Debye source representation for type-I superconductors, I

In this note, we analyze the classical magneto-static approach to the theory of type I superconductors, and a Debye source representation that can be used numerically to solve the resultant equations. We also prove that one of the fields, $\boldsymbol{J}^-$, found within the superconductor via the London equations, is the physical current in that the outgoing part of the magnetic field is given as the Biot-Savart integral of $\boldsymbol{J}^{-}$. Finally, we compute the static currents for moderate values of London penetration depth, $\lambda_L,$ for a sphere, a stellarator-like geometry and a two-holed torus.