Convective magnetic fields in a dispersive half-space

Abstract

It is known that inside a material half-space the magnetic field B owing to the currents generated there by a slowly moving exterior charge (velocity u ) is almost the same whether the material is a good Ohmic conductor or a highly refractive non-dispersive/non-dissipative insulator. By contrast, the drag force experienced by the charge is completely different for conductors and insulators. To gain insight into the somewhat surprising coincidence regarding B fields, we study a microscopic model whose macroscopic Drude-type dielectric function ε ( ω ) can fit a fair variety of dispersion and dissipation. We look for B only to first order in u / c , but with otherwise arbitrary u . Then, B is given by the Biot–Savart rule. The term linear in u follows directly from the polarization produced as if electrostatically by the charge in its instantaneous position, and depends only on ε (0), the strictly static (zero frequency) response function; only the corrections of higher order in u depend on just how ε varies with ω , and we determine the first such corrections.