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This paper demonstrates the application of a polarization current approach to estimate the yield of soft x-ray Cherenkov radiation when a charged particle travels in the vicinity of a finite-size screen. To account for resonant anomalous dispersion of the complex permittivity, we apply the formalism for the atomic scattering factor. We derived the simple analytical expression for the radiation field, which allows us to determine all properties for a wide energy range of emitted radiation of a charged particle. The influence of screen sizes on Cherenkov radiation characteristics is determined by macroscopic and microscopic approaches.