 Force on a current loop in a bilinear magnetic field

For a square spanning $0\le x,y\le a$, with current counterclockwise as seen from positive $z$, an applied field $B_z=\alpha xy$ vanishes on two edges. Integrating $Id\mathbf{\ell }\times \mathbf{B}$ on the other two gives the displayed force. Its direction reverses with the current. The formula concerns the loop, not unspecified supply leads, and cannot be replaced by a small-loop approximation without checking the field variation.