 Force between distant coaxial current loops

For coaxial circular loops with radii $a_1,a_2\ll d$, the first loop's on-axis magnetic field of a circular current loop has far value $B_0\simeq {\mu }_0{I}_1{a}_1^2/(2d^3)$. Its near-axis radial magnetic field at the second loop is $B_{\rho }\simeq 3{\mu }_0{I}_1{a}_1^2{a}_2/(4d^4)$. Integrating the axial Lorentz force from $I_2{a}_{2}d\varphi \hat{\mathbf{\varphi }}\times B_{\rho }\hat{\mathbf{\rho }}$ gives the displayed magnitude. Equal current senses attract; opposite senses repel. This is the leading dipole limit and is not an exact close-coil force law.