Electromagnetic Field Equations for a Moving Medium with Hall Conductivity

Abstract

In an isotropically conducting medium the electric field and current density are associated by Ohm's law. This, together with Maxwell's equations, leads to $${\nabla}^{2}H=4\pi \sigma{\frac{\partial H}{\partial t}\,-\text{curl}\,(\upsilon \times H)}+{c}^{-2}\frac{{\partial}^{2}H}{\partial{t}^{2}},$$ where H , υ and σ are the magnetic field, velocity of the medium and conductivity. This equation describes any electromagnetic field in a moving (or fixed) conductor. It includes the “telegraph equation” of radio waves and also describes any hydromagnetic wave. Ionized gas in a sufficiently strong magnetic field is anisotropically conducting and σ is replaced by a tensor. An equation analogous to the above is derived for these conditions, allowing the study of electromagnetic disturbances in a medium with Hall conductivity. The equation is used on some simple problems of astrophysical interest, including hydromagnetic waves.