Electromagnets are a relativistic phenom?

[DaveC426913]

This is a new one one me:

If a charged object sits still next to an electromagnet, then nothing happens to it. Even though the electrons are flowing, they occupy a similar amount of space to the protons so that over all the electrified metal has no effect on it.

However, if this charged object moves alongside the wire, then it starts to feel the effects of length contraction in the moving electrons. This means that the density of stationary protons becomes larger than the flowing electrons and the metal exhibits a positive charge, causing the object to be attracted or repelled.

http://www.iflscience.com/physics/4-examples-relativity-everyday-life

 

[SlowThinker]

This is a popular "explanation" of magnetism, but it's hard to apply to, e.g., individual electrons affecting each other, or electrons moving towards/away from a wire.
It may be correct, but don't take it too seriously.

 

[DaveC426913]

I don't see how it can be.

It suggests that "even if the electrons are flowing" there is a negligible amount of length contraction, yet I can wave the electromagnet fast enough (like, 2mph) to create length contraction?

 

[SlowThinker]

I don't see how it can be.

It suggests that "even if the electrons are flowing" there is a negligible amount of length contraction, yet I can wave the electromagnet fast enough (like, 2mph) to create length contraction?

The important thing is that if you just wave a piece of copper, both electrons and nuclei are contracted in the same way, so you have no effect.
Only if a current is flowing through a wire, the speeds, and thus contractions, of electrons and nuclei are different, and you get a magnetic field.

 

[DaveC426913]

The important thing is that if you just wave a piece of copper, both electrons and nuclei are contracted in the same way, so you have no effect.
Only if a current is flowing through a wire, the speeds, and thus contractions, of electrons and nuclei are different, and you get a magnetic field.

There's something screwy about this explanation.
Electrons flowing in a wire don't generate length contraction, but moving the wire by hand does. That seems to be what the paragraph is saying.

 

[Nugatory]

This is a new one one me:
...
http://www.iflscience.com/physics/4-examples-relativity-everyday-life

Edward Purcell used this model in his classic E&M textbook (I believe he came up with it in the first place), and it's mentioned in the FAQ linked from our sticky thread on experimental verification of SR - perhaps as close to a practical demonstration of length contraction that we can find, as macroscopic objects moving at relativistic speeds are (thankfully) scarce. All that's necessary to make it work is that the charge density be frame-dependent.

Google for "Purcell magnetism relativity" and you'll find many hits. This one is good: http://physics.weber.edu/schroeder/mrr/MRRtalk.html

 

[DaveC426913]

All right. Fair enough.

I think the problem I'm having is with the murky explanation in the article. It seems to suggest that an electrified wire will not have an attractive force, but waving the wire with my hand will.

 

[Nugatory]

All right. Fair enough.

I think the problem I'm having is with the murky explanation in the article.

That's a problem with the article, not the concept.

 

[DaveC426913]

That's a problem with the article, not the concept.

Yes. That's what I said.

 

[bcrowell]

I don't see how it can be.

It suggests that "even if the electrons are flowing" there is a negligible amount of length contraction, yet I can wave the electromagnet fast enough (like, 2mph) to create length contraction?

There is length contraction in the lab frame, and in this frame the wire is electrically neutral with the length contraction taken into account. In the moving frame, there is a different amount of length contraction for the electrons, and there is also length contraction for the protons. The net effect of these two changes in length contraction is to produce an attraction or repulsion.

The original quote is a somewhat garbled and incorrect explanation, but the general idea is correct. Magnetism is a purely relativistic phenomenon. For a (hopefully) correct version of the explanation, see http://www.lightandmatter.com/lm/ , section 23.2.

 

[vanhees71]

It's put in a bit an unclear way. It's all about Lorentz-transformation properties of the electromagnetic field and the Lorentz force.

The situation discussed is the following: In the inertial reference frame A you have a wire with a DC current density $\vec{j}$ producing a magnetic field. Neglecting the very small relativistic effects (or the Hall effect for the conduction electrons) in this reference frame the charge density is $\rho=0$. Then you have no electric field $\vec{E}=0$ but a magnetic field $\vec{B}=B \vec{e}_{\varphi}$, where I use cylinder coordinates $(\rho,\varphi,z)$ with the wire along the $z$ axis.

A charged particle moving with a velocity $\vec{v}=v \vec{e}_z$ along the wire feels a purely magnetic Lorentz force,
$$m c \mathrm{d}_{\tau} \vec{u}=q \vec{u} \times \vec{B}.$$
Here $\tau$ is the proper time of the charged particle and $\vec{u}=\gamma \vec{v}$ are the spatial components of the four-velocity
$$u^{\mu} = \frac{1}{c} \mathrm{d}_{\tau} x^{\mu}.$$
The forth component of the equation of motion follows from $u_{\mu} u^{\mu}=1=\text{const}$, which implies
$$u_{\mu} \mathrm{d}_{\tau} u^{\mu}=0 \; \Rightarrow \; u_0 \mathrm{d}_{\tau} u^0=\vec{u} \cdot \mathrm{d}_{\tau} \vec{u}=0$$
in our case, i.e., the 4D invariant Minkowski force on the particle at the initial time is
$$K^{\mu}=\begin{pmatrix} 0 \\ q \vec{u} \times \vec{B} \end{pmatrix}=\begin{pmatrix}0 \\ -\frac{q B v \gamma}{c} \vec{e}_{\rho} \end{pmatrix}.$$
Here we've used the usual notation $\gamma=1/\sqrt{1-v^2/c^2}=u^0$.

Now consider the situation in another inertial frame B, where the particle is initially at rest. This is described by a Lorentz boost in $z$ direction with boost velocity $v$.

The current density together with the charge density makes up a four-vector field. In frame A it has the components
$$(j^{\mu})=\begin{pmatrix} 0 \\ 0 \\ 0 \\ j \end{pmatrix},$$
and in frame B, according to the Lorentz transformation the components
$$(\tilde{j}^{\mu})=\begin{pmatrix} -\beta \gamma j \\ 0 \\ 0\\ \gamma j \end{pmatrix}.$$
Since in this frame the charged particle is at rest at the moment it moves parallel to the wire in A it only fields the electric force. Since the wire is charged now, in this frame is also an electric field, which is also given by the Lorentz transformation (the electromagnetic field is represented as an antisymmetric four-tensor field in relatitivity which determines its transformation properties). In our case we have
$$\vec{\tilde{E}}=\frac{\gamma}{c} \vec{v} \times \vec{B}=\frac{\gamma v B}{c} \vec{e}_z \times \vec{e}_{\varphi}=-\frac{\gamma v B}{c} \vec{e}_{\varrho}.$$
The Minkowski force on the particle at that moment as measured in frame B thus is
$$\vec{\tilde{K}}=q \vec{\tilde{E}} = -\frac{q\gamma v B}{c} \vec{e}_{\varrho}.$$
The temporal component follows again from
$$\tilde{u}_0 \mathrm{d}_{\tilde{t}} \tilde{u}^0=\mathrm{d}_{\tilde{t}} \tilde{u}^0 = \vec{u} \cdot \mathrm{d}_{\tilde{t}} \vec{u}=0.$$
This you get also directly from the Lorentz boost of $K^{\mu}$.

Note that for the usual non-covariant Force $\vec{F}=\vec{K}/u^0$ there's a $\gamma$ factor
$$\tilde{\vec{F}}=\vec{\tilde{K}}{\tilde{u}^0}=\vec{\tilde{K}}=\gamma \vec{F}.$$

The relation of all this to Lorentz contraction is a bit indirect. You can derive everything from the Lorentz transformation properties of the charge-current-density four vector. In frame A the positively charged ions making up the wire are at rest and the conduction electrons are moving. In our approximation, neglecting the Hall effect, in this frame the total charge density (positive ions + conduction electrons) is set to 0. Now in frame B the positively charged ions are moving and thus their charge density gets a $\gamma$ factor compared to frame A because of the length contraction of the volume elements used to define the charge density. The conduction electrons are moving in frame A and thus their charge density does not only get a Lorentz factor in frame B but there's also a component from the spatial components of the corresponding current density, and thus the charge density of the conduction electrons does not compensate precisely the charge density of the positive ions when measured in frame B. The net result is the above given negative charge density of the wire as a whole, which implies the existence of the radial electric field, which is the cause of the purely electric Lorentz force in frame B.

A very nice discussion of the relativistic formulation of electromagnetism (which simplifies a lot compared to the usual non-relativistic heuristics used in conventional textbooks) can be found in

M. Schwartz, Principles of Electrodynamics, Dover (1987)

or

L. D. Landau, Course of Theoretical Physics, vol. 2, Butterworth-Heinemann (1996)

 

[A.T.]

This is a new one one me:

See here:



And this diagram by DrGreg:

From this old thread:
https://www.physicsforums.com/threa...-of-electrostatics.577456/page-3#post-3768045

 

[Demystifier]

Edward Purcell used this model in his classic E&M textbook (I believe he came up with it in the first place)

I don't know who was first, but a very similar approach is also described in the book
W.G.V. Rosser, Classical Electromagnetism via Relativity (1968)
https://www.amazon.com/dp/1489962581/?tag=pfamazon01-20

Unfortunately, for some reason the Rosser's book is not so well known.

 

[vanhees71]

Yep, that's something I never understood: Why is the oldfasioned historical approach to electromagnetism so much favored compared to the relativistic approach? Usually you go through a long pretty dull set of special cases, starting from electrostatics, magnetostatics (sometimes then even the quasi-stationary approximation), and finally the full Maxwell equations and waves. This does not bring the entire beauty of electromagnetism into sight until the very last part of the lecture. Usually there's not even time to discuss the fully relativistic formulation (including relativity for the mechanical part, i.e., the charges and currents making up the sources of the field, let alone the very exciting topic of radiation reaction). The relativistic approach as in Schwartz's book is much more lucid. However, the standard textbook (at least in Germany) for the theory lecture on E&M is usually still Jackson, which is of course very good, but follows this traditional approach. Then there are newer books like Griffiths, who emphasizes the apparent paradoxes like "hidden momentum", instead of doing the full relativistic treatment right away. Then it's very clear that there is no hidden momentum but just momentum, and the apparent "problems" or "paradoxes" have been already well understood by von Laue in 1911 (if not even earlier in 1904-1906 by Poincare)!

 

[Demystifier]

Yep, that's something I never understood: Why is the oldfasioned historical approach to electromagnetism so much favored compared to the relativistic approach?

That's probably because most students who learn classical electrodynamics will eventually be something like condensed-matter experimentalists rather than something like particle-physics theorists. For them (the former ones) it probably makes a lot of sense to think of electricity and magnetism as two different phenomena.

 

[Demystifier]

The relativistic approach as in Schwartz's book is much more lucid.

A very non-traditional relativistic-from-the-start textbook of classical electrodynamics, ideal for (future) particle physicists, is
A.O. Barut, Electrodynamics and Classical Theory of Fields and Particles
https://www.amazon.com/dp/0486640388/?tag=pfamazon01-20

 

[harrylin]

This is a new one one me:http://www.iflscience.com/physics/4-examples-relativity-everyday-life

As far as I know, in the general case such as a coil electromagnet, the magnetic field can not be transformed away (contrary to Einstein's suggestion in 1905).

Check out this old thread, in which we had a lengthy discussion:
https://www.physicsforums.com/threa...relativistic-effect-of-electrostatics.577456/

 

[vanhees71]

A very non-traditional relativistic-from-the-start textbook of classical electrodynamics, ideal for (future) particle physicists, is
A.O. Barut, Electrodynamics and Classical Theory of Fields and Particles
https://www.amazon.com/dp/0486640388/?tag=pfamazon01-20

That's a great book (also bit plagued with typos), but I'd not recommend it as a primary source for the introductory E&M-theory lecture.

You may be right that the reason for doing the "non-relativistic electrodynamics" (i.e., treating the matter part non-relativistically) is that for many physicists that's what they primarily need in applications. However, I think the relativistic consistent picture should be taught to all physicists, not only those specializing in some research dealing with relativistic issues.

 

[vanhees71]

As far as I know, in the general case such as a coil electromagnet, the magnetic field can not be transformed away (contrary to Einstein's suggestion in 1905).

Check out this old thread, in which we had a lengthy discussion:
https://www.physicsforums.com/threa...relativistic-effect-of-electrostatics.577456/

It's clear that you cannot always transform away the magnetic field. This is clear without any specific example from the fact that the two scalar invariants you can construct from the electromagnetic field are $\vec{E} \cdot \vec{B}$ and $\vec{E}^2-\vec{B}^2$, which are the contraction of the Faraday tensor with its Hodge dual and with itself, respectively.

If you have $\vec{E}^2-\vec{B}^2<0$ you never can transform to a system where there is no magnetic field, because there you'd have $\vec{E}'^2-\vec{B}'^2=\vec{E}'^2 \geq 0$.

That's why you need to start from the full electromagnetic field when treating at as a relativistic field theory. The arguments, leading from electrostatics via the demans of Lorentz invariance to the necessity of the additional magnetic field components is not a "derivation" in the strict sense but a heuristic argument. As such it is pretty convincing (see, e.g., the above cited book by Schwartz).

 

[bcrowell]

As far as I know, in the general case such as a coil electromagnet, the magnetic field can not be transformed away (contrary to Einstein's suggestion in 1905).

In the Purcell pedagogy, we're talking about transforming from a frame in which a particular *force* is purely magnetic to another frame in which the *force* is purely electric.

 

[vanhees71]

I'm not so much a fan of Purcell's book. Even the above example with the current in a wire and a particle around it, is quite overcomplicated. I think, it's much simpler to just do the Lorentz transformation, and that's it.

 

[bcrowell]

I'm not so much a fan of Purcell's book. Even the above example with the current in a wire and a particle around it, is quite overcomplicated. I think, it's much simpler to just do the Lorentz transformation, and that's it.

I think you're oversimplifying. The material has to be developed in a logical way, with some physical motivation at each step. At the stage we're referring to, we're dealing with questions like how we even know that a phenomenon like magnetism should exist.

 

[vanhees71]

Ok, but you cannot derive it simply from electrostatics. You can make it very plausible but not more. If you are at that level of sophistication, i.e., if you are acquaint with special relativity (which you admittedly should get before (sic!) starting with electromagnetism) you can much simpler ask for the possible realization of field equations for a massless vector field, leading to electromagnetism. See, e.g., Landau Lifshitz vol. II. I got quite confused with Vol. II of the Berkeley physics course as a beginner.

 

[atyy]

Ok, but you cannot derive it simply from electrostatics. You can make it very plausible but not more. If you are at that level of sophistication, i.e., if you are acquaint with special relativity (which you admittedly should get before (sic!) starting with electromagnetism) you can much simpler ask for the possible realization of field equations for a massless vector field, leading to electromagnetism. See, e.g., Landau Lifshitz vol. II. I got quite confused with Vol. II of the Berkeley physics course as a beginner.

Yes, of course. It's just a question of how much some handwavy explanation you like. I myself found the traditional order incredibly confusing, and found it much easier to understand the complete Maxwell's equations like the commandments to Moses from heaven :) David Dugdale's, and Haus and Melcher's text do it that way.

But there are two things in Purcell that I think are great handwavy derivations - the derivation of magnetism from relativity, and the incredible derivation of the Lamor formula (I don't think I can do that off the top of my head). It's the same balance the extent to which one has to do difficult problems involving constraints in elementary mechanics without the Lagrangian.

 

[vanhees71]

I prefer Schwartz's text over Purcell's, but that's just personal taste. Both are, no doubt, great textbooks. The most simple one is Landau-Lifshitz vol. II. That's relativistic field-theory (including general relativity) for physicists with the exact right amount of "geometry" and emphasis of physics.

The only flaw in Landau Lifshitz is the apparently simple "derivation" of the retarded potential. He makes a great shortcut, by just invoking a handwaving argument which is finally something like Huygen's principle. This, of course, works, but only because the space-time dimension 4 is even in the real world. For the wave equation in odd space-time dimensions it's wrong!

 

[atyy]

The only flaw in Landau Lifshitz is the apparently simple "derivation" of the retarded potential. He makes a great shortcut, by just invoking a handwaving argument which is finally something like Huygen's principle. This, of course, works, but only because the space-time dimension 4 is even in the real world. For the wave equation in odd space-time dimensions it's wrong!

Huygen's principle only works in even spacetime dimensions?

 

[robphy]

Huygen's principle only works in even spacetime dimensions?

Although this is off-topic...
P. Ehrenfest, In that way does it become manifest in the fundamental laws of physics that space has three dimensions? (1918)
http://www.dwc.knaw.nl/DL/publications/PU00012213.pdf

http://math.stackexchange.com/quest...ence-between-waves-in-odd-and-even-dimensions

 

[bcrowell]

Ok, but you cannot derive it simply from electrostatics. You can make it very plausible but not more.

It might depend on what you mean by this, but basically Purcell would be a counterexample to this claim. And Purcell is quite logically and mathematically rigorous. He doesn't do hand-wavy derivations or resort to plausibility arguments.

You've made a lot of vague and hand-wavy complaints about Purcell, but you haven't said anything specific.

 

[Demystifier]

However, I think the relativistic consistent picture should be taught to all physicists, not only those specializing in some research dealing with relativistic issues.

I certainly agree, but I am not aware of any textbook on classical electrodynamics where the relativistic picture is not taught. The question is how much that picture is emphasized, and that certainly varies from a textbook to a textbook.

 

[Demystifier]

And Purcell is quite logically and mathematically rigorous. He doesn't do hand-wavy derivations or resort to plausibility arguments.

I think he does. He derives the existence and properties of the magnetic field for some special case (which is rigorous), and then assumes that these properties are valid generally (which is plausible, but not rigorous).

But that's OK. A first textbook on classical electrodynamics (or, for that matter, on any topic) should not be too rigorous.