We present a Chandra and XMM-Newton study of X-ray emission from the lobes of 33 classical double radio galaxies and quasars. We report new detections of lobe-related X-ray emission in 11 sources. Together with previous detections, we find that X-ray emission is detected from at least one radio lobe in $75% of the sample. For all of the lobe detections, we find that the measured X-ray flux can be attributed to inverse Compton scattering of the cosmic microwave background radiation, with magnetic field strengths in the lobes between 0.3B eq and 1.3B eq , where the value B eq corresponds to equipartition between the electrons and magnetic field, assuming a filling factor of unity. There is a strong peak in the magnetic field strength distribution at B $ 0:7B eq . We find that more than 70% of the radio lobes are either at equipartition or electron dominated by a small factor. The distribution of measured magnetic field strengths differs for narrow-and broad-line objects, in the sense that broad-line radio galaxies and quasars appear to be further from equipartition; however, this is likely to be due to a combination of projection effects and worse systematic uncertainty in the X-ray analysis for those objects. Our results suggest that the lobes of classical double radio sources do not contain an energetically dominant proton population, because this would require the magnetic field energy density to be similar to the electron energy density rather than the overall energy density in relativistic particles.