Charge expulsion and electric field in superconductors

The theory of hole superconductivity predicts that when a metal goes superconducting negative charge is expelled from its interior towards the surface. As a consequence the superconductor in its ground state is predicted to have a non-homogeneous charge distribution and an outward pointing electric field in its interior. Here we propose equations to describe the behavior of the charge density and electric field in superconductors, and solve them for a spherical geometry. The magnitude of the predicted interior electric field depends on superconducting parameters such as the condensation energy and the London penetration depth and is found to be of order $10^6 V/cm$. A physical interpretation of the result is given. It is predicted that for small superconducting bodies (compared to the penetration depth) an electric field $outside$ the superconductor should result from this physics. This may explain a recent experimental observation in $Nb$ metal clusters.