  We report on ROSAT HRI observations of the nearby powerful radio galaxies
3C33 and 3C111, which both have detected optical hot spots. We find nuclear
X-ray sources in both objects, but no X-ray emission from the hot spots. This
confirms the presence of a high-energy cutoff in the spectrum of
synchrotron-emitting electrons. Since these electrons necessarily scatter the
synchrotron photons by the inverse-Compton process, our upper limits on the
X-ray fluxes of the hot spots allow us to set lower limits of a few nanotesla
on their magnetic flux density, close to or greater than the fields implied by
equipartition of energy between radiating particles and magnetic field.
