Analytic solution for the critical state in superconducting elliptic films

A thin superconductor platelet with elliptic shape in a perpendicular magnetic field is considered. Using a method originally applied to circular disks, we obtain an approximate analytic solution for the two-dimensional critical state of this ellipse. In the limits of the circular disk and the long strip this solution is exact, i.e., the current density is constant in the region penetrated by flux. For ellipses with arbitrary axis ratio the obtained current density is constant to typically ${10}^{-3}$, and the magnetic moment deviates by less than ${10}^{-3}$ from the exact value. This analytic solution is thus very accurate. In increasing applied magnetic field, the penetrating flux fronts are approximately concentric ellipses whose axis ratio $b/a<~1$ decreases and shrinks to zero when the flux front reaches the center, the long axis staying finite in the fully penetrated state. Analytic expressions for these axes, the sheet current, the magnetic moment, and the perpendicular magnetic field are presented and discussed. This solution applies also to superconductors with anisotropic critical current if the anisotropy has a particular, rather realistic form.