Are the value of the magnetic field of a current carrying coil and the magnetic dipole moments of the involved electrons comparable?

Question

The magentic dipole moment of electrons is a intrinsic property. To get the macroscopic effect of their common magnetic field this moments have to be aligned, like in permanent magnets or in current carrying coils. I'm wondering is it possible to compare the value of the magnetic field of a coil with the sum of the magnetic dipole moments of the involved electrons?

Answer 1

Dipole fields fall as $1/r^3$. The magnetic field due to a long linear current falls as $1/r$ near the wire.

So you can ask yourself "Could I get the long wire behavior by integrating a bunch of dipole in a line?" If you try it \begin{align} E(r) &\propto \int_{-\infty}^{\infty} \frac{\mathrm{d}x}{\left(x^2 + r^2\right)^{3/2}} \\ &= \left. \frac{x}{r^2 \sqrt{x^2 + r^2}}\right|_{x=-\infty}^{\infty}\\ &= \frac{2}{r^2} \,, \end{align} you get the wrong power law.

So, the magnetic field due to a current is not just the sum of a bunch of intrinsic dipole fields. Indeed the field due to the motion of charges must be sufficiently larger than that due to the combined dipole to dominate the field measurement at the laboratory scale.