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Two equal circular current loops are placed coaxially with each other. The loops have equal but opposite currents $I$.

$$ A \propto r^{-l} $$ $$ B \propto r^{-k} $$

, where $A$ is the vector potential, and $B$ is the magnetic field. What is $l$ and $k$, when you are at a distance far away?

I thought that I could approximate this system as a dipole system, which should imply that $A \propto r^{-2}, B \propto r^{-3}$, however the correct answer is $l=3, k=4$, and I can't seem to understand why.

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A single current loop is already a magnetic dipole, thus having a far field $$A \propto r^{-2},$$ $$B \propto r^{-3}.$$ (See especially the section Magnetic dipole - External magnetic field produced by a magnetic dipole moment.)

Hence, when combining two opposite current loops (i.e. two opposite magnetic dipoles) separated by a small distance, you get a magnetic quadrupole. The far fields of the two dipoles nearly cancel, and you get one more power of $r^{-1}$ in the resulting quadrupole field: $$A \propto r^{-3},$$ $$B \propto r^{-4}.$$

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