  The existing theoretical framework for the energies stored in the
synchrotron-emitting lobes of radio galaxies and quasars doesn't properly
account for the curved spectral shape that many of them exhibit. We
characterise 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
characterisation 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 cutoff frequencies to calculate these
important physical quantities. This framework readily yields the form of the
X-ray emission due to inverse-Compton (IC) scattering of Cosmic Microwave
Background (CMB) photons by the electrons in the plasma having Lorentz factors
of $\sim$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.
