Mean Lorentz force in a magnetic skin layer (${\overline{F}}_y=C^2{e}^{-2y/\delta }/(2{\mu }_0\delta )$)

For real boundary amplitude $C$ and $q=(1+i)/\delta$, the leading electric current density is $j_z=qC{e}^{-qy}/{\mu }_0$. Averaging products of real harmonic fields gives $\overline{\mathbf{F}}=\mathrm{Re}[(\nabla \times \mathbf{B})\times {\mathbf{B}}^{\ast }]/(2{\mu }_0)$. Its leading normal component is the displayed positive force. The first candidate tangential contribution is proportional to $\mathrm{Re}(q/q^{\ast })C{C}^{\prime }=0$, so retaining the small normal field does not introduce a tangential force of that order.