Ampère force law, action, reaction

[lalbatros]

This is apparently a well known topic, but I did not know it before today.
Let us consider the Ampère law for the force experience by a current element (1) in the magnetic fields of another current elment (2):

$$\mathbf{dF}_{12} = \frac {\mu_0} {4 \pi} I_1 I_2 \frac {d \mathbf{s_2}\ \mathbf{ \times} \ (d \mathbf{s_1} \ \mathbf{ \times } \ \hat{\mathbf{r}}_{12} )} {r_{12}^2}$$

You can easily check that the "action and reaction" are not balanced by the Ampère law since:

$$\mathbf{dF}_{12} + \mathbf{dF}_{21} <> 0$$

How should we understand that?

 

[Meir Achuz]

That force law is only valid if ds_1 and ds_2 are integrated over closed circuits.
Then a little vector calculus can be used to put it into a different form that does have
F_12=F_21. This form is called the Neumann form. It depends on the dot product of
ds_1 and ds_2.
This is shown in most EM textbooks.

 

[confinement]

It's a fascinating fact that the electromagnetic field itself carries momentum in classical electrodynamics, and this causes Newton's third law to appear to fail. For this reason it is intractable to use Newton's laws in electrodynamics (it is not impossible, there are pseudo-mechanical formulations of E&M, including Maxwell's own model, that allow us to apply Newtonian mechanics to find the missing reaction forces and momentum, but no one does this). Instead it is more convenient to use Lagrangian methods and/or the covariant formulation of E&M i.e. the faraday tensor, etc.

 

[Meir Achuz]

But NIII does apply for constant currents in closed loops.

 

[confinement]

But NIII does apply for constant currents in closed loops.

Thanks for clarifying this point, I did not mean to imply that NIII is never valid in electromagnetism.

 

[lalbatros]

Thanks clem, I found the symmetric form in Jackson.
I realize of course that a current element can never exist outside of a closed circuit.
Therefore, in itself, the force on a current element alone could never be measured and could probably not be defined without ambiguity.
On the other hand microscopic forces can be clearly defined and verified experimentally.
Therefore, one can clearly say that current element are not really part of "fundamental physics".

However, I would be interrested by a extended discussion of this topic.
For example, would it not be possible to modelize the forces in a wire from a miscroscipic point of view?
Also, I will go back to my MHD notes and see if it could improve my understanding of this topic.
In MHD, would this ambiguity remain as such or would it be easier to understand?

Thanks