Cross product of magnetic fields

[wofsy]

Is there any physical meaning to the cross product of two magnetic fields e.g. two fields generated in two different current loops?

 

[Born2bwire]

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.

 

[Phrak]

Two loops will interact, if that's what you mean, so that the results are more than the sum of each.

 

[wofsy]

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.

 

[Born2bwire]

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.

 

[wofsy]

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.

 

[Phrak]

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?

 

[wofsy]

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.