# Explanation of EM-fields using SR

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.

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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", :uhh: , 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.

"Contracted repulsive fields are still repulsive", :uhh: , 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.

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Dale
Mentor
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.

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.

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...

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 wich 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
Mentor
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.

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.

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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.