# Why is $A \propto 1/r^3, B \propto 1/r^4$, far away from circular loops

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.

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: $$A \propto r^{-3},$$ $$B \propto r^{-4}.$$