In this paper, inspired by the success of recent experiments, we discuss a new possible type of sources of extreme ultraviolet radiation for the semiconductor industry, based on the radiating plasma with multiply charged ions supported in a mirror magnetic trap by high-power microwaves. We propose a simple theory that describes the main features of such source, perform modeling for a wide range of plasma parameters and magnetic configurations, compare the results to the existing experimental data, and study the prospects of the new scheme in present technological circumstances.