Flux penetration in a thin superconducting disk

Pearl's description of a quantized flux line in an infinite thin superconducting film with an effective penetration depth $\Lambda$ is generalized to a thin finite disk of radius $R$. The presence of the boundary precludes an exact analytic solution; instead, the problem is reduced to an integral equation. An approximate solution provides a convenient interpolation between a neutral superfluid film ($R\ll \Lambda$) and an unbounded superconducting film ($R\gg \Lambda$). The lower critical field $H_{c1\mathrm{}}$ and the interaction function between two flux lines are evaluated as functions of $\frac{R}{\Lambda }$. In the limit of many flux lines with nonoverlapping cores, the flux density is shown to be uniform for arbitrary $\frac{R}{\Lambda }$.