Mother of all equations for the Lorentz force

[Jaaanosik]

TL;DR Summary

Lorentz force equation invariance leads to different Lorentz force values in different inertial frames. Is this a problem for conservation of momentum?

David J. Griffiths Introduction to Electrodynamics page 460:

Lorentz force equation invariance leads to different Lorentz force values in different inertial frames.
Is this a problem for conservation of momentum? More specifically conservation of angular momentum?

 

[Sagittarius A-Star]

TL;DR Summary: Lorentz force equation invariance leads to different Lorentz force values in different inertial frames. Is this a problem for conservation of momentum?

David J. Griffiths Introduction to Electrodynamics page 460:

View attachment 348382


Lorentz force equation invariance leads to different Lorentz force values in different inertial frames.
Is this a problem for conservation of momentum? More specifically conservation of angular momentum?

No. Components of Forces and momentum are frame-dependent. Invariant is the norm of 4-vectors.

The cited equation of Griffiths shows a 3-vector force. They transform like this between inertial frames:
http://www.sciencebits.com/Transformation-Forces-Relativity

Conservation is a different thing than invariance.

 

[Jaaanosik]

No. Components of Forces and momentum are frame-dependent. Invariant is the norm of 4-vectors.

The cited equation of Griffiths shows a 3-vector force. They transform like this between inertial frames:
http://www.sciencebits.com/Transformation-Forces-Relativity

Conservation is a different thing than invariance.

Angular momentum is absolute, if a body rotates clockwise in one inertial frame it has to rotate clockwise in all other inertial frames.
Would you agree?

If Lorentz force causes opposite rotation in different inertial frames then it is a problem. It seems.

 

[Ibix]

If Lorentz force causes opposite rotation in different inertial frames then it is a problem.

Do you have a circumstance where you think that happens?

 

[Nugatory]

Angular momentum is absolute

No. The angular momentum of an object moving inertially is frame-dependent. Even for rotating bodies

if a body rotates clockwise in one inertial frame it has to rotate clockwise in all other inertial frames.

That depends on the sign of the axis of rotation (viewed from the above the North Pole the earth rotates counterclockwise, viewed from above the South Pole it’s clockwise), so also frame-dependent.

 

[Jaaanosik]

Do you have a circumstance where you think that happens?

Yes, I do. I'll prepare a figure in about an hour.

 

[Jaaanosik]

No. The angular momentum of an object moving inertially is frame-dependent. Even for rotating bodiesThat depends on the sign of the axis of rotation (viewed from the above the North Pole the earth rotates counterclockwise, viewed from above the South Pole it’s clockwise), so also frame-dependent.

Magnitude yes, but the direction is supposed to be agreed upon by all inertial observers.
The direction is absolute.
Sign of the axis does not contradict the agreement between the inertial frames on the direction of the rotation of a body.

 

[Jaaanosik]

Do you have a circumstance where you think that happens?

Here is a thought experiment:

An electron flies from the middle of the cathode plate to the middle of the anode plate.

Now here is what is happening mid flight:

... and the K'₂ frame:

We pick electron Q charges as per the figures above.
When the calculation is done then the result is following:
The repulsive Lorentz force between qQ₁ is bigger than qQ₂ in K'₁ frame.
The repulsive Lorentz force between qQ₂ is bigger than qQ₁ in K'₂ frame.

The Lorentz force is causing clockwise rotation of the plates in K'₁ frame.
The Lorentz force is causing counter clockwise rotation of the plates in K'₂ frame.

This appears to be a problem.

 

[Ibix]

Are both plates supposed to be charged, or do you really just have three charges? And have you checked what momentum is carried by the EM field?

 

[Jaaanosik]

Are both plates supposed to be charged, or do you really just have three charges? And have you checked what momentum is carried by the EM field?

Both plates are charged, the Q₁ and Q₂ are just an example to show the Lorentz force delta between the 'left' and the 'right' side of plates.
As Griffiths says: That, together with the superposition principle, would tell us the force exerted on a test charge Q by any configuration whatsoever.
If we do calcs for all charges the delta between the left and the right side is present.

The EM field is a response to charges.
The EM field of moving and accelerating charges is frame dependent.
The same book:

So essentially the EM field is in agreement with the Lorentz force:

The EM field shows why there is a delta between Q₁ and Q₂.
The EM field of accelerated electron q is 'rotating' counter clockwise in K'₁ and in order to conserve the angular momentum this field repels plates clockwise in K'₁.
The opposite is happening in K'₂.

 

[Ibix]

One obvious point is that from the symmetry expressed in the rest frame, where the electron is moving perpendicular to the plates and in the center, there can be no rotation. And all quantities can be expressed in a covariant form, so there's no rotation in any frame.

I don't know what all the symbols used in the extract from Griffiths are, and I can't even read one of them. So I can't use his formula. It is not clear to me that you have done so either - you don't seem to have shown any algebra.

Can you define all the terms in Griffiths' 10.74 apart from $c$ and the constants in the first prefactor, which I understand?

 

[Sagittarius A-Star]

One obvious point is that from the symmetry expressed in the rest frame, where the electron is moving perpendicular to the plates and in the center, there can be no rotation. And all quantities can be expressed in a covariant form, so there's no rotation in any frame.

In the inertial "rest frame", the charged plated in the isolated system, which must be mechanically connected by a distance piece, accelerate upwards when the electron accelerates downwards. Because of relativity of simultaneity, this happens in frame $K'_1$ first on the right side and then on the left side.

 

[Jaaanosik]

One obvious point is that from the symmetry expressed in the rest frame, where the electron is moving perpendicular to the plates and in the center, there can be no rotation. And all quantities can be expressed in a covariant form, so there's no rotation in any frame.

I don't know what all the symbols used in the extract from Griffiths are, and I can't even read one of them. So I can't use his formula. It is not clear to me that you have done so either - you don't seem to have shown any algebra.

Can you define all the terms in Griffiths' 10.74 apart from $c$ and the constants in the first prefactor, which I understand?

Here are the definitions:

The calculation, middle column is Q₁ and the right column is Q₂:

 

[Jaaanosik]

The rest frame calculation:

 

[Jaaanosik]

The derivation of the Lorentz force equation starts at page 456 with this:

... and it leads to the equation of the original post on page 460.

 

[Ibix]

I can't read all of your numbers, but I think @Sagittarius A-Star nailed it:

In the inertial "rest frame", the charged plated in the isolated system, which must be mechanically connected by a distance piece, accelerate upwards when the electron accelerates downwards. Because of relativity of simultaneity, this happens in frame $K'_1$ first on the right side and then on the left side.

The plate doesn't rotate, but it does curve in the moving frame (in a stress-free manner). The reason is that "the plate" is the intersection of the (2+1)d volume of the plate and your chosen frame's constant-time surface. If the plate is accelerating its (2+1)d volume is a curved shape, and generally its intersection with a constant-time surface will also be curved.

I think you've calculated the Lorentz force on your charges at some time $t_0$ in the rest frame and some time $t'_0$ in the primed frame. But due to the relativity of simultaneity, $t'_0$ isn't the same as $t_0$ everywhere. So the charges $Q_1$ and $Q_2$ ought to have different $y$ coordinates and different $y$ velocities. That they don't means that you are modelling them as constrained to constant $y$, and whatever is doing the constraining is where you will find your missing momentum.

 

[PeterDonis]

The OP is on a permanent vacation, so this thread is closed. For the benefit of future readers, the OP's claims are not correct.