Cross product of magnetic fields

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Discussion Overview

The discussion revolves around the physical meaning and implications of the cross product of two magnetic fields, particularly in the context of fields generated by current loops. Participants explore theoretical interpretations, potential applications, and the mathematical properties of the cross product in this scenario.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants question the physical significance of taking the cross product of two different magnetic fields, suggesting that the geometrical interpretation of the cross product applies universally to any pair of vectors.
  • Others propose that two current loops will interact, implying that their combined effects are more complex than a simple sum of their individual fields.
  • One participant mentions that the cross product of the fields from two current loops can indicate the work done by one field on a magnetic monopole moving along the other loop.
  • Another participant notes that work is a scalar quantity, which raises questions about the interpretation of the cross product in this context.
  • A later reply introduces the idea that dividing the cross product by the work could yield a fundamental generator of the second cohomology of R^3 minus the two loops, suggesting a deeper mathematical relationship.
  • There is a clarification regarding the nature of the loops involved, with questions about whether they are electric-current loops or magnetic monopole-current loops.
  • One participant clarifies that the monopole does not need to move uniformly but must complete a circuit of the second loop, indicating a specific approach to the integral involved.

Areas of Agreement / Disagreement

Participants express differing views on the relevance and application of the cross product of magnetic fields, with no consensus reached on its physical meaning or practical scenarios where it might be applied.

Contextual Notes

There are unresolved assumptions regarding the definitions of the magnetic fields involved and the conditions under which the cross product is considered. The discussion also highlights the complexity of interpreting work in relation to vector quantities.

wofsy
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Is there any physical meaning to the cross product of two magnetic fields e.g. two fields generated in two different current loops?
 
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I do not think so, I am at a lost as to a situation where we would take the cross of two different magnetic fields. There is of course the usual geometrical interpretation of the cross product, indicating the anti-parallelism of the vectors, a normal to the vectors, etc., but that applies to any pair of vectors independent of their physical interpretations.
 
Two loops will interact, if that's what you mean, so that the results are more than the sum of each.
 
Born2bwire said:
I do not think so, I am at a lost as to a situation where we would take the cross of two different magnetic fields. There is of course the usual geometrical interpretation of the cross product, indicating the anti-parallelism of the vectors, a normal to the vectors, etc., but that applies to any pair of vectors independent of their physical interpretations.

The reason I asked is that the cross product of the fields of two current loops tells you the work that one field would do on a magnetic monopole moving uniformly along the other loop.
 
wofsy said:
The reason I asked is that the cross product of the fields of two current loops tells you the work that one field would do on a magnetic monopole moving uniformly along the other loop.

But work is a scalar quantity, not a vector.
 
Born2bwire said:
But work is a scalar quantity, not a vector.

True. The work is a scalar factor in the cross product. If you divide out by the work you get a fundamental generator of the second cohomology of R^3 minus the two loops. For instance if the work is zero the the cross product is the curl of another vector.
 
What do you mean by "moving uniformly along the other loop." Are these loops supposed to be electric-current loops or magnetic monopole-current loops?
 
Phrak said:
What do you mean by "moving uniformly along the other loop." Are these loops supposed to be electric-current loops or magnetic monopole-current loops?

Sorry - the monopole doesn't need to move uniformly. It just needs to make a complete circuit of the second loop. It's just that I am used to doing the integral using the unit of arc length.
 

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