

[particlezoo]

BTW, when working the problem yourself, it is important to use the fully relstivistic constituent equations, particularly in your case Ohm's Law, which reads
$$\vec{j}=\sigma(\vec{E}+\vec{v} \times \vec{B}/c).$$
It's also a good exercise to think, why this is the correct law by thinking about the current (density) as the flow of charged particles (usually electrons for metals) in the wire.

I know it has been over a year since I posted this thread, but here is a thought that I had been holding back from this discussion:

The following quote is from Prof. Akira Hirose's lecture notes from course P812:

http://physics.usask.ca/~hirose/p812/notes/Ch10.pdf

Since the momentum $$\mathbf{p}$$ and energy $$\mathcal{E}$$ are related through

$$\mathbf{p} = \frac{\mathbf{v}}{c^2}\mathcal{E},$$

the acceleration $$\mathbf{a} = \frac{d\mathbf{v}}{dt}$$ can be readily found,

$$\mathbf{a} = \frac{q}{m\gamma}\left[\mathbf{E} + \boldsymbol{\beta} \times \mathbf{B} - \boldsymbol{\beta} \left(\boldsymbol{\beta}\ \dot\ \mathbf{E}\right)\right].$$

So, if I had a conducting loop moving relative to an electric charge, in the electric charge's rest frame, the electric field produced by that charge would integrate to zero over a closed loop, however, the path integral of ma of electrons along the loop would not integrate to zero, as would the path integral of ma for the positive ions along the loop. The result would be an EMF even in the absence of any magnetic field in this frame, and more importantly, it would increase with the square of the velocity of the loop in the charge's rest frame as given by the term $$- \boldsymbol{\beta} \left(\boldsymbol{\beta}\ \dot\ \mathbf{E}\right).$$

 

[Dale]

I am not sure if you consider that a problem or a feature of the solution. Did you ever work it out explicitly as I recommend?

 

[particlezoo]

I am not sure if you consider that a problem or a feature of the solution. Did you ever work it out explicitly as I recommend?

I consider it to be a feature of the solution.

The main crux as it turned out was how the voltage was going to scale with the square of the relative velocity between the electric charge and the conductive loop, meaning that one may expect the induced current in the loop to increase with the square of the relative velocity. An induced electric polarization on the surface of the conductive loop due to the external charge would change at a rate varying in proportion to their relative velocity (for v << c). It appears the difference between the specific Lorentz force and the actual acceleration of charges inside the loop is able to account the induced EMF. If this difference is indeed what is needed to be accounted for, then it would certainly need to not be omitted when working it out explicitly through the series of problems you described.