  We study the origin of the non-thermal emission from the intracluster medium,
including the excess hard X-ray emission and cluster-wide radio haloes, through
fitting two representative models to the Coma cluster. If the synchrotron
emitting relativistic electrons are accelerated in situ from the vast pool of
thermal electrons, then a quasi-stationary solution of the kinetic equation
with particle acceleration through turbulence at high energies (>200 keV)
naturally produces a population of supra-thermal electrons responsible for the
excess hard X-ray emission through bremsstrahlung. Inverse Compton scattering
is negligible at hard X-ray energies in this case. The radio halo flux density
constrains the magnetic field strength to a value close to that of
equipartition ~1 uG. Alternatively, if the relativistic electrons are injected
from numerous localised `external' sources then the hard X-rays are best
explained by inverse Compton scattering from GeV electrons, and little of the
hard X-radiation has a bremsstrahlung origin. In this case, the magnetic field
strength is constrained to ~0.1-0.2 uG. Both models assume that the non-thermal
emissions are generated by a single electron spectrum, so that only two free
parameters, well constrained by the observed hard X-ray and radio halo spectra,
are needed in either case. Measurements of the cluster magnetic field will
distinguish between the models.
