= Mean Lorentz force in a magnetic skin layer
{title2=$\overline F_y=C^2e^{-2y/\delta}/(2\mu_0\delta)$}
For real boundary amplitude $C$ and $q=(1+i)/\delta$, the leading <electric current density> is $j_z=qCe^{-qy}/\mu_0$. Averaging products of real harmonic fields gives $\overline{\mathbf F}=\operatorname{Re}[(\nabla\times\mathbf B)\times\mathbf B^*]/(2\mu_0)$. Its leading normal component is the displayed positive force. The first candidate tangential contribution is proportional to $\operatorname{Re}(q/q^*)CC'=0$, so retaining the small normal field does not introduce a tangential force of that order.
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