Understanding the Equation B = del x A in Electromagnetic Theory

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SUMMARY

The equation B = ∇ x A is a fundamental relationship in electromagnetic theory, specifically relating the magnetic field B to the vector potential A. The discussion clarifies that A represents the vector potential of B and is essential for satisfying the condition ∇ · B = 0, which is a manifestation of Gauss's Law for magnetic fields. The participants emphasize the importance of understanding the physical interpretation of A, particularly in the context of vector calculus and Stokes' theorem. The conversation highlights the necessity of grasping these concepts to solve related problems effectively.

PREREQUISITES
  • Understanding of vector calculus, particularly Stokes' theorem
  • Familiarity with electromagnetic theory concepts, including Gauss's Law
  • Knowledge of curl and divergence operations
  • Basic understanding of magnetic vector potential
NEXT STEPS
  • Study the physical interpretation of the magnetic vector potential A
  • Learn about the implications of Gauss's Law for magnetic fields
  • Explore the application of Stokes' theorem in electromagnetic contexts
  • Investigate the relationship between magnetic fields and vector potentials in various scenarios
USEFUL FOR

Students of physics, particularly those studying electromagnetism and vector calculus, as well as educators looking to clarify concepts related to magnetic fields and vector potentials.

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Homework Statement



Not going to write out the whole problem (yet). It's a "find the error in the incorrect proof" type of question in a section on curl and divergence.

Homework Equations



B = \nablax A is given as an equation of "electromagnetic theory" and used in the proof. It's stated that the error is not in this equation. The other equation used in the proof is Gauss's Law for magnetic fields, but I get that one.

The Attempt at a Solution



Haven't really tried; I'd like to know the physical interpretation of the above equation before I start staring at integrals. I just got through electricity and magnetism in my general physics course, but this doesn't look familiar. I'm assuming that A is some type of force field, but what field would satisfy the equation?
 
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Well μH = ∇ X A

Magnetic Force
 
Thanks Pion, but can you elaborate? What is μH and how does it relate to B?
BTW, this is for a course that basically covers vector calculus that our math department doesn't in early calc classes. The question is really just about applying Stoke's theorem, which I understand pretty well. It's just the one equation that I don't get is keeping me from even starting the problem.
I can try and scan the problem if more context is needed, but it's nothing I want to write out here.
 
Ok, A is the vector potential of B. Does A have any physical meaning or is it just an arbitrary vector that must exist because \nabla\bulletB=0?
 

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