How Do Magnetic Dipole Moments Interact at a Distance?

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SUMMARY

The interaction energy between two magnetic dipole moments, p1 and p2, separated by a distance r, is given by the formula (-μ₀ / 2πr³) p1 ∙ p2. This equation accounts for the orientation of the dipoles relative to each other but does not fully incorporate the orientation of each dipole with respect to the vector joining them. To accurately assess the magnetic interaction, both dipoles must be considered in the context of their mutual influence, leading to a more comprehensive understanding of their energy dynamics.

PREREQUISITES
  • Understanding of magnetic dipole moments
  • Familiarity with the concept of magnetic fields
  • Knowledge of vector calculus
  • Basic principles of electromagnetism
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  • Study the derivation of the magnetic dipole interaction energy formula
  • Explore the effects of dipole orientation on magnetic interactions
  • Learn about the role of external magnetic fields in dipole interactions
  • Investigate advanced topics in electromagnetism, such as multipole expansions
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Physicists, electrical engineers, and students studying electromagnetism who are interested in the interactions of magnetic dipoles and their applications in various fields.

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Say you have two magnetic dipole moments, say p1 and p2, which are separated by a distance r, with no external magnetic fields.

If you want to figure out the energy of their magnetic interaction, is it valid to figure out the energy of p1 in the magnetic field generated by p2, or vice versa? Or will this not work?
 
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Actually, sorry, I think I was being stupid!
Do I consider it both ways round, so treat p1 as being in an external mag field (from p2), then p2 in an external field (from p1) and then sum the energies together?
 
I have an answer:
(-mu_0 / 2 pi r^3) p1 ∙ p2

I have a problem with this answer though; while it takes into account the orientation of the two with respect to each other, shouldn't it also take into account the orientation of each with the vector joining them?

My equation seems to agree with a general one I found, but only when assuming that p1 and p2 are parallel to to the vector joining them!
 

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