E & M Magnetism Relativity Paradox

[fysicsandphol]

E & M Magnetism Relativity "Paradox"

In theory the laws of magnetism are derivable from coulomb's law and special relativity. Right. While my physics homework group were doing a problem set, I came up with this seeming paradox. (This isn't a homework question). There are 2 senarios:

Senario 1:
The stationary lab frame is that of a positive charge. We'll call this p.
Some distance above p there is a theoretical wire with an equal amount of positive ions and electrons flowing in opposite directions with a velocity v relative to p.

Senario 2:
An equivalent way of stating this is that there is a wire with electrons moving with speed u (where u is just the relative velocity of the electrons in the frame the positive ions). The positive ions are stationary (The lab frame is that of the + ions). At point p there is a positive charge moving with speed v parallel to the wire.

It is obvious that these are qualitativly the same scenario from 2 reference frames.
In scenario 2, it is obvious that since the wire has a current, it creates a magnetic field B. Since the charge at p is moving, this implies there is a magnetic force on particle at p pointing either up, or down.

Using only coulomb's force and the laws of relativity, derive the magnetic force on the particle at p in scenario 2?

This isn't a real paradox. But it seems counter-intuitive because of the parity between the positive flow and the negative flow in case 2.

 

[JesseM]

I assume in scenario 1 the spacing between positive ions is the same as the spacing between electrons as seen in the frame of the positive charge, so there is no net electric force on the positive charge? If so, this means that in scenario 2 where the positive ions are at rest, the spacing between the electrons is smaller than the spacing between the positive ions due to length contraction, so there is more negative charge then positive charge in a given section of wire and thus there's a net attractive electric force from the wire on the positive charge, which must exactly balance out the repulsive magnetic force in this frame (since both frames must agree the positive charge doesn't accelerate towards or away from the wire).

 

[Born2bwire]

I assume in scenario 1 the spacing between positive ions is the same as the spacing between electrons as seen in the frame of the positive charge, so there is no net electric force on the positive charge? If so, this means that in scenario 2 where the positive ions are at rest, the spacing between the electrons is smaller than the spacing between the positive ions due to length contraction, so there is more negative charge then positive charge in a given section of wire and thus there's a net attractive electric force from the wire on the positive charge, which must exactly balance out the repulsive magnetic force in this frame (since both frames must agree the positive charge doesn't accelerate towards or away from the wire).

Yeah, pretty much. This appears to be a textbook example of relativity in electrodynamics. Many electrodynamics texts, like Griffiths or Purcell, deal with the problem of the force between two parallel wires of currents in this same manner.

You treat the wire as having a positive charge density flowing in say the -x direction and an equal but opposite charge density flowing in the +x direction. The net charge is still zero but you have a current flowing in the wire. Let us take some test charge and let it travel parallel to our currents. In the lab frame, the moving charge will experience a Lorentz force from the magnetic field produced by the wire's current. If you were to observe the wire in a moving reference frame that moves such that the test charge that was moving in the lab frame is now stationary, then the negative and positive charge densities undergo a Lorentz contraction. But since the two densities are traveling in opposite directions, the length of the negative charge is different then that of the positive density. Thus the wire now has a net electric charge in this frame which will produce the exact same force observed in the lab frame but now due to an electric field.

 

[Dale]

Hi physicsandphol, welcome to PF!

Here is a good place to start. The example you are talking about is covered in the second section.

http://physics.weber.edu/schroeder/mrr/MRRtalk.html

 

[fysicsandphol]

Well see there is my issue. I have gone through my textbook (Purcell actually) several times and they give more or less the same answer. After thinking about it some more I have a better way of expressing my issue:

There is a current wire and a frame stationary relative to the positive ions. According to the laws of physics, there is no force in this case, therefore the charge densities for the positive and negative ions must appear the same.
Now there is a current wire and a frame stationary relative to the negative ions. According to the laws of physics, since there is a current and motion relative to the positive ions, there must be a force. Therefore the charge densities must be different.

But the only difference between the two frames is which ions I took to be the stationary frame. Why do the charge densities balance each other in the positive frame but not the negative frame? It seems like the laws of physics magically change depending on which is positive and which is negative.

Relativity alone has never been an issue for me so I think I must be missing some underlying assertion about how currents and charge work.

 

[fysicsandphol]

Hi physicsandphol, welcome to PF!

Here is a good place to start. The example you are talking about is covered in the second section.

http://physics.weber.edu/schroeder/mrr/MRRtalk.html

Looking at the diagram showing the test frame and the lab frame,
It all makes perfect sense if I accept that the ions are equally spaced out in the lab frame.
Couldn't I just as easily say that the charge densities are unequal in the lab frame such that when transforming them to the test frame they appear equal?
Why does current flow so that charge density cancels out for the positive frame but not the negative frame?

 

[Dale]

Why do the charge densities balance each other in the positive frame but not the negative frame? It seems like the laws of physics magically change depending on which is positive and which is negative.

There is nothing magical about it. If the positive and negative charge carriers are moving relative to each other then there can only be one frame where the wire is uncharged. In all other frames the wire is charged. Just work out the math. In any other frame the differing amounts of length contraction make the wire charged.

Looking at the diagram showing the test frame and the lab frame,
It all makes perfect sense if I accept that the ions are equally spaced out in the lab frame.
Couldn't I just as easily say that the charge densities are unequal in the lab frame such that when transforming them to the test frame they appear equal?

Sure, you could do that. Then the wire would necessarily be charged in the lab frame.

 

[fysicsandphol]

No I realize that the charge densities can only be equivalent in one frame. My question is why this is always the lab frame. I realize that we are only considering an uncharged wire, implying that that is a standard case in nature. My problem is that I don't see why this is the standard case. It makes just as much sense to me to say that if you hooked up a battery to a resistor that the current densities should be equal in the moving frame as it does for the current densities to be equal in the lab frame.

 

[Dale]

No I realize that the charge densities can only be equivalent in one frame. My question is why this is always the lab frame.

Because the lab is grounded usually.

I don't know what you mean "standard case in nature". Batteries and wires are human designed, and they are traditionally designed to be uncharged in their rest frame. This is not some mysterious cosmic coincidence, it is an engineering design choice.

 

[Saw]

I was going to post a question that seems to be the same as the one posed by fysicsandphol. So I have thought I should put it here.

I refer to the example mentioned in http://galileo.phys.virginia.edu/classes/252/rel_el_mag.html:

As observed in the ground frame, there is:

- A stationary wire with a row of fixed +ions and a current of electrons flowing to the right with velocity +v.
- A + charge Q on a train moving also with velocity +v.

As observed in the train frame, there is:

- A stationary charge Q and also a stationary row of electrons hovering in the wire.
- A row of + ions flowing to the left with velocity -v.

What happens is that Q is accelerated away from the wire.

According to the text, the explanations for this fact are:

In the ground frame:

- There is no electrostatic force, because the wire is neutral, as it contains equal densities of + and – charges.
- There is *magnetic force* because there is a - current in the wire and Q is in motion, with velocity v, wrt the wire.

In the train frame:

- There is no magnetic force, because Q is stationary.
- It seems as if there should be no electrostatic force, since we said the wire is neutral… But wait a minute. In this frame, the row of +ions is in motion, so it is length contracted, which in turn means higher + density. That implies the wire has + charge and hence it exerts an *electrostatic force* repelling the +Q.

My doubt is: in the ground frame, the electrons are in motion. So one could say that their row is length contracted, that there is higher - density, that the wire has – charge and hence it exerts an *electrostatic force* repelling the +Q. Why don’t we say so?

 

[JesseM]

My doubt is: in the ground frame, the electrons are in motion. So one could say that their row is length contracted, that there is higher - density, that the wire has – charge and hence it exerts an *electrostatic force* repelling the +Q. Why don’t we say so?

Because they assumed in the problem that in the lab frame the spacings were equal, meaning the rest distance between electrons must be large than the rest distance between positive ions. You could of course imagine a different problem where the spacings were unequal in the lab frame, though as DaleSpam suggested the fact that the wire has zero net charge in the lab frame is probably a physical consequence of the fact that the wire is grounded to some mass at rest in the lab frame.

 

[Dale]

My doubt is: in the ground frame, the electrons are in motion. So one could say that their row is length contracted

Their row is length contracted to the point that the spacing is equal to the + charge spacing. Therefore, in the electron's rest frame their spacing must be greater, because the rest frame always has the greatest length.

 

[Saw]

Because they assumed in the problem that in the lab frame the spacings were equal, meaning the rest distance between electrons must be large than the rest distance between positive ions. You could of course imagine a different problem where the spacings were unequal in the lab frame, though as DaleSpam suggested the fact that the wire has zero net charge in the lab frame is probably a physical consequence of the fact that the wire is grounded to some mass at rest in the lab frame.

Their row is length contracted to the point that the spacing is equal to the + charge spacing. Therefore, in the electron's rest frame their spacing must be greater, because the rest frame always has the greatest length.

Ok, thanks, understood. Let me check it with numbers:

v = 0.5 c
In the + ions' frame, the latter are spaced out by 1.
In the electrons' frame, the latter are spaced out by 1.154.
In the + ions' frame, the electrons -after considering LC- are spaced out by 1, same as the +ions, hence the wire has no charge.
In the electrons' frame, the +ions -after considering LC- are spaced out by 0.866, quite less than the electrons themselves (1.154), hence the wire has +charge.

So "charge of an extended object" (like the wire) is also a relative, frame-dependent concept... What about the charge of a subatomic particle?

 

[Dale]

So "charge of an extended object" (like the wire) is also a relative, frame-dependent concept... What about the charge of a subatomic particle?

yes. Charge is the timelike component of the four-current. So it is relative the same way that the components of any four-vector is.

 

[bcrowell]

yes. Charge is the timelike component of the four-current. So it is relative the same way that the components of any four-vector is.

Wait, you mean the charge *density*, right? Charge is a Lorentz scalar; this is verified to extremely high precision because the hydrogen atom is electrically neutral.

 

[Dale]

Oops, yes.

 

[Saw]

Wait, you mean the charge *density*, right? Charge is a Lorentz scalar; this is verified to extremely high precision because the hydrogen atom is electrically neutral.

Noted. Yes, actually, after re-reading, I see that the text I was referring to made it clear. What is relative or frame-dependent is the charge density and hence the denomination of the field as magnetic (in the frame of the wire and +ions, where + and - densities are equal) or electric (in the frame of the electrons and the outside charge, where the row of +ions is denser) and hence the denomination of the force in action, although not its magnitude.

And what if the outside + charge had been at rest wrt the wire?

- In the frame of the +ions, the charge would be subject neither to electric force (the wire is neutral) nor magnetic force (the charge is stationary).
- In the frame of the electrons, the wire still emanates a + electric field (that is intrinsic to the wire, isn't it?) and the outside +charge is in motion, so it suffers a magnetic force... so that both forces cancel out?

 

[JesseM]

And what if the outside + charge had been at rest wrt the wire?

- In the frame of the +ions, the charge would be subject neither to electric force (the wire is neutral) nor magnetic force (the charge is stationary).
- In the frame of the electrons, the wire still emanates a + electric field (that is intrinsic to the wire, isn't it?) and the outside +charge is in motion, so it suffers a magnetic force... so that both forces cancel out?

In the frame of the electrons the + charge would be moving to the left, while the magnetic field would be circling around the wire as shown here...and by convention current is considered to be the direction of positive charge flow, so since the positive charges are moving to the left in the electron rest frame, if you point the thumb of your right hand to the left your fingers curl in the direction of the field vectors, meaning the field vectors are pointing out of the plane of the screen below the wire and into the plane of the screen above the wire where the lone + charge is. So you can use the "right hand rule #1" on this page to show that if your pointer finger goes to the left (the direction the lone + is moving in the electron frame) while your middle finger is pointing into the screen (the direction of the magnetic field vector at the position of the + charge), then your thumb is pointing down, meaning the magnetic force on the + charge is down towards the wire. But meanwhile the wire is positively charged in this frame and like charges repel, so the electric force on the + charge would be away from the wire in the opposite direction, and presumably detailed calculation would show the two forces exactly cancel out, to be consistent with the fact that there are no electric or magnetic forces on the + charge in the lab frame.

 

[Saw]

In the frame of the electrons the + charge would be moving to the left, while the magnetic field would be circling around the wire as shown here...and by convention current is considered to be the direction of positive charge flow, so since the positive charges are moving to the left in the electron rest frame, if you point the thumb of your right hand to the left your fingers curl in the direction of the field vectors, meaning the field vectors are pointing out of the plane of the screen below the wire and into the plane of the screen above the wire where the lone + charge is. So you can use the "right hand rule #1" on this page to show that if your pointer finger goes to the left (the direction the lone + is moving in the electron frame) while your middle finger is pointing into the screen (the direction of the magnetic field vector at the position of the + charge), then your thumb is pointing down, meaning the magnetic force on the + charge is down towards the wire. But meanwhile the wire is positively charged in this frame and like charges repel, so the electric force on the + charge would be away from the wire in the opposite direction, and presumably detailed calculation would show the two forces exactly cancel out, to be consistent with the fact that there are no electric or magnetic forces on the + charge in the lab frame.

Thanks for all the details and I was going to comment again on the concept of "charge" as relative or not... But then another more elementary doubt assaulted me. We were considering here a wire with a current flowing through it and a charge in front of it. In the last variation, we assumed that the charge was stationary wrt the wire. Hence we also assumed that it suffered no force, since in the wire frame (+ ions frame), to start with, the wire is neutral. But then I remembered what I read http://www.physicsclassroom.com/Class/estatics/u8l1c.cfm" , that is to say, that neutral objects (like the wire) also attract charged objects (like the outside charge). Should we alter the conclusion?

 

[Saw]

I read more in that site and realized they refer to the fact that a charge may cause polarization in an insulator and, once the opposite charge of the dipole is closer, then it is attracted by the charge in question. I do not know if that could happen here, in a conductor, as well: could the hovering outside charge deflect the electrons current?