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

In summary, the direction of the curl of the magnetic field lines can be derived from the direction of the current and the convention of integrating over loops counterclockwise. This is a result of the mathematical models used to understand electromagnetic phenomena and the choice of emphasizing force fields over potentials. The concept of handedness in the magnetic field is a human construct and an alternative model using potentials is often more flexible and easier to understand.
  • #1
msandler
2
0
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?

===========================
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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  • #2
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. ;)
 
  • #3
You could equally well ask why it is that all corkscrews turn in exactly the same direction as the magnetic field.

Oooooh Scary! :wink:
 
  • #4
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.
 
  • #5
AJ Bentley said:
You could equally well ask why it is that all corkscrews turn in exactly the same direction as the magnetic field. Oooooh Scary! :wink:

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... :bugeye:

(Assuming the corkscrew is kept completely isolated while in the drawer and that you keep it there for a very, very, long time.)
 
  • #6
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?
 
  • #7
Pythagorean said:
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.
 
  • #8
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.
 
  • #9
AJ Bentley said:
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.
 
  • #10
Pythagorean said:
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! :rofl:)
 
  • #11
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
 

1. What is the direction of the curl of magnetic field lines?

The direction of the curl of magnetic field lines is always perpendicular to both the direction of the magnetic field and the direction of the current or changing electric field that is creating the magnetic field.

2. How do you determine the direction of the curl of magnetic field lines?

The direction of the curl of magnetic field lines can be determined using the right-hand rule. If you point your right thumb in the direction of the current or changing electric field and curl your fingers, the direction your fingers point is the direction of the curl of the magnetic field lines.

3. What factors affect the direction of the curl of magnetic field lines?

The direction of the curl of magnetic field lines is affected by the direction and strength of the current or changing electric field, as well as the presence of any nearby magnetic materials or other external magnetic fields.

4. What is the significance of the direction of the curl of magnetic field lines?

The direction of the curl of magnetic field lines is important because it indicates the direction of the force that a charged particle would experience if it were placed in the magnetic field. This is known as the Lorentz force and is key to understanding the behavior of charged particles in magnetic fields.

5. Can the direction of the curl of magnetic field lines be reversed?

Yes, the direction of the curl of magnetic field lines can be reversed by reversing the direction of the current or changing electric field that is creating the magnetic field. This is known as the right-hand grip rule, where the direction of the curl of the magnetic field lines is opposite to the direction your fingers curl in when gripping the current-carrying wire with your right hand.

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