Special Relativity Paradox

[swansont]

8 hours ago, Markus Hanke said:

But I don’t think that’s what the OP has in mind, unless I’m still misunderstanding him

I think you’re right. With their last post it’s clearer to me what their misconception is. The wire’s length is only contracted in the electron’s frame, not the lab frame, so there’s no reason to expect the result they claim.

[studiot]

The speed of electrons in copper noted in my previous post imply you would have to apply 10¹¹ volts to boost their flow speed to 0.1c

[joigus]

\[ \gamma=\frac{1}{\sqrt{1-v²/c²}} \]

\[ \gamma\times0=0 \]

Last comment by @Markus Hanke & @swansont spot on, I think. You cannot take electron density in proper frame for electrons while proton density in "rest" frame.

[M S La Moreaux]

Does anyone really need a diagram of a straight current-carrying wire?

Length contraction applies not only to moving objects but also to the spaces between those moving with the same velocity.

[joigus]

15 minutes ago, M S La Moreaux said:

Does anyone really need a diagram of a straight current-carrying wire?

Apparently you.

[swansont]

33 minutes ago, M S La Moreaux said:

Does anyone really need a diagram of a straight current-carrying wire?

Length contraction applies not only to moving objects but also to the spaces between those moving with the same velocity.

The wire is not length-contracted, because it is not moving. The electrons enter and leave at the end of the wire. So the number of electrons in that section is the same, regardless of this length contraction.

This might be related to the rotating disk paradox; the radius is not changed but the circumference is not 2*pi*r. The solution is because there’s an acceleration involved (when the electrons turn the corner, in our case) which means you can’t naively apply Euclidean geometry to the description.

https://simple.m.wikipedia.org/wiki/Rigidly_rotating_disk_paradox

[joigus]

Detailed calculations:

Nuclei's rest frame:

\[ ρ = n p - n e ⁻ V = 0 \]

 

Electrons' rest frame: 

\[ \rho=\frac{n_{p}-n_{e⁻}}{V}=0 \]

Protons' rest frame:

\[ \rho’=\frac{n_{p}-n_{e⁻}}{V’}=0 \]

Where \( V’=\gamma^{-1}V \) 

Charge and number are special-relativity invariants. It's only volume that varies with an inverse gamma factor. So, as @swansont said,

9 minutes ago, swansont said:

the number of electrons in that section is the same, regardless of this length contraction.

Same applies to the number of protons.

I hope that helped.

[Markus Hanke]

4 hours ago, M S La Moreaux said:

Length contraction applies not only to moving objects but also to the spaces between those moving with the same velocity.

No, that’s where your mistake lies. The lengths between the electrons are part of the circuit frame, not the electron frame, and hence not contracted. The total electron number (charge) in the wire is conserved.

13 hours ago, swansont said:

The wire’s length is only contracted in the electron’s frame, not the lab frame, so there’s no reason to expect the result they claim.

Exactly.

6 hours ago, studiot said:

The speed of electrons in copper noted in my previous post imply you would have to apply 10¹¹ volts to boost their flow speed to 0.1c

Indeed. 

[studiot]

23 hours ago, swansont said:

I think you’re right. With their last post it’s clearer to me what their misconception is. The wire’s length is only contracted in the electron’s frame, not the lab frame, so there’s no reason to expect the result they claim.

I don't believe that there is only a single misconception here.

Apart from mixing up the frames of reference, there is there plainly incorrect statement

14 hours ago, M S La Moreaux said:

Length contraction applies not only to moving objects but also to the spaces between those moving with the same velocity.

 

@M S La Moreaux  claiming this is equivalent to saying that if a stream of spaceships ply between Earth and Alpha Centauri, the distance from Earth to AC (as viewed from Earth) magically changes because of the spaceships.
It doesn't change for the Earth observer, but does for the shipborne observer. However the shipborne observer also 'sees' himself as stationary and AC rushing towards him.

 

Then there is the simple issue should SR be applied at all.

As I have noted the actual electron velocities in copper are such a tiny fraction of the speed of light. 

Further there is the failure to appreciate the difference between the velocities as a result of an electric field, the random velocities of the electrons in the absence of an electric field, the random velocities of the electrons  and the velocities of an electric signal or disturbance in the copper or other medium.

The electric signal is actually quite fast, faster than any electron can travel, But is not steady state.

I have already worked out the velocity due to an electric field, so

Here are some interesting facts and figures about the random walk velocities of an electron gas and other gases.

Note the electron gas is a truly free gas, the electrons in copper or other medium are often called 'free electrons' but that is not really the case. They are better named unbound electrons, they are not free as they are confined to a giant molecular orbital called a band.

 

 

 

We all make mistakes so a correction to this is in order.

Apologies

The voltage should be should be 10⁸ not 10¹¹ V

16 hours ago, studiot said:

The speed of electrons in copper noted in my previous post imply you would have to apply 10¹¹ volts to boost their flow speed to 0.1c

Edited Friday at 08:52 PM by studiot

[swansont]

36 minutes ago, studiot said:

@M S La Moreaux  claiming this is equivalent to saying that if a stream of spaceships ply between Earth and Alpha Centauri, the distance from Earth to AC (as viewed from Earth) magically changes because of the spaceships.
It doesn't change for the Earth observer, but does for the shipborne observer. However the shipborne observer also 'sees' himself as stationary and AC rushing towards him.

Not that distance, but the spacing between the ships (much like the circumference of the rotating disk does not require a solid rim; it would be the same with just spokes). But this ignores the fact that the distance doesn’t change, and as one ship lands another is launched, and the new ship is not moving at v. So there are always N ships in the distance L. Neither N nor L changes. N because it’s invariant, and L because the path itself is not moving, and that’s what matters.

Quote

Then there is the simple issue should SR be applied at all.

Since there is motion, there’s no issue with applying it. As the citations in my link address, it gives the explanation for the repulsion/attraction of parallel wires. Or you can apply relativity to come up with magnetism. Different explanations for the result.