We present a rigorous analysis of the eigenvalue problem associated with the onset of superconductivity for a thin domain in the presence of a large applied magnetic field. We prove the validity of Richardson and Rubinstein's formal result which reveals that in this double limit of thin domain and large field, the appropriate Rayleigh quotient differs from the standard one for order 0(1) applied fields through the addition of a potential depending on the field. This also demonstrates a parabolic background for the oscillatory phase transition curve between the normal and superconducting state.