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After reading the Chapter 35 of Halliday Resnick and Krane's Physics, which is the section on magnetism in which magnetic dipoles are introduced, I still don't understand why the particular quantity $$\mu:= iA$$ of a loop of wire with area $A$ and current $i$ is important.

Searching online gives not very satisfying answers like "The torque on the loop is written in a compact way as $\mu \times B$" but that shouldn't be the reason the quantity was defined in the first place.

Why the quantity is important ?

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  • $\begingroup$ "HRK" is not a sufficient bibliographic reference. $\endgroup$ – Emilio Pisanty Oct 8 '18 at 13:28
  • $\begingroup$ HRK means Halliday Resnick Krane. The other things are way too much over my head. $\endgroup$ – cdt Oct 8 '18 at 13:29
  • $\begingroup$ I have edited in the bibliographic reference; in future, you should always provide complete references that can be understood by people who are not aware of your immediate context. $\endgroup$ – Emilio Pisanty Oct 8 '18 at 13:32
  • $\begingroup$ Further, I'm deleting your second question as it is unrelated to your first question - you should ask it separately. This format works well when you have one core question per thread, and multiple core questions just mess things up. $\endgroup$ – Emilio Pisanty Oct 8 '18 at 13:32
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You can always define whatever quantity you want, so long as you take appropriate care to justify its appropriateness in the cases where you're (completely or partially) using existing terminology. What really matters is your ability to take the concepts you've defined and put them to use in an effective manner.

Thus, when you define the concept of the magnetic dipole moment of a current loop and then you go on to show something like

The torque on the loop is written in a compact way as $\mu \times B$,

i.e. you go on to show that all of your system's interactions of a given type (in this case, the torque exerted by an external field) are not only (i) compactly encapsulated by your new concept, but also (ii) completely encapsulated by your new concept (i.e. you've proved that the spatial layout of your current loop is irrelevant, and that the only thing that matters is the dipole moment), then your new concept has already earned its place as a relevant concept in the theory.

The dipole moment of a current loop, of course, is indeed more important than just that isolated property. The thing that makes it really important is that it is the initial member of a series of moments, known as the multipole expansion of your current distribution, which is able to completely capture your current distribution's interactions with the outside world.

In other words, for any given interaction $I$, you're generally able to decompose it as $$ I = \sum_\ell \mathrm{moment}_\ell[\mathbf J] \times \mathrm{property}_\ell[\mathbf B] $$ where

  • $\mathrm{moment}_\ell[\mathbf J]$ is some function of your current distribution $-$ generally some integral of the current density times some polynomial of the position's cartesian coordinates $-$ not unlike the dipole moment,
  • $\mathrm{property}_\ell[\mathbf B]$ is the magnetic or electric field, or some combination of its derivatives,
  • the sum goes from the dipole case $\ell=1$ upwards, and most importantly
  • if your circuit is small enough compared to the length-scale of the variation of the external field, then the sum is dominated by the dipole component, so long as the dipole moment is nonzero.

That's plenty to earn a central place in the theory.

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