Tiny core or thin layer as a perturbation in scattering by a single-layered sphere

Analytic expressions for scattering by a single-layered sphere when either the core is tiny or the shell thickness is small are presented within the framework of a perturbative approach. In this approach, the contribution of the core or the shell (each acting as a perturbation) to the scattering is separated out, thus facilitating a deeper insight into the nature of its effect. This is illustrated by the application of the analytic expressions to the well-known practical case of absorption of visible light by water droplets contaminated with graphitic carbon (soot) in the atmosphere.