I Explaining the magnetic (cable) paradox

[psychics_xxx]

Hi,

I was wondering about explanation of magnetic paradox which is about different frames of reference. When we put a negative electric charge close to cable with current flowing and observe it moves along the cable (with exact speed and direction as electrons in cable) we say that the magnetic force is dragging it to the cable. We are assuming that total charge of the cable is 0.

But when change our reference to that charge (it is no longer moving for us) then we say that due to the Lorentz's contraction on positive charges, they are squished together and thus total charge is no longer 0. Now we see extra eletric force (which was previously magnetic in first refference).

So my question is:
Why does the contraction appears to effect only positive charges? What i mean by that is that the contraction should also effect the cable itself and make both negative and positive charge's density bigger. Also, let's say there are N protons and neutrons inside of cable. The contraction cannot change that number so total charge is still 0?

Where I got wrong? Thanks

 

[A.T.]

Why does the contraction appears to effect only positive charges?

Lorentz contraction affects both, positive and negative charges. But only the positive charges (atom cores) have fixed proper distances (distances in their rest frames), while the negative charges (electrons) can change their proper distances to keep the same density in the wire rest frame, while accelerating.

Lorentz contraction relates the distances in different frames for a given relative motion, not distances in a single frame before and after acceleration. When you assume that the proper distance remains constant, then the later follows from the former, but you cannot always assume that.

Here is a good explanation and diagram by @DrGreg which indicates what is related by Lorentz contraction, and what isn't:

This has been asked many times:

L

B Thread 'Stationary charge next to a current-carrying wire'

Dec 25, 2023

(1) https://www.youtube.com/results?search_query=relativity+and+electromagnetism

(2)

If I understand this correctly, then, well, shouldn't all current-carrying wires exhibit a small amount of positive charge?

• lightlightsup
• Charge Current Wires
• Replies: 68
• Forum: Special and General Relativity

L

I Thread 'Electric currents and length contraction'

Nov 8, 2017

Hi people, me bother yo again!

I understand that relativity theory length contraction is the explanation of magnetics fields.
See the image:

https://dl.dropboxusercontent.com/s/vuy69jhig3kni3n/densidadcargasencorriente.jpg?dl=0

1) In the S system, if q is moving at the same velocity than the electrons that made the current, the -I current into a neutral wire produces a magnetic field B that make a force Fm on q.
2) It can be explained what happens in 1) from what happens in a system S´ that is moving at the same velocity than electrons and q. The length contraction of...

• Luis Babboni
• Contraction Currents Electric Length Length contraction
• Replies: 2
• Forum: Special and General Relativity

K

B Thread 'Relativistic explanation of electromagnetism'

Jan 9, 2020

I am intrigued by the special-relativity explanation of magnetic force discussed here (linked from the physicsforums FAQ): http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html#Length_Contraction

Naively, from this explanation, it seems that a test charge at rest in the lab frame should experience a force from a current-carrying wire, since the electrons' fields are Lorentz-contracted relative to the test charge, but the nuclei fields are not. And, that the test charge should experience no force only if the positive and negative charges in the wire are...

• Karl Coryat
• Electromagnetism Explanation Relativistic
• Replies: 34
• Forum: Special and General Relativity

G

B Thread 'How To Consistently Explain Electromagnetism With Relativity'

Nov 21, 2017

Superconducting Ring

In a superconducting ring does the contraction of space between electrons cause them to move inwards? Like in this animation.

Force Between Parallel Wires With Current

In the proton frame of two parallel wires with identical current, I've been told they attract and this is because the motion and length-contraction of the electrons increases the negative charge density. Is this correct? Because it seems this explanation accounts for the protons in one wire being attracted to the excess electrons in the other wire, but ignores the excess electrons...

• Geocentricist
• Electromagnetism Explain Relativity
• Replies: 161
• Forum: Special and General Relativity

 

[div_grad]

Where I got wrong?

@psychics_xxx Its just the same case as with any other type of paradox: the premise of the question is wrong. There is a thinly veiled assumption that one can derive the magnetic field from the static Coulomb's law and the kinematics of special relativity - which is not true.

What is true is that the electric current flowing through a wire produces a magnetic field $\mathbf{B}$ around it (determined by the "right-hand-rule"). It is also true that if a particle with electric charge $q$ moves in this field with velocity $\mathbf{v}$, then the direction of its motion is changed by the magnetic part of the Lorentz force, $\mathbf{F}_L = q \mathbf{v}\times\mathbf{B}$.

But It is not true that the electrically neutral wire acquires a non-zero electric charge when the reference frame is changed (by means of a Lorentz transformation). The electric charge is a Lorentz scalar - it is invariant under Lorentz transformations. What does change under Lorentz transformation, however, is the form of the force vector $\mathbf{F}_L$.

When you transform to a rest frame of the charged particle (which in the original frame was moving along the wire), then the form of the appropriate force vector changes. If for some reason you take this new expression for the force in the new frame and try to re-express the involved variables back in terms of the variables from the original frame, then you will obtain an additional term which formally may look like a magnetic force. But this pops up only because you tried to re-express part of the expression for the transformed force back in terms of the variables used for writing the force in the original frame.

This is explained, for example, in the textbook: J. D. Jackson, Classical Electrodynamics (2nd edition), see §12.2 "On the Question of Obtaining the Magnetic Field, Magnetic Force, and the Maxwell Equations from Coulomb's law and Special Relativity" (page 578). If you want to understand what is going on, you need to take look at this chapter and not rely solely on pictures and pop-sci videos. In particular, the charge carriers are treated as point particles and thus there is no Lorentz contraction affecting them.

 

[A.T.]

But It is not true that the electrically neutral wire acquires a non-zero electric charge when the reference frame is changed (by means of a Lorentz transformation). The electric charge is a Lorentz scalar - it is invariant under Lorentz transformations.

The total charge in the wire loop is invariant, but the charge distribution is not, so a segment of the loop can have non-zero charge in some frames. See the diagram by @DrGreg in post #2.

In particular, the charge carriers are treated as point particles and thus there is no Lorentz contraction affecting them.

Lorentz contraction applies to the field of a moving charge, which is the physically relevant thing about it:
https://www.feynmanlectures.caltech.edu/II_26.html

 

[div_grad]

The total charge in the wire loop is invariant, but the charge distribution is not, so a segment of the loop can have non-zero charge in some frames.

Sure, that is why I wrote what I wrote, referring the OP question and statements.

Lorentz contraction applies to the field of a moving charge, which is the physically relevant thing about it:
https://www.feynmanlectures.caltech.edu/II_26.html

This is not stressed nor shown in the diagram in post #2. On the contrary, it suggests that point-like charge carriers are affected by Lorentz contraction, which is why I wrote what I wrote.

 

[A.T.]

This is not stressed nor shown in the diagram in post #2. On the contrary, it suggests that point-like charge carriers are affected by Lorentz contraction

That potential misinterpretation of the diagram never occurred to me, because it seems obvious that a point cannot be further contracted. Thus I took the squeezed balls to mean contracted fields of the charges. But that's besides the main point of diagram anyway, which is to show the frame dependent charge densities.