= Force between distant coaxial current loops
{title2=$F\simeq3\pi\mu_0I_1I_2a_1^2a_2^2/(2d^4)$}
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 $B_0\simeq\mu_0I_1a_1^2/(2d^3)$. Its <near-axis radial magnetic field> at the second loop is $B_\rho\simeq3\mu_0I_1a_1^2a_2/(4d^4)$. Integrating the axial <Lorentz force> from $I_2a_2\,d\phi\,\widehat{\boldsymbol\phi}\times B_\rho\widehat{\boldsymbol\rho}$ 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.
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