Solutions to Axion Electrodynamics in Various Geometries

Recently there has been a surge of new experimental proposals to search for ultra-light axion dark matter with axion mass, $m_a\lesssim1\,\mu$eV. Many of these proposals search for small oscillating magnetic fields induced in or around a large static magnetic field. Lately, there has been interest in alternate detection schemes which search for oscillating electric fields in a similar setup. In this paper, we explicitly solve Maxwell's equations in a simplified geometry and demonstrate that in this mass range, the axion induced electric fields are heavily suppressed by boundary conditions. Unfortunately, experimentally measuring axion induced electric fields is not feasible in this mass regime using the currently proposed setups with static primary fields. We show that at larger axion masses, induced electric fields are not suppressed, but boundary effects may still be relevant for an experiment's sensitivity. We then make a general argument about a generic detector configuration with a static magnetic field to show that the electric fields are always suppressed in the limit of large wavelength.