  As superconductors are cooled below their critical temperature, stray
magnetic flux can become trapped in regions that remain normal. The presence of
trapped flux facilitates dissipation of ac current in a superconductor, leading
to losses in superconducting elements of microwave devices. In type II
superconductors, dissipation is well-understood in terms of the dynamics of
vortices hosting a single flux quantum. In contrast, the ac response of type I
superconductors with trapped flux has not received much attention. Building on
Andreev's early work [Sov. Phys. JETP 24, 1019 (1967)], here we show
theoretically that the dominant dissipation mechanism is the absorption of the
ac field at the exposed surfaces of the normal regions, while the deformation
of the superconducting/normal interfaces is unimportant. We use the developed
theory to estimate the degradation of the quality factors in field-cooled
cavities, and we satisfactorily compare these theoretical estimates to the
measured field dependence of the quality factors of two aluminum cavities.
