Induced-current magnetophoresis

When an electrically conducting non-magnetic particle is subjected to a spatially varying and oscillating applied magnetic field of amplitude $\mathcal{H} + \mathcal{G} \cdot x$ and frequency $\omega$, an oscillating eddy current is induced. The Lorentz force density, the cross product of the current density and the magnetic field, consists of a steady component and a component with frequency $2 \omega$. If there is a spatial variation in the applied field, there is a steady force on a sphere of radius $R$ proportional to $- \mu_0 R^3 \mathcal{G} \cdot \mathcal{H}$, and a steady force on a thin rod of radius $R$ and length $L$ proportional to $- \mu_0 R^2 L (\mathcal{G} \cdot \mathcal{H} - \tfrac{1}{2} (\mathcal{G} \cdot \hat o)(\mathcal{H} \cdot \hat o))$, where $\mu_0$ is the magnetic permeability. There is torque proportional to $\mu_0 R^2 L (\hat o \times \mathcal{H} ) (\hat o \cdot \mathcal{H} )$ on a thin rod which tends to align the rod direction of the magnetic field. The coefficients in the force and torque expressions are functions of the dimensionless ratio of the radius and the penetration depth of the magnetic field, $\beta R = \sqrt{\mu_0 \omega \kappa R^2}$, where $\kappa$ is the electrical conductivity. It is shown that the effect of particle interactions can be expressed as an anisotropic diffusion term in the equation for the particle number density. The diffusion coefficient is negative, and concentration fluctuations are amplified, in the plane perpendicular to the magnetic field.