Superconducting strip in an oblique magnetic field

As an example for a seemingly simple but actually intricate problem, we study the Bean critical state in a superconducting strip of finite thickness $d$ and width $2w⪢d$ placed in an oblique magnetic field. The analytical solution is obtained to leading order in the small parameter $d∕w$. The critical state depends on how the applied magnetic field is switched on, e.g., at a constant tilt angle, or first the perpendicular and then the parallel field component. For these two basic scenarios we obtain the distributions of current density and magnetic field in the critical states. In particular, we find the shapes of the flux-free core and of the lines separating regions with opposite direction of the critical currents, the detailed magnetic field lines (along the vortex lines), and both components of the magnetic moment. The component of the magnetic moment parallel to the strip plane is a nonmonotonic function of the applied magnetic field.