Biot-Savart: Why symmetry-break?

[Dale]

I was instead interested in the asymmetry that arises as the electrons start to move *in comparison with* the case when they stand still.

For clarification: Are you thinking of the electrons in the wire (i.e. when the current is zero and when it is non-zero), or are you thinking of an electron moving through the field (i.e. at rest 1 m away from the wire or 1 m away and moving but both cases with the same current in the wire)

 

[mordechai9]

The current exhibits this same asymmetry, therefore the field must exhibit it also. The field has every symmetry that the current has. In fact, in the sense that you are thinking both the current and the field exhibit an "odd" symmetry.

OK, now that it has taken 25 posts for everyone to acknowledge the asymmetry:

Why does that asymmetry occur? It is not enough to say that it exhibits the same asymmetry of the wire. The fact that the current goes in one direction and the magnetic field goes in one direction is a correlation, but it is not an explanation. What is it about the magnetic field which makes it "one-way" rotational? As far as I know, this is simply a physical phenomena which has no further explanation, e.g., in terms of electron spin effects, or something of that nature. Is there a microscopic or generally more elaborate explanation for this physics?

 

[Dale]

Why does that asymmetry occur? It is not enough to say that it exhibits the same asymmetry of the wire. The fact that the current goes in one direction and the magnetic field goes in one direction is a correlation, but it is not an explanation.

I don't understand your reasoning here at all. The magnetic field has every symmetry that the current does. Why would you expect the field to display a symmetry that the current does not?

EDIT: The more I think about it the more I realize that this behavior is, in fact, another symmetry. If you take the current and rotate it 180º about any axis perpendicular to the current then you have a reversed current ("odd" symmetry). Therefore the field must exhibit this same odd symmetry, which it does. If the field were to have the property you are looking for it would, in fact, lack a symmetry of the current.

 

[turin]

Why does that asymmetry occur?

It sounds like you're asking why electric current has a direction. Asymmetry is the very essence of nontrivial behavior. I cannot tell you why it occurs, but I think that if the universeve were in a state of perfect symmetry w.r.t. every possible imaginable transformation, then problably there would be no physics to speak of.

It is not enough to say that it exhibits the same asymmetry of the wire. The fact that the current goes in one direction and the magnetic field goes in one direction is a correlation, but it is not an explanation.

How can the explanation be anything more than a correlation? I believe that this "correlation" nails it, because it simultaneously emphasizes that the choice of clockwise vs. counterclockwise is irrelevant, but at the same time it must be made consistently and so the choice is asymmetric.

What is it about the magnetic field which makes it "one-way" rotational? As far as I know, this is simply a physical phenomena which has no further explanation, e.g., in terms of electron spin effects, or something of that nature. Is there a microscopic or generally more elaborate explanation for this physics?

I would say that it does have a further explanation, i.e. relativity, but you may just consider this a correlation.

 

[Phrak]

Let me first say that I was already sufficiently satisfied with the answer in post #6. I still am. I write to hopefully clean up some of the mess that followed post #7.

I understand that there is no rotational asymmetry in the B-field (which I already admitted in post #11). That was never really an issue to me.

Understood. Chiral is the name of the symmetry to which you refer.

 

[geonat]

Ok, good. Now, from everyday life, one does not expect that a pointlike mass (in vacuum) undergoing uniform motion or accelerating along a straight line should lead to any rotation-like effects. But this is what seems to happen for charge-carriers such as electrons, since a chiral B-field is produced.

It was then very tempting to think something like "Aha, perhaps the spins of the electrons prefer a certain spin orientation as they are accelerated, and that they then maintain that state. That would perhaps explain the chirality from microscopic phenomena. I.e. that the energy levels for spin up and spin down are different in an accelerated frame. Or something like that."

Again, trying to adopt a mechanical everyday life approach, that preferred spin orientation might be "explained" by some sort of mass imbalance in the structure of the electron itself. I know that this is very speculative, but I just wanted to share the line of thought that initiated my decision to post a message here in the first place. As I have understood it though, the chirality cannot be deduced from any spin properties of charge carriers. And then the mystery remains - at least for me.

 

[Dale]

Again, trying to adopt a mechanical everyday life approach, that preferred spin orientation might be "explained" by some sort of mass imbalance in the structure of the electron itself.

The electron is thought to be a fundamental particle (no internal structure), but the electron does have intrinsic spin so you probably don't need to postulate an internal structure anyway.

However, if you want to adopt a "mechanical" approach then you should probably focus not on fields but on "mechanical" things like forces. That was one of Vanadium's points in posts 2 and 4. I would recommend calculating the force between two wires in a variety of different orientations, and between a wire and a free electron moving in different directions and at different speeds. Once you have a good understanding of the mechanics of the situation then you will be in a much better position for making an alternative mechanical explanation.

By the way, if you would like a mechanical explanation in terms of relativity you can essentially http://physics.weber.edu/schroeder/mrr/MRRtalk.html" [Broken].

 

[Phrak]

It's fairly amazing that electromagnetism displays no measurable chirality. The electric and magnetic fields can be expressed as the antisymmetric derivative of the 4 vector potential. This is called the Maxwell tensor among other names. Maxwell's equations, relating charge and fields, are the antisymmetric derivative of the Maxwell tensor. The charge continuity equation is an antisymmetric derivative of Maxwell's equations. The wave equation is another antisymmetric derivative of Maxwell's equations.

[It seems that the inclusion of magnetic monopoles into electromagnetism would display chiral variance, if there were such things. I'm not quite sure.]

 

[turin]

... one does not expect that a pointlike mass (in vacuum) undergoing uniform motion or accelerating along a straight line should lead to any rotation-like effects.

Ah, but such linear phenomena CAN lead to rotation-like effects. Consider the point mass moving along the z-axis, travelling in the +z-direction. I suggest that this situation has axial symmetry about the z-axis. However, if I calculate the angular momentum about, for example, the spatial points

(x,y,z)={(1,0,0),(0,1,0),(-1,0,0),(0,-1,0)},

I find that the angular momentum points in the respective directions

{(0,1,0),(-1,0,0),(0,-1,0),(1,0,0)}.

In other words, you can think of an "angular momentum field" that encircles the z-axis in a CLOCKWISE sense. But this encricling character is not physical; it is only a calculational tool. Does this mean that the angular momentum itself is not physical. No, the angular momentum is physical. You can see this by placing an object in the path of the point particle, and observing that the object will rotate when the particle collides with it, depending on where is the center of mass of the object. The direction of the rotation is consistent with the direction of the angular momentum that I calculated if I use the same right-hand-rule convention. That is, if the center of mass of the object is at (x,y,z)=(1,0,0), then the rotation will result in an angular momentum in the (0,1,0) direction, etc. You can think of this object as the "angular momentum field" probe.

If you just consider the linear motion of the rod and the rotation of the object, you see that there is no seemingly strange assymetry, and everything is just a straightforward consequence of balancing forces and momentum. The point is that it is simply more convenient to calculate the angular momentum about the center of mass, using a convention that makes the situation seem assymetric, even though it actually isn't. Analogously, it is usually more convenient to use the magnetic field of a linear current rather than considering the relativistic contraction of linear charge densities in "neutral" wires.

 

[turin]

[It seems that the inclusion of magnetic monopoles into electromagnetism would display chiral variance, if there were such things. I'm not quite sure.]

If I'm not mistaken, magnetic monopoles and EM guage invaraince cannot be simulaneously true, or at least a magnetic monopole introduces some wierd topological branch in space, but I guess this wouldn't be so catastrophic if they always came in pairs.

 

[Phrak]

If I'm not mistaken, magnetic monopoles and EM guage invaraince cannot be simulaneously true, or at least a magnetic monopole introduces some wierd topological branch in space, but I guess this wouldn't be so catastrophic if they always came in pairs.

Defining the electric and magnetic fields as spacetime derivatives of a potential field precludes magnetic monopole fields on a simply connected manifold. (All closed forms are exact.) Something about deRham cohomology. This is probably what you've heard, but it's not true. I've disproved it--though I cheated. I get a low energy regime where everything looks normal with one kind of charge. Higher energies allow monopole fields.