Radio Measurements of the Magnetic Field in the Solar Chromosphere and the Corona

The structure of the upper solar atmosphere, on all observable scales, is intimately governed by the magnetic field. The same holds for a variety of solar phenomena that constitute solar activity, from tiny transient brightening to huge Coronal Mass Ejections. Due to inherent difficulties in measuring magnetic field effects on atoms (Zeeman and Hanle effects) in the corona, radio methods sensitive to electrons are of primary importance in obtaining quantitative information about its magnetic field. In this review we explore these methods and point out their advantages and limitations. After a brief presentation of the magneto-ionic theory of wave propagation in cold, collisionless plasmas, we discuss how the magnetic field affects the radio emission produced by incoherent emission mechanisms (free-free, gyroresonance, and gyrosynchrotron processes) and give examples of measurements of magnetic filed parameters in the quiet sun, active regions and radio CMEs. We proceed by discussing how the inversion of the sense of circular polarization can be used to measure the field above active regions. Subsequently we pass to coherent emission mechanisms and present results of measurements from fiber bursts, zebra patterns, and type II burst emission. We close this review with a discussion of the variation of the magnetic field, deduced by radio measurements, from the low corona up to ~ 10 solar radii and with some thoughts about future work.