# Biot-Savart: Why symmetry-break?

## Main Question or Discussion Point

What determines the sense of rotation of the magnetic field around an electric current?
Current through a straight wire is a problem with rotational symmetry, so what is it that breaks the symmetry?
Does the spin of a moving electron have a preferred orientation - or what could it be?

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Staff Emeritus
It's convention. You could have used a left-hand rule instead of a right-hand rule and, provided you did it everywhere, all the physical observables would remain the same.

Ok, I can vaguely understand that. But that doesn't change the fact that there exists an asymmetry. It is possible to measure the (relative) orientation of the magnetic field - a simple test magnet would flip if the current direction was reversed.

I guess that I am looking for a microscopic explanation for the presence of the asymmetry (regardless of actual orientations to start with), and the only possible circular-asymmetric about an electron traveling along a straight path that I can come to think of, is the electron spin.

But do you mean that the Biot-Savart law (and the symmetry-break) has not yet been derived from microscopic phenomena (physics of moving electrons)?

Staff Emeritus
What asymmetry? The only physical asymmetry is that two parallel currents in the same direction are attractive and not repulsive. Like currents attract, just like like charges repel. Everything else is a mathematical convention.

An analogy may perhaps help:
Let's say that each time you flush, the water rotates in the same direction. Wouldn't you say that this is an asymmetric situation, and that there should be an underlying explanation?

It has of course absolutely nothing to do with any conventions.

The same goes with the magnetic field around a current. The magnetic field is a vector field with magnitude and direction at every point. My question again is: Why isn't the vector field oriented in the opposite direction?
Without any prior experimental evidence, one would perhaps assume one of the following results:
1) There would be no magnetic field at all, or
2) The magnetic field would be directed radially (which *would* preserve symmetry), or
3) The magnetic field would be directed clockwise sometimes and counterclockwise sometimes, with no apparent trend.

None of the above happens of course, since Biot-Savart's law applies. But where does this asymmetry stem from?

Staff Emeritus
There is a difference between watching a toilet swirl, where you can directly see the direction of the flow, and the magnetic field, which can be observed only by its effects. Do the field lines point from north to south? Or from south to north? That's convention - the only thing that's actually observable is the force on a current loop. We could have just as easily picked a convention where we use a left-hand rule instead of a right-hand rule, and the force on the current loop would be the same - because we would have used the left-hand rule twice.

The only asymmetry is the force on two parallel currents in the same direction is attractive, not repulsive, and that comes from experiment.

geonat...good question...I do not know the answer but I do know many times we understand what happens, far less often why...

and you can also wonder why the electric and magnetic fields are orthogonal to each other?? Who required THAT?? and why perpendicular to the current flow?? I've read some scientists don't even believe there is a magnetic field...just electric...although Maxwell's equations can be interpreted to show the fields are orthogonal!! Likely they are formulated that way because Faraday's experiments showed forces in those directions and Maxwell was smart enough to be able to formulate the underlying math...

"we know much, we understand little"

Scientist know these things to be true. Just because you don't understand them doesn't mean scientist just make things up. When a natural phenomenon occurs while studying completely separate things it tends to add some truth to the theory. The reason for the direction of magnetic fields is described by the alignment of the magnetic dipole moments. If you put a ferromagnetic material in a field of opposite direction you can cancel the magnetism of the material. This would, like the parallel wire set up, suggest the B fields have direction. This also explains why ferromagnetic material strengthen the B field because of u alignment. You could go on and on. It is again proven when you study electromagnetic wave propagation. E fields and B fields are perpendicular that's why they can propagate without a medium. They have direction, look at the Poynting vector. Like Vanadium said the direction assigned is convention. Your question is like saying is an electron really negatively charged.

My initial main worry was that a symmetrical situation seemed to give an asymmetric result. I am happy that this isn't the case.

So the magnetic field is essentially just a mathematical crutch, that could possibly be replaced by a simpler alternative? One that does not introduce asymmetries where there are none?

Dale
Mentor
There is no asymmetry in this situation (even if the right-hand rule were something physical rather than the convention it is): If you have an infinitely long straight wire with a current and you solve the equation for the B field, then if you rotate that solution you have another B field which is also a solution for the same current distribution. That is what is meant by rotational symmetry.

Ok, I could have been more careful in my use of the phrase 'rotational symmetry'. Perhaps 'clock-/counterclockwise ambiguity' would have been more to the point.

Dale
Mentor
Well, again, even if you take the right-hand rule to be something physical rather than simply a convention, then there is still no ambiguity. The clockwise or counter-clockwise direction of the B-field is unambiguously determined by the direction of the current.

So the direction of the B-field is unambiguous (last post in this thread) but unobservable (post #6 in this thread)?
That requires a good portion of faith I would say.

Your right it's all FM theory. People just make it up. Particle accelerators work on magic and faith.

Hmm... I think this is just (justified) confusion about a lot of conventions.

We use the right hand rule, and we call the point of a magnetized needle that points in the direction of the field "north".

We could have the field the other way, or we could reverse the names of the poles. Since there are two degrees of freedom in the conventions, it really does not make a difference. It is symmetric; it's symmetric inasmuch as there are two poles and two ways to take the direction of the field.

The electric field doesn't suffer from such a duality of conventions, because there is only one direction that's parallel to a given direction.

The magnetic field suffers because there are infinitely many directions perpendicular to a given direction... we're forced to pick one.

Dale
Mentor
So the direction of the B-field is unambiguous (last post in this thread) but unobservable (post #6 in this thread)?
Vanadium50 and I are making different points, but they are not in conflict in any way. Vanadium is correctly pointing out the conventionality of the right-hand rule as it applies to the physics of magnetism and also pointing out the experimental distinction between the field and the forces.

I was pointing out the meaning of the mathematical term "symmetry" in order to correct your misunderstanding that there was asymmetry involved. I also corrected your misunderstanding about ambiguity. But my comments were intended only to help you understand the math concepts that you were missing, and were not in any way specific to the physics of magnetism.

That requires a good portion of faith I would say.
Your ignorance of the basic math terminology and physics here does not constitute faith on our part. You should spend more time learning and less time making silly accusations like this.

turin
Homework Helper
The reality of the magnetic field is on the same footing as the reality of inertia or angular momentum. In fact, I wonder, do you have the same problem with angular momentum? I suggest to think of the fundamental source of magnetism as a loop of current rather than a straight line of current, analogously to thinking of the source of angular momentum as a spinning mass rather than a mass moving in a straight line. Then, consider the effects of this current loop on other currents. Clearly, the direction of the current in the loop is physically unambiguous, and you can unambiguously talk about "clockwise" and "counterclockwise" directions about a given direction through the loop (in so much as you have established a convention for the sign of the charges). The hard part to understand is that, unlike electric phenomena whose fundamental effect is to push and pull radially, the fundamental effect of magnetic phenomena is to cause rotation. The attraction of an unmagnetised paper clip to a permanent magnet, for example, is a secondary phenomenon due to induced magnetism, analogously to the attraction of a neutral piece of paper to a staticly charged rod due to induced polarization.

Regarding the right-hand-rule, ask yourself, why should the cross-product, which isn't even physical but merely mathematical, be defined so that
$$\hat{x}\times\hat{y}=\hat{z}$$
etc.
What if you mathematically define the cross-product so that
$$\hat{x}\times\hat{y}=-\hat{z}$$
etc.? This will indeed change the sign in the Biot-Savart Law, but it will also change the sign in the magnetic force law, so the physical result will be the same. Can you physically distinguish between this mathematical change of sign in the cross-product vs. the quasi-physical change of sign of the B-field itself? The magnetic field isn't actually a vector field; it is a pseudo-vector field, which basically means that it is a cross-product field. This is a fundamentally different kind of field than, say, the electric field. What this amounts to, physically, is that the direction of the B-field should really be represented as
$$\hat{r}\times\hat{x}$$
etc., rather than simply as
$$\hat{x}$$
etc..

All right, I will stop. I think that some of you misunderstood my intentions.
I will treat the B-field as a non-unique mathematical tool, since it is not directly observable. This was a good lesson.

I think your question was answered in a way earlier in the thread. There is a kind of asymmetry in the fact that two parallel lines of current attract rather than repulse. This fact comes from charges and how they attract, whereby we introduce conventions, which in the end leads to the convention of the right hand rule.

Dale
Mentor
There is a kind of asymmetry in the fact that two parallel lines of current attract rather than repulse.
Again, that is not an asymmetry. Here, the setup does not have symmetry about an axis, instead if the currents are equal then it is symmetrical about the plane that is normal to a perpendicular line connecting the equal currents. So, if you solve and get a solution for the B-field, then if you flip it across this plane you get a solution for the same currents. This is symmetry.

I think many of the responses to this post (as is usual for physics forums) have been derogatory and are more concerned with "shutting down" the question rather than finding out what's really being asked, and providing a clear, personal interpretation. I don't see how these people are shouting "no asymmetry exists!", providing "be all, end all" answers, as if this an academic conference, and people are even entitled or qualified to make these opinions.

Anyways, barring my disgust for all of that, allow me to provide my own interpretation.

The way I see it, there is clearly an asymmetry in the B field about a current carrying wire. As the OP said, it is a rotational/counter-rotational symmetry. For example, in providing a torque ("moment") force in non-electrical mechanics, a force in the counter-clockwise direction generates a moment in the +z direction, whereas a force in the clockwise direction generates a moment in the -z direction. Hence you see that there is no asymmetry in this case; the direction of the moment "flips" sign when you provide a corresponding "flip" to the direction of the applied force. There is no "innate" direction to the moment; it completely depends on the direction of the applied force.

However, in the magnetic case, there is an asymmetry in the sense that the B field about the current is unidirectional. For a current going in one direction, the B field goes EITHER clockwise OR counterclockwise. It is true that this direction depends on your choice of convention; however, regardless of your convention, the B field will be unidirectional only. Hence there is a rotational asymmetry to the field; in upper echelon physics, I believe they call this "chirality", although to be honest I am unfamiliar with the lingo of the term.

Now, where does this come from? I don't know.....

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Dale
Mentor
Anyways, barring my disgust for all of that
It is rather telling that you think it is OK for genoat to accuse me and Vanadium of accepting things on faith but think it is disgusting for me to correct his misconceptions or respond to his accusations. Seems like a double-standard to me.

The way I see it, there is clearly an asymmetry in the B field about a current carrying wire. As the OP said, it is a rotational/counter-rotational symmetry.
The remainder of your post shows the same misunderstanding of symmetry. There is no asymmetry here:

An infinite straight current-carrying wire is axisymmetric about the wire. That means that it has a fundamental domain which is a half-plane starting at the axis. As long as a given vector field is the same relative to any half-plane starting at the axis then the field is also axisymmetric.

Here we find that all of the B-field vector field is at every point normal to any half-plane starting at the axis. This means that the B-field has also has a half-plane fundamental domain and thus has the same axisymmetry that is displayed by the current.

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The OP wasn't "accusing" you of anything, he was simply questioning the basis for your assessment. You are the one interpreting his dissent as a personal attack (which it isn't) and then responding aggressively. It's not a double standard because the OP never acted aggressively, whereas the responders have.

Now, again, you claim that the current carrying wire is an axisymmetric case. Allow me to be clear: I agree that it is axisymmetric about the wire. You can rotate the wire about its axis and the field remains the same; also, I understand your point about the half-plane behavior (I think). However, this is only a -partial- symmetry. You cannot say that the B field is -symmetric- just because it has this one type of symmetry. If you stand at the middle of a very long current wire and look both left and right, you will see that the B field rotates in opposite directions (clockwise/counter-clockwise). This is an asymmetry about the plane which cuts the wire at that point. You will always find this asymmetry for the wire; there is no escaping it, unless the current stops, and the field disappears entirely.

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.

I was instead interested in the asymmetry that arises as the electrons start to move *in comparison with* the case when they stand still. When they stand still there is no interesting distinction between clockwise and counterclockwise. But afterwards it is, because there then exists a B-field. Or maybe not, if it is unobservable. Then it might as well not exist. Or?

Dale
Mentor
The OP wasn't "accusing" you of anything, he was simply questioning the basis for your assessment. You are the one interpreting his dissent as a personal attack (which it isn't) and then responding aggressively. It's not a double standard because the OP never acted aggressively, whereas the responders have.
There is nothing more personal and accusatory to a scientist than the assertion that their scientific claim is based on faith instead of evidence. Such comments are inappropriate. I think geonat understands that now and I am satisfied with his response in post 18.
Now, again, you claim that the current carrying wire is an axisymmetric case. Allow me to be clear: I agree that it is axisymmetric about the wire. You can rotate the wire about its axis and the field remains the same; also, I understand your point about the half-plane behavior (I think).
Excellent, you understand the symmetry.
However, this is only a -partial- symmetry. You cannot say that the B field is -symmetric- just because it has this one type of symmetry.
Certainly, there are many different kinds of symmetries and a solution that exhibits one may not exhibit another, so it is probably best to always specify the kind of symmetry. However, since it is not possible to have every kind of symmetry at once, I would call a solution symmetric if it exhibits any symmetry. That is a semantic disagreement rather than a substantive disagreement.
If you stand at the middle of a very long current wire and look both left and right, you will see that the B field rotates in opposite directions (clockwise/counter-clockwise). This is an asymmetry about the plane which cuts the wire at that point. You will always find this asymmetry for the wire; there is no escaping it, unless the current stops, and the field disappears entirely.
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.

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