## what's the causal model for direction of curl of magnetic field lines?

untruncated version of question: As electrons move through a wire, is the direction of the curl of the magnetic field lines observed derivable from an underlying property?

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more detail (only to further clarify my question as useful):
> looking for a mechanistic understanding, if possible, much like 'temperature' is derivable from the mean kinetic velocity of the molecules of the medium

> another way to ask the question, would be that if you point your thumb in the direction of the current, and use the right-handed-rule, why are the magnetic field lines curling toward you, instead of away from you? (and it's not a question about sign convention, in that, if you flip the sign convention, and point your thumb the opposite way, in the direction of the electron travel, and use a left-handed-rule, same question about why the field lines are curling as they are, instead of the other way? )

> one friend of mine speculated it might derivable somehow, from the 'spin' of electrons....?

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 What you are asking IS a sign convention question. Spin has nothing to do with it. The direction of the magnetic field can be defined by the direction of the force that acts on charges inside it. If you change the sign on a few of the Maxwell equations then then everything works out again. If you do not change the definitions, then the answer to your question is Lenz's law. If the induced fields were reversed, currents could cause field's that would produce more current ad infinitum and the universe would explode. ;)
 You could equally well ask why it is that all corkscrews turn in exactly the same direction as the magnetic field. Oooooh Scary!

## what's the causal model for direction of curl of magnetic field lines?

My understanding is that given Coulomb's law for static charges and special relativity you can show that there must exist "magnetic" forces in order for relatively moving observers to predict the same results for experiments. That derivation would tell you what direction the magnetic force must point.

Recognitions:
 Quote by AJ Bentley You could equally well ask why it is that all corkscrews turn in exactly the same direction as the magnetic field. Oooooh Scary!
It is spooky! The Hamiltonian of a corkscrew is symmetric with respect to coordinate-inversion, so the corkscrew wave function should evolve into the superposition of right-handed and left-handed states. So you don't really know which direction you'll need to turn your corkscrew until you open the drawer and observe it...

(Assuming the corkscrew is kept completely isolated while in the drawer and that you keep it there for a very, very, long time.)

 Recognitions: Gold Member I kinda of felt he was asking the more fundamental question: Where does handedness come from in the magnetic field? What mechanism is behind the asymmetry that gives rise to handedness?

 Quote by Pythagorean What mechanism is behind the asymmetry that gives rise to handedness?
The 'handedness' is entirely a human fiction.

If we use a different mathematical formulation of the electromagnetic field, we can do away with the E and B fields entirely and just use the scalar and vector potentials.
In that schema, the direction of the vector (magnetic) potential is simply parallel to the direction of current flow and the resultant force is perpendicular - there is no 'curl of a vector' to worry about.

 Corresponding to every vector field, there is an associated field of differential 2-forms (the adjoint under stokes theorem). These are basically little weighted planes sitting at every point in space roughly normal to the direction of the current. For these forms there is a clear orientation (plane normal in the same direction as the current rather than the opposite). Combining this with the convention that you integrate over loops counterclockwise, and the direction of the magnetic field is completely determined. So really the right handedness is really a convoluted encoding of the direction that the current is going, taking into account the convention that you integrate over loops counterclockwise rather than clockwise.

Recognitions:
Gold Member
 Quote by AJ Bentley The 'handedness' is entirely a human fiction. If we use a different mathematical formulation of the electromagnetic field, we can do away with the E and B fields entirely and just use the scalar and vector potentials. In that schema, the direction of the vector (magnetic) potential is simply parallel to the direction of current flow and the resultant force is perpendicular - there is no 'curl of a vector' to worry about.
Hrm.... I see.

Not that I dispute the statement, but in my EM classes, I was given the impression that the magnetic vector potential was a fiction (a mathematical convenience) itself.

 Quote by Pythagorean vector potential was a fiction (a mathematical convenience) itself.
All mathematics applied to physical phenomena is in essence a 'fiction'.
We use these logical models to understand or create a conceptual basis for our understanding but the model is not 'real' nor is another model rendered a 'fiction'.

Maxwell chose to emphasise the force field approach to EM because he was firmly convinced of the existence of the aether and that model suited the concept better. He was aware of the potentials as an alternative model but chose to sideline that approach.
We now know that the aether is a seriously flawed concept so we prefer not to use it.

In fact, because the vector and scalar potentials together make a four-vector which is invariant under Lorentz transform, in many ways the model is more flexible - it is certainly a lot easier to understand and use in practice.

Unfortunately, the education system is firmly wedded to a historical approach to learning which means that the potentials are only introduced at an advanced level and then only as a 'mathematical abstraction' from the E and B fields. In doing so, much of the simplicity is lost (curl of a curl indeed! - you just finish up going parallel to the original direction - Isn't that a clue that you took a wrong turn somewhere! )

 thanks all for your comments!! Been thinking a bunch since original post (catalyzed by your helpful comments), and with the add of cracking my physics texts to the right pages (spurned by your comments), I think I see my original question was a bit ill posed. In short, there's no real 'asymmetry' or chirality (like in biological molecules) -- rather the 'curl' of the fingers is about which direction the 'polarity' of the affected charge will have forces induced on it. a bit more illustration further below.... -------------------------- nevertheless, I realize a slightly deeper question arises, about why we see induction of any magnetic field (or charge-related forces induced). I think before I can post a usefully intelligently coherent question on it, need to do more reading-- thank you again all!! :) yeaahh physicsforums.com /////////////////////////////////////////////////////////////////////////// bit more illustration on what getting at. using positive ions say, w/curl option1, they go directionA, whereas w/curl option2 (say the other way from option1) and the positive ions go directionB (the opposite from directionA). conversely, negative ions, are affect the 'opposite way'. That is, under curl option1, go directionA, and under curl option2, go directionB. in a quickiee table: curl: option1 option2 neg_ion directionA directionB pos_ion directionB directionA

 Tags magnetic field lines