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This paper presents a quantitative description of field emission in superconducting rf cavities. The goal is to fully reconstruct the physics involved in the phenomenon, from electron current generation at the emitter to the radiation pattern measured by the x-ray detector. The field emission process in the cavity was reconstructed using CST Particle Studio, and then the generation of x-ray radiation and its propagation toward the detectors were modeled analytically. The entire model was then applied to the specific case of the European Spallation Source medium beta superconducting cavities, and the theoretical predictions were cross-checked with experimental data collected during cavity vertical tests.