B Relativistic explanation of electromagnetism

[Orodruin]

a neutral current-carrying wire creates an electrostatic force for a test charge moving relative to the wire/current, which we call the magnetic force.

We call it electromagnetic force. This is because, as you have noted, how the electromagnetic field splits into an electric and a magnetic field is frame dependent.

If we replace the test charge with a chunk of ferromagnetic material, we find that it experiences a magnetic force even if both it and the wire are uncharged, and even if it is stationary relative to the wire.

The force is created due to how the material reacts to the external magnetic field by creating induced currents.

 

[Karl Coryat]

The force is created due to how the material reacts to the external magnetic field by creating induced currents.

Ordinarily we associate induction with a changing magnetic field. Here, we have a static magnetic field. Is it that the field is always changing for the moving electrons in the ferromagnetic material? And, that the resulting currents set up a calculable electric-charge buildup on the surface of the material, and an opposite charge on the wire, which causes the wire to attract the material?

I realize this is more complicated than the test-charge case, so if anyone knows of a good online resource, I'd love to save you the time of having to explain the details.

 

[vanhees71]

This thread has shown how a neutral current-carrying wire creates an electrostatic force for a test charge moving relative to the wire/current, which we call the magnetic force. Excellent, thank you.

But I seek a more complete understanding.

If we replace the test charge with a chunk of ferromagnetic material, we find that it experiences a magnetic force even if both it and the wire are uncharged, and even if it is stationary relative to the wire.

How is this explained in terms of electrostatic forces and the Lorentz transform? The PF threads I've reviewed, and the linked resources (such as http://physics.weber.edu/schroeder/mrr/MRRtalk.html), only discuss test charges; and all explanations I've found of ferromagnetic materials appeal directly to the magnetic force, with none of that relativistic goodness. Perhaps Purcell covers it, but I don't have that book. Does it have to do with eddy currents?

Ferromagnetism is explained by the fact that elementary particles (in this case the electron) do not only carry electric charge but also a magnetic dipole moment, related to their spins. A permanent magnet is a material, where a macroscopic number of spins is oriented in one direction, because (at the given temperature) it is energetically more favorable for the associated magnetic moments being directed in one direction than being in random orientation as is the case in usual materials. To understand this completely from first principles you need quantum many-body theory (in this case the non-relativistic version is sufficient).

I also do not think that one can derive in a logical way from electrostatics the complete electrodynamics just using the relativistic spacetime structure. A more convincing argument is the analysis of relativistic quantum field theory in view of the symmetry group of Minkowski space, which is Poincare symmetry (i.e., symmetry under translations in space and time, rotations of space, and Lorentz boosts; to be more precise the here relevant symmetry group is the proper orthochronous symmetry group since it's known that the weak interaction breaks the discrete symmetries of space reflections and time reversal; only the "grand reflection" CPT is to the best of our knowledge a symmetry in accordance with the predictions from local relativistic QFT). Then you find out that causal theories can be built via representations with local fields of a given mass with $m^2 \geq 0$ and spin $s \in \{0,1/2,1,\ldots \}$.

The massless case $m=0$ is special, and for $s=1$ you necessarily get a gauge theory, if you don't want to have continuous intrinsic polarization-degrees of freedom. Since such a thing has never been observed, that's a plausible additional assumption, but as soon as you have the necessity of a gauge theory electromagnetism follows (together with using only the minimal number of necessary field-derivatives in the Lagrangian, i.e., keeping the corresponding QFT Dyson-renormalizable) quite inevitably. Also the generalization to non-Abelian gauge groups (a la Yang and Mills) is pretty obvious. These considerations, together with a lot of empirical input from the last decades of experimental HEP physics, lead to the Standard Model of elementary particle physics, which is the most robust theory of matter ever.

Classical electrodynamics thus indeed follows quite convincingly from the mathematical structure of Minkowski space, i.e., the special-relativistic spacetime model, but not from electrostatics alone though electrostatics gives a good hint at the fact that the electromagnetic field should be most simply be describable by a (Lorentz-)vector field. That it is massless is an empirical fact and cannot be derived from more fundamental (symmetry) assumptions.

 

[Hans de Vries]

Summary:: Why a test charge at rest in the lab frame does not experience a force from a current

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 moving in equal and opposite directions relative to the test charge, i.e., when the test charge is moving along the wire at 1/2 the drift velocity. But that's not what happens. What am I missing?

This question is a simpler version of the always recurring question:

"How to explain Magnetism as a relativistic side effect of the Electric Field"

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The answer to the OP's question:

- A test-charge at rest is only subject to an Electric Lorentz force.

- In the rest-frame the Lorentz force is calculated by integrating over all relativistic transformed Electric fields.

- Charge is Lorentz invariant. A wire with an equal number of negative and positive charges has a net charge of zero

- The Electric field of a moving charge changes under Lorentz transform.

- But the integral over all Lorentz transformed electric fields of all electrons in an infinitely long straight wire does not change regardless of the velocity of the electrons as long as the electron density in the wire stays the same. (The relativistic calculation is http://www.physics-quest.org/Magnetism_from_ElectroStatics_and_SR.pdf in section 2)

- Electrons move in principle independently from each other through the wire. The electron density is therefore not necessarily subject to Lorentz contraction. The field of each individual electron is transformed but the electron density is in principle arbitrary. The wire is neutral if the electron density is the same as the positive charge (proton) density.

This why the test charge at rest in the lab frame does not experience a force from a neutral current.

Also note this logical fallacy: The drift-speed of electrons is spread over a wide range of different velocities and the speed of each individual electron changes all the time. A Lorentz contraction of the electron density based on some average electron velocity makes no sense.

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Next:

Explain the Magnetic Lorentz force on a test-charge moving in parallel with a neutral current carrying wire. The by far simplest way to explain this (using pure electric fields) is non-simultaneity and going to the rest-frame of the test-charge- In its rest-frame the test-charge is only subject to Electric Lorentz forces.

- Due to non-simultaneity one end of the wire lays in the future and the other end in the past

- Therefore a net current has streamed into (or out from) the wire when viewed from the rest-frame of the test-charge.

- The wire is thus not electrically neutral anymore in the rest-frame of the test-charge

- The integral over all electric fields gives us the non-zero Lorentz force. (See http://www.physics-quest.org/Magnetism_from_ElectroStatics_and_SR.pdf in section 1)

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we can also calculate the Lorentz force on a test-charge moving perpendicular to a neutral current carrying wire by integrating over all electric fields as seen in the test-charge's rest-frame. (Again http://www.physics-quest.org/Magnetism_from_ElectroStatics_and_SR.pdf in section 3)

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A detailed derivation of the Lorentz transform of the Electromagnetic Potentials and Fields can be found here in my book:
http://www.physics-quest.org/Book_Chapter_EM_LorentzContr.pdf
following the original work of Liénard and Wiechert in 1898-1900.

 

[vanhees71]

As I said, you only can derive the Lorentz force from the force in a static electric field if you know the transformation rules for the electromagnetic field components under Lorentz transformations, which you cannot derive from electrostatics alone. I think it's best to start from the full set of Maxwell's equations to understand electrodynamics and then deriving the Lorentz-transformation properties of the fields from the Lorentz-transformation properties of the four-current density, which follows from the assumption that electric charge is a Lorentz scalar (as it must be for any intrinsic quantity of matter, which are nowadays always defined as scalars and operationally in the (local) rest frame of the matter) and the transformation properties of space-time four-vector components.