  We calculate the flux of radio, hard X-ray and UV radiation from clusters of
galaxies as produced by synchrotron emission and Inverse Compton Scattering of
electrons generated as secondaries in cosmic ray interactions in the
intracluster medium. Both the spatial distribution of cosmic rays due to their
diffusion and the spatial distribution of the intracluster gas are taken into
account. Our calculations are specifically applied to the case of the Coma
cluster. The fluxes and spectra of the radio halo emission and of the hard
X-ray excess from Coma can be explained in this model if an average magnetic
field $B \sim 0.1\mu G$ is assumed. However, such a low value for the
intracluster magnetic field implies a large cosmic ray energy density which in
turn is responsible, through neutral pion decay, for a gamma ray flux above 100
MeV which exceeds the EGRET upper limit. This gamma ray bound can be relaxed if
the hard X-ray excess and the radio halo emission from Coma are not due to the
same population of electrons. We finally stress the unique role that the new
generation gamma ray satellites will play to discriminate among different
models for the non thermal emission in clusters of galaxies.
