The existing theoretical framework for the energies stored in the synchrotron-emitting lobes of radio galaxies and quasars does not properly account for the curved spectral shape that many of them exhibit. We characterize these spectra using parameters that are straightforwardly observable in the era of high-resolution, low-frequency radio astronomy: the spectral curvature and the turnover in the frequency spectrum. This characterization gives the Lorentz factor at the turnover in the energy distribution (we point out that this is distinctly different from the Lorentz factor corresponding to the turnover frequency in a way that depends on the amount of curvature in the spectrum) and readily gives the equipartition magnetic field strength and the total energy of the radiating plasma obviating the need for any assumed values of the cut-off frequencies to calculate these important physical quantities. This framework readily yields the form of the X-ray emission due to inverse-Compton scattering of cosmic microwave background (CMB) photons by the electrons in the plasma having Lorentz factors of ∼1000. We also present the contribution to CMB anisotropies due to relativistic plasmas such as giant radio galaxy lobes, expressed in terms of the extent to which the lobes have their magnetic field and particle energies are in equipartition with one another.