We investigate the non-thermal X-ray emission from Cas A , using BeppoSAX , CGRO -OSSE , and Chandra data . For the hard X-ray continuum we test the model proposed by Laming , which invokes non-thermal bremsstrahlung from electrons accelerated by lower hybrid plasma waves . The justification for this model comes from our determination of a lower limit to the average magnetic field of B > 0.5 mG. For such high magnetic fields the synchrotron losses are severe enough that most of the electron populations responsible for the radio emission have maximum electron energies well below the limit for which X-ray synchrotron emission is important . However , we do suggest that the rim surrounding Cas A , seen in Chandra continuum images , is X-ray synchrotron emission . The width of this rim of 1.5″ to 4″ , can be used to infer the magnetic field near the shock front , for which we estimate B = 0.08 - 0.16 mG , and electron energies of \sim 57 - 40 TeV . This magnetic field strength is lower than the average magnetic field , but higher than what may be expected from shocked interstellar medium , suggesting either a high magnetic field in the wind of the progenitor , or rapid , post shock , field amplification by non-linear growth of plasma waves . Combining the two magnetic field measurements we have constructed a simple two zone model . Most of the radio emission comes from inside Cas A where the magnetic field is strong . In contrast , the inverse Compton emission is dominated by emission from near the shock front . Only for our lower limit on the magnetic field strength near the shock front is it possible to explain the recent detection of TeV emission by HEGRA with inverse Compton , for which , in addition , we have to assume a rather high far infrared photon density that should be twice as high as our best estimate of \sim 70 cm ^ { -3 } . Pion decay is therefore likely to be the dominant emission from Cas A at TeV energies .