Two coaxial circular loops of radii $0.5\text{m}$ and $5\times {10}^2\text{m}$ are separated by a distance $0.5\text{m}$ and carry currents $2\text{A}$ and $1\text{A}$ respectively. What is the force between the loops?

The potential energy of a coil in a magnetic field is 

where $M_2$ is magnetic moment of coil $2$ and $B_1$ magnetic field on it. 

The force between the loops

$F=-\frac{dU}{dx}$ and U = - $M_2.B_1$

   $=\frac{d}{dx}\left[I_2\pi R_2^2\right]\left[\frac{{\mu }_0{I}_1{R}_1^2}{2(R_1^2+x^2{)}^{3/2}}\right]=-\frac{3{\mu }_0{I}_1{I}_2{R}_1^2{R}_2^{2}x}{2(R_1^2+x^2{)}^{5/2}}$

    $=\frac{3\times ({10}^{-7}\times 4\pi )2\times 1\times (0.5{)}^2\times (0.05{)}^2\times (0.5)}{2\left[\right(0.5{)}^2+(0.5{)}^2{]}^{5/2}}$

$F=-2.09\times {10}^{-8}\text{N}$ Ans.

The minus sign shows that the force is attractive.