  We study the emission of X-rays from lobes of FR-II radio galaxies by inverse
Compton scattering of microwave background photons. Using a simple model that
takes into account injection of relativistic electrons, their energy losses
through adiabatic expansion, synchrotron and inverse Compton emission, and also
the stopping of the jet after a certain time, we study the evolution of the
total X-ray power, the surface brightness, angular size of the X-ray bright
region and the X-ray photon index, as functions of time and cocoon size, and
compare the predictions with observations. We find that the radio power drops
rapidly after the stopping of the jet, with a shorter time-scale than the X-ray
power. The X-ray spectrum initially hardens until the jet stops because the
steepening of electron spectrum is mitigated by the injection of fresh
particles, for electrons with $\gamma \ge 10^3$. This happens because of the
concurrence of two times scales, that of the typical jet lifetimes and cooling
due to inverse Compton scattering ($\sim 10^{7\hbox{--}8}$ yr), of electrons
responsible for scattering CMB photons into keV range photons (with $\gamma
\sim \sqrt{1 \, {\rm keV}/ kT_{CMB}}$). Another finding is that the ratio of
the X-ray to radio power is a robust parameter that varies mostly with redshift
and ambient density, but is weakly dependent on other parameters. We also
determine the time-averaged ratio of X-ray to radio luminosities (at 1 keV and
151 MHz) and find that it scales with redshift as $\propto (1+z)^{3.8}$, for
typical values of parameters. We then estimate the X-ray luminosity function of
FR-II radio galaxies and estimate the number of these diffuse X-ray bright
objects above a flux limit of $\sim 3 \times 10^{-16}$ erg cm$^{-2}$ s$^{-1}$
to be $\sim 25$ deg$^{-2}$.
