  The time-dependent nonlinear kinetic theory for cosmic ray (CR) acceleration
in supernova remnants (SNRs) is applied studying the properties of the
synchrotron emission from SNRs, in particular, the surface brightness-diameter
($\Sigma-D$) relation. Detailed numerical calculations are performed for the
expected range of the relevant physical parameters, namely the ambient density
and the supernova explosion energy. The magnetic field in SNRs is assumed to be
significantly amplified by the efficiently accelerating nuclear CR component.
Due to the growing number of accelerated CRs the expected SNR luminosity
increases during the free expansion phase, reaches a peak value at the
beginning of the Sedov phase and then decreases again, since in this stage the
overall CR number remains nearly constant, whereas the effective magnetic field
diminishes with time. The theoretically predicted brightness-diameter relation
in the radio range in the Sedov phase is close to $\Sigma_\mathrm{R}\propto
D^{-17/4}$. It fits the observational data in a very satisfactory way. The
observed spread of $\Sigma_\mathrm{R}$ at a given SNR size $D$ is the result of
the spread of supernova explosion energies and interstellar medium densities.
