Explanation of EM-fields using SR

[harrylin]

Get it now. Thanks A.T. Must have been tired last night. The initial loss of electrons from the section of wire is not recovered later in the steady state phase when the amount of electrons entering is the same as the amount of electrons leaving. As a side note, generally speaking the electrons do not all start moving simultaneously in the rest frame of the wire when the switch is closed. I am checking out the situation when the switch on event propagates down the wire at the speed of light which is probably faster than in reality. The end result is the same as far as charge density is concerned, but the explanation appears to be a little different.

It appeared that you were not tired but lucid; see my earlier reply to you. If I'm not mistaken then RoS merely gives you in an indirect way the electron distance as transformed from the stationary frame, and which can be found directly. Isn't that what you meant? The transformation doesn't contain information about final force equilibrium according to the cat's reckoning, other than that according to PoR the cat should obtain that result.

 

[TrickyDicky]

I think the problem with the clip's explanation is that it doesn't clarify why(and I'm referring to the current switch on) for observers at rest wrt the protons(like the guy who explainss it) the electrons are not length contracted while for observers at rest wrt the electrons (like the cat moving along them) protons are length contracted.

 

[yuiop]

I think the problem with the clip's explanation is that it doesn't clarify why(and I'm referring to the current switch on) for observers at rest wrt the protons(like the guy who explainss it) the electrons are not length contracted while for observers at rest wrt the electrons (like the cat moving along them) protons are length contracted.

In DrGregs excellent diagram, the electrons are shown as length contracted in the rest frame of the wire when the current is on (bottom left drawing). It is just the gaps between the centres of the electrons that remains the same as when the current was off in the rest frame of the wire (top left drawing). I think it is also debatable whether electrons length contract if they are considered as quantum point particles and the length contraction of the electrons themselves is not important to the basic theme of this thread. It is only the inter electron gaps that has any significance.

I think a better question is why the battery (which is a store of excess electrons) does not take the opportunity to pack extra electron into the wire when the circuit is switched on. I am sure there is a good reason that does not happen, but it is just sort of assumed without explanation.On the other hand, it would be possible to construct a circuit with a coil at one end and induce a current, so that we are certain no additional electron are added to the circuit.

 

[Dale]

I think the problem with the clip's explanation is that it doesn't clarify why(and I'm referring to the current switch on) for observers at rest wrt the protons(like the guy who explainss it) the electrons are not length contracted while for observers at rest wrt the electrons (like the cat moving along them) protons are length contracted.

The video may not draw it correctly (not sure), but they are both length contracted in fact. The spacing between the electrons is greater in the electrons' frame than in the protons' frame. The spacing between the protons is greater in the protons' frame than in the electrons' frame. Length contraction occurs for both protons and electrons.

 

[Dale]

I think a better question is why the battery (which is a store of excess electrons) does not take the opportunity to pack extra electron into the wire when the circuit is switched on. I am sure there is a good reason that does not happen, but it is just sort of assumed without explanation.

It is not really an assumption, it is more like part of the specification of the problem. It is what I called earlier a boundary condition.

You could pack additional electrons on if you used a very high voltage battery (MV or more) and a big resistor in series with the wire so that very little voltage is dropped across the wire. But that would be a different problem.

If you were working a projectile problem and the problem description says that the projectile leaves the barrel at 45º angle and 1000 m/s then I wouldn't call the 1000 m/s an assumption, it is just part of the specification of the problem. Would you?

 

[yuiop]

It is not really an assumption, it is more like part of the specification of the problem. It is what I called earlier a boundary condition.

Fair comment, but I don't think the video took much time to define the boundary conditions clearly. I was just making sure that the door was firmly closed on allowing extra electrons into the circuit. With a fixed number of electrons I am not sure how anyone can claim the gap between the electrons should contract in the rest frame of the wire, unless they imagine the electrons going around in a small clump with a massive gaps between the ends of the clump.

 

[TrickyDicky]

In DrGregs excellent diagram, the electrons are shown as length contracted in the rest frame of the wire when the current is on (bottom left drawing). It is just the gaps between the centres of the electrons that remains the same as when the current was off in the rest frame of the wire (top left drawing). I think it is also debatable whether electrons length contract if they are considered as quantum point particles and the length contraction of the electrons themselves is not important to the basic theme of this thread. It is only the inter electron gaps that has any significance.

The video may not draw it correctly (not sure), but they are both length contracted in fact. The spacing between the electrons is greater in the electrons' frame than in the protons' frame. The spacing between the protons is greater in the protons' frame than in the electrons' frame. Length contraction occurs for both protons and electrons.

I know that, I said that the video din't clarify it well enough, it gave the impression that it was that asymmetry that explained what makes magnets work, but it isn't that asymmetry since both electrons and protons length contract from each other's frame, instead they should have emphasized the fact that the proton's rest frame and the wire's is the same.

 

[universal_101]

The individual electrons are length contracted in the wire's frame. But their spacing doesn't decrease in the wire's frame, when the current starts flowing.

Then why do you not apply this "individual electron contraction and no change in spacing" to the video this thread is based on and to the picture that DrGreg has posted, and see what comes out. Let me guess, it does not give correct results !?

 

[universal_101]

The comparison of distances at different times in one frame is not length contraction. Length contraction is a disagreement between two frames, not a change over time in one frame. Because of that, your comment about length contraction all in one frame is a self-contradiction. There is no such thing as length contraction in one frame.

You could have said it far too earlier in this thread and could have saved every one a lot of typing! And it is not over yet, although it is nice to have the modified definition of Length contraction yet again, but this definition makes me wonder, why do you then not apply the same definition to the Time Dilation, Mass Increment etc. in a simple scenario of twin paradox(where you watch everything from the frame of staying twin, and I hope that where there is time Dilation there is length Contraction i.e all the relativistic effects are present) or when you work with the results of accelerators!

 

[universal_101]

Fixed it for you. This the key element that people often forget, when assuming "length contraction" in that historical sense. And it is a good reason to avoid that historical usage, because it is based on this often not explicitly stated assumption, that the proper length is constant which is not always true. This leads to confusion like we see here with universal_101, and generally in Bell-Spaceship-Paradox threads.

First of all the word rigid has NO meaning here or anywhere where relativity is discussed(that is as long as EM forces are concerned), if it means anything then it is how particles were related to each other before there was any relative motion, and you are using it to define Length Contraction.

"Since electrons are not rigidly connected to each other, there is NO length Contraction", I don't know what does the underlined statement means, because as far as I know electrons are connected to each other by Coulomb forces(mostly) which according to relativity are liable to change if there is relative motion, which should result in electrons coming closer to each other. Just like Twin Paradox!

 

[universal_101]

In DrGregs excellent diagram, the electrons are shown as length contracted in the rest frame of the wire when the current is on (bottom left drawing). It is just the gaps between the centres of the electrons that remains the same as when the current was off in the rest frame of the wire (top left drawing). I think it is also debatable whether electrons length contract if they are considered as quantum point particles and the length contraction of the electrons themselves is not important to the basic theme of this thread. It is only the inter electron gaps that has any significance.

What is the meaning of Length contraction if instead you are contracting individual points(which by definition cannot be contracted). And why do you not apply it to the whole picture of DrGreg's and instead of contracting the inter electron gaps just contract the electrons. So you can't have Length Contraction and same spacing both in one case, and Length contraction and different spacing in the other case, just so it suits the observed facts!

 

[TrickyDicky]

Universal, instead of bringing further cofusion by questioning everyone's relativity knowledge thru your distorting misconceptions, why don't you tell us if you finally understood why in the presence of current length contraction in the wire's frame retains neutrality while in frames moving wrt the wire it doesn't.

 

[A.T.]

First of all the word rigid has NO meaning here or anywhere where relativity is discussed(that is as long as EM forces are concerned), if it means anything then it is how particles were related to each other before there was any relative motion, and you are using it to define Length Contraction.

By "rigid" I meant: Keeping a constant proper length (length measured in the object rest frame) over time. That is the premise, based on which you can assume that an accelerating object will shorten in the original rest frame.

as far as I know electrons are connected to each other by Coulomb forces(mostly)

No, they are not connected by Coulomb forces. They are repulsing each other by Coulomb forces

which according to relativity are liable to change if there is relative motion,

Yes, the E-fields of moving electrons are contracted in the wire's frame.

which should result in electrons coming closer to each other.

No, contracted repulsive fields are still repulsive. They don't become attractive. The electrons still try to distribute as far as possible from each other.

 

[harrylin]

Universal, instead of bringing further cofusion by questioning everyone's relativity knowledge thru your distorting misconceptions, why don't you tell us if you finally understood why in the presence of current length contraction in the wire's frame retains neutrality while in frames moving wrt the wire it doesn't.

All the counterproductive quibbling over words may have made you overlook post #74, #75 and #103. When we do find the answer, we may also be able to explain the steady state charge imbalance according to the physics from the cat's perspective.

 

[A.T.]

Then why do you not apply this "individual electron contraction and no change in spacing" to the video this thread is based on and to the picture that DrGreg has posted, and see what comes out.

This is exactly what DrGreg's diagram shows.

 

[Dale]

You could have said it far too earlier in this thread and could have saved every one a lot of typing!

So does that mean that you understand it now?

but this definition makes me wonder, why do you then not apply the same definition to the Time Dilation, Mass Increment etc.

I do apply the same defintion to time dilation (I don't use relativistic mass at all).

 

[TrickyDicky]

All the counterproductive quibbling over words may have made you overlook post #74, #75 and #103. When we do find the answer, we may also be able to explain the steady state charge imbalance according to the physics from the cat's perspective.

Find the answer to what question exactly, can you phrase it in specific terms?

 

[yuiop]

In this pape http://www.chip-architect.com/physics/Magnetism_from_ElectroStatics_and_SR.pdf the charge density measured in the rest frame of a test charge moving at v relative to the wire, is given as:

$g_L = I v/c^2$

I might be wrong, but I suspect that is just for the special case when the drift velocity of the electrons $(v_e)$ is equal to the velocity of the test charge. If that is the case, then the more general expression should be:

$g_L = I (1/v_e)* (v^2/c^2)$

which reduces to the previous expression when $v_e = v$

Expressed like this, it is easy to see that the slow drift velocity of the electrons (about 1mm per second) relative to the speed of light actually magnifies the effect and explains why the EM effect is one of the few relativistic effects readily observable at everyday velocities.

Any thoughts?

 

[Dale]

So, if you start with a four-current $(0,I,0,0)$ and boost to any arbitrary frame moving at velocity v you get a four-current $(\gamma I v/c, \gamma I,0,0)$ regardless of what the drift velocity of the charge carriers are. So I get the charge density is $\rho=\gamma I v/c^2$, which is the same as their expression to first order in v.

 

[yuiop]

So, if you start with a four-current $(0,I,0,0)$ and boost to any arbitrary frame moving at velocity v you get a four-current $(\gamma I v/c, \gamma I,0,0)$ regardless of what the drift velocity of the charge carriers are. So I get the charge density is $\rho=\gamma I v/c^2$, which is the same as their expression to first order in v.

In their derivation they equate the magnetic force with the electric force, but they do not allow for the fact that the forces are measured in different reference frames. If we allow for this using the Lorentz transformation of transverse force, then I get $F_{mag} = F_{elec}/\gamma$ and this gives the same result as yours, $Q_L=\gamma I v/c^2$. This of course implies that when the electric force is measured in the same reference frame as the magnetic force is measured in (two separate experiments alongside each other in the same lab) the equation for the electric force is gamma greater than the equation quoted in the paper. My argument is partly supported by equation 29 of this text if I understood it correctly. This can thought of as the difference between the transverse force on a particle that has relative motion, compared to the force on a static particle.

Equation (0) of this article uses an approximation, which when carried out accurately also indicates the charge density and electric force is greater by a factor of gamma than the result they obtain.

Trouble is I guess I am still a bit confused because this additional gamma factor is well hidden in this https://www.mtholyoke.edu/courses/tdray/phys310/electromag.pdf. Also, the implication is that the Lorentz Force law (Eq 8.4) should actually be $F = q(\gamma*E + v \times B)$ although that seems unlikely as it has been tested to high velocities, or the equation for the Electric force field is gamma greater than the usually quoted formula as in (Eq 8.1) of the text?

As for the relevance of the drift velocity, I will have to give that some more thought. It is just that to me, the derivations of electric force require that the electrons are stationary in the rest frame of the test charge and so the derivation is only valid for the case where the test charge is moving at the drift velocity.

 

[universal_101]

Universal, instead of bringing further confusion by questioning everyone's relativity knowledge thru your distorting misconceptions, why don't you tell us if you finally understood why in the presence of current length contraction in the wire's frame retains neutrality while in frames moving wrt the wire it doesn't.

I don't feel like turning a blind eye and start bookkeeping, in favor of observed facts and the standard theory. So, no I still don't know why there are two definitions of Length Contraction one for the current on-off, and other for the regular Lorentz transform of a current.

 

[universal_101]

So does that mean that you understand it now?

You haven't presented any logical argument yet, and I don't even know what do you want me to understand. But to me it seems that you don't want to apply the length contraction to the current, and at the same time you want to keep it under the domain of applicability of SR.

I do apply the same definition to time dilation (I don't use relativistic mass at all).

Really! Then why do we have a net Time Dilation in Twin paradox and accelerators, and net length contraction in regular Lorentz transform, but not in the case of a current.

And please don't say, that since in a conductor, every charge wants to spread all over the volume and since the number of electrons are equal to the number of protons, so there is NO net charge due to Length contraction, because instead of contracting the inter gaps electrons tend to contract themselves at their previous spacing before the current. Because the ditto condition is always present for the regular Lorentz transform of a current.(i.e. a conductor...same volume...same no. of charges...no net charge...) where you changed the nature of Length contraction in order to get all the observed facts correct.

 

[universal_101]

In this pape http://www.chip-architect.com/physics/Magnetism_from_ElectroStatics_and_SR.pdf the charge density measured in the rest frame of a test charge moving at v relative to the wire, is given as:

$g_L = I v/c^2$

I might be wrong, but I suspect that is just for the special case when the drift velocity of the electrons $(v_e)$ is equal to the velocity of the test charge. If that is the case, then the more general expression should be:

$g_L = I (1/v_e)* (v^2/c^2)$

which reduces to the previous expression when $v_e = v$

Expressed like this, it is easy to see that the slow drift velocity of the electrons (about 1mm per second) relative to the speed of light actually magnifies the effect and explains why the EM effect is one of the few relativistic effects readily observable at everyday velocities.

Any thoughts?

There is an inherent $v_e$ in the $I$ of your expression, which i think you missed. And I don't think a slow drift velocity magnifies relativistic effects, if it is anything then its the huge number of electrons, which is also inherently included in the $I$ and that is why you get the relativistic effects in everyday velocities.

 

[harrylin]

Find the answer to what question exactly, can you phrase it in specific terms?

Sure. What we so far couldn't answer is the following basic question, as viewed from the rest frame of the electrical circuit: As the electrons are drifting in one direction, one would expect that their electric field parallel to the wire is reduced by γ². In contrast, this doesn't happen for the electric field of the ions, nor for the electric field of the electrons in the battery. Why doesn't the wire suck in electrons from the power source (and from earth, if there is an Earth connection)?
[edit:] For a large power source I would expect the electron density in the wire to be increased by a factor γ. If you know the answer -as you seemed to suggest- please tell!

 

[A.T.]

I still don't know why there are two definitions of Length Contraction one for the current on-off, and other for the regular Lorentz transform of a current.

Because people are lazy in inventing new names. And because under the assumption of constant proper-length the two have the same result. But that assumption doesn't hold true for the electron distances.

 

[A.T.]

And please don't say, that since in a conductor, every charge wants to spread all over the volume and since the number of electrons are equal to the number of protons, so there is NO net charge due to Length contraction, because instead of contracting the inter gaps electrons tend to contract themselves at their previous spacing before the current.

Not every charge, just the free electrons, because they can change their proper distances. Contracted repulsive fields are still repulsive. So why should the electrons move closer together?

Because the ditto condition is always present for the regular Lorentz transform of a current.(i.e. a conductor...same volume...same no. of charges...no net charge...) where you changed the nature of Length contraction in order to get all the observed facts correct.

This is incomprehensible. And using bold font doesn't give it any more sense.

If you wonder why atoms in a lattice behave differently than free electrons when they start moving in some frame, then consider the different interaction in the two cases: Lennard-Jones potential between atoms can be attractive or repulsive. Coulomb forces between electrons are always repulsive.

The cations are fixed in the lattice, so they cannot change their proper distance. The electrons are free to move, so they can change their proper distance, while keeping the distance in the wire frame constant.

 

[universal_101]

Not every charge, just the free electrons, because they can change their proper distances. Contracted repulsive fields are still repulsive. So why should the electrons move closer together?

"Contracted repulsive fields are still repulsive", , but a decreased repulsive field is basically an attraction compared to the situation when field is not decreased, and it essentially means less distance between electrons(attraction).

This is incomprehensible. And using bold font doesn't give it any more sense.

If you wonder why atoms in a lattice behave differently than free electrons when they start moving in some frame, then consider the different interaction in the two cases: Lennard-Jones potential between atoms can be attractive or repulsive. Coulomb forces between electrons are always repulsive.

The cations are fixed in the lattice, so they cannot change their proper distance. The electrons are free to move, so they can change their proper distance, while keeping the distance in the wire frame constant.

It should be, otherwise you would have to answer the question that I'm repeatedly asking.

And we are discussing just the relative motion under SR, and every thing about the internal structure is meaningless.

 

[A.T.]

"Contracted repulsive fields are still repulsive", , but a decreased repulsive field is basically an attraction compared to the situation when field is not decreased, and it essentially means less distance between electrons(attraction).

Wrong. As long as they repulse each other, they will distribute uniformly. Reducing the repulsive force doesn’t change that.

And we are discussing just the relative motion under SR,

Wrong. We are discussing a specific physical situation.

and every thing about the internal structure is meaningless.

Wrong. The fixed lattice structure is the reason why there is no symmetry between the cations and the electrons. So it is not meaningless, but actually the answer to your question.

 

[Dale]

You haven't presented any logical argument yet, and I don't even know what do you want me to understand.

I am wondering if you understand the arguments and explanations provided here: http://physics.weber.edu/schroeder/mrr/MRRtalk.html

But to me it seems that you don't want to apply the length contraction to the current, and at the same time you want to keep it under the domain of applicability of SR.

I don't know why it seems that way to you. I thought I was very clear that length contraction does apply here:
https://www.physicsforums.com/showpost.php?p=4531669&postcount=94
and here:
https://www.physicsforums.com/showpost.php?p=4528325&postcount=22

Really! Then why do we have a net Time Dilation in Twin paradox and accelerators, and net length contraction in regular Lorentz transform, but not in the case of a current.

We do have length contraction in the case of the current, as described above.

Because the ditto condition is always present for the regular Lorentz transform of a current.(i.e. a conductor...same volume...same no. of charges...no net charge...) where you changed the nature of Length contraction in order to get all the observed facts correct.

I have no idea about the way that you think that I "changed the nature of Length contraction". I don't know what you mean. Please be explicit. The different frames are related to each other via the Lorentz transform, the same as always and in all cases.

 

[yuiop]

I decided to ascertain what part the drift velocity of the electrons play, by deriving the change in charge density from scratch.

Consider a wire at rest in a lab frame S with a current flowing in the x direction with drift velocity $v_e$. The total charge density in this reference frame is

$p = (p_+) + (p_-)$

where $p_+$ is the charge density of the cations and $p_-$ is the charge density of the electrons. Since they have opposite signs and equal magnitude in this frame the total charge density in this frame is zero.

Now we transform to a new reference frame S' with velocity v relative to the lab frame and parallel to the wire, that is chosen to be not the same as the drift velocity $v_e$ so neither the electrons or the cations are at rest. The positive charge density in this reference frame increases by a factor of $\gamma$ due to the length contraction of the gaps between the positive charges. The transformation of the negative charge density is a little trickier because the electrons are moving in both S and S', but with the use of the relativistic velocity addition formula it can be ascertained that the gap increases by a factor of $\sqrt{1-v^2}/(1-v_ev)$ and the negative charge density decreases by the inverse of that factor, so the transformed total density is:

p' = (p_+)\gamma + (p_-)(1-v_e)\gamma
p' = (p_+) \left(\frac{1 -(1-v_ev)}{\sqrt{1-v^2}}\right)
p' = (p_+)\frac{v_ev}{\sqrt{1-v^2}}
p' = - (p_-)v_ev\gamma

The current in the lab frame is given by $I = -(p_-)v_e$ where the negative sign of the current follows the convention that current flows in the opposite direction to the negative charges. Inserting this value for I into the above equation yields:

p' = Iv\gamma
So yes, as Dalespam said, the drift velocity of the electrons does not play a part and the correct charge density should contain the $\gamma$ factor.

 

[harrylin]

In their derivation they equate the magnetic force with the electric force, but they do not allow for the fact that the forces are measured in different reference frames. If we allow for this using the Lorentz transformation of transverse force, then I get $F_{mag} = F_{elec}/\gamma$ and this gives the same result as yours, $Q_L=\gamma I v/c^2$. This of course implies that when the electric force is measured in the same reference frame as the magnetic force is measured in (two separate experiments alongside each other in the same lab) the equation for the electric force is gamma greater than the equation quoted in the paper. [..]

Yes indeed - well spotted!

However, it's unclear to me what you want to do with that... The only way forward that I see is to use the full equations - as de Vries also does in section 2 - and that looks pretty independent of the section with the error. Strangely enough, I don't see where he actually does the "full derivation" of the wrong "correct derivation" of section 1...

 

[TrickyDicky]

The spacing of the electrons in the wire frame is determined by the observed fact that the wire is uncharged in the wire frame. This is a "boundary condition" that can be experimentally controlled.

This was actually the only explanation universal_101 needed all along and it basically makes the length contraction explanation moot, since it is actually the premise of the problem what forces the existence or not of neutrality in the different frames rather than length contraction by itself which is always present wrt the situation without current(see posts by samshorn about this in the parallel thread "are the transformations just observed...").
There seems to be a problem with this explanation and maybe it's what's been bothering him, this apparently innocent "boundary condition" is itself outside the scope of relativity by preferring one frame over the others wrt the effects of length contraction on neutrality(no effect on neutrality in the frame of the wire, that is cation's or Earth's
frame), but the fact that if we put the charge at rest with Earth's frame and move the wire instead will make lose the wire's neutrality too renders the condition seem totally ad hoc for the experiment and not relativistic at all.

 

[Dale]

There seems to be a problem with this explanation and maybe it's what's been bothering him, this apparently innocent "boundary condition" is itself outside the scope of relativity by preferring one frame over the others wrt the effects of length contraction on neutrality(no effect on neutrality in the frame of the wire, that is cation's or Earth's
frame), but the fact that if we put the charge at rest with Earth's frame and move the wire instead will make lose the wire's neutrality too renders the condition seem totally ad hoc for the experiment and not relativistic at all.

I do agree with your characterization here. As you said, boundary conditions (in general) are outside the scope of the corresponding theory. And you are completely correct that asymmetric boundary conditions can disrupt the symmetry of the corresponding law of physics. You are completely correct in both of those points, and those points apply to all physics and not just relativity and not just this specific problem.

 

[yuiop]

To make it clearer, we can put a positively charged dog in rest with the wire, next to it. Now the cat, using the frame that is co-moving with the electrons as rest frame, has to explain the lack of net force on the dog despite the electric field. The cat can only explain this by the magnetic field of the moving ions and which must exactly compensate the electric field force.

This is a good question. The force acting on the dog should be neutral in both the rest frame of the dog and in the rest frame of the cat when the dog is moving relative to the cat. The analysis of the forces on the dog is a little more complicated than that of the cat so I will take Dalespam's advice and use the transformation of the four-current which can be expressed as:

$[\rho, \mathbf{I_x,I_y,I_z}]$ using units where c=1.

I only want to analyse the case where the current and relative motion are always parallel to the wire which in turn remains parallel to the x axis. This simplifies things as $\mathbf{I_y = I_z} = 0$ so the four current can be abbreviated to $[\rho, \mathbf{I}]$ where I have defined $\mathbf{I_x}$ as $\mathbf{I}$.

After performing a Lorentz boost with velocity v in the x direction:

$\rho' = \gamma_v(\rho - \mathbf{v I})$

$\mathbf{I'}= \gamma_v(-\mathbf{v}\rho +\mathbf{I})$

where $\gamma_v$ is the gamma factor for velocity v.

The Lorentz force acting on a test particle with charge q and velocity $v_0$ is defined as:

$\mathbf{F} = q(\mathbf{E} + \mathbf{v_0 \times B})$

In our simplified case this can be expressed in a non vector form as:

$F = q(E - v_0*B)$

(Note the change of sign when we use ordinary multiplication rather than the cross product.)

The electric and magnetic fields are defined in terms of various constants but we can conveniently use units such that $E = \rho$ and $B = I$ so that

$F = q(p - v_0*I)$.

In the rest frame of the dog, p=0 and $v_0 = 0$ so F =0.

In the general case, when carrying out the Lorentz transformation of the force, we should use the the transformed four velocity of $v_0$ or the relativistic velocity subtraction formula $v_0' = (v_0 -v)/(1-v_0*v)$ For the case of our positively charged dog which is at rest in S, the velocity of the dog in the rest frame of the cat (S') is $v_0' = -v$.

The force acting on the dog in the cat's reference frame is:

$F' = q\rho' - qv_0'*I'$

$F' = q\rho' + qv*I'$

$F' = q(\gamma_v \rho - \gamma_v vI ) + qv(-\gamma_v v\rho +\gamma_v I)$

$F' = q\gamma_v(\rho - vI ) + q\gamma_v(- v^2\rho + vI)$

In the rest frame of the dog, the boundary conditions specified neutral charge density $\rho =0$ so the above reduces to:

$F' = q\gamma_v( - vI ) + q\gamma_v(v I)$

It can be seen that the electrostatic force on the dog is equal in magnitude and opposite in sign to the magnetic force on the dog as measured in the cat's rest frame, so there is still a total force of zero acting on the dog.

 

[Strafespar]

I don't immediately get why the separation of the negatively charged particles doesn't contract from the man's reference frame, as they are moving relative to him, and therefore there would be a negative overall charge.

the negative particles do contract on the man's reference! It's just that they contract in a way that yields a zero electric field. The normal (rest) length between electrons is seen when in the moving observer- so we are always looking at a contracted density instead of a rest density when we observe.