  For the system of Maxwell equations of electromagnetism in an $l$-periodic
composite medium of overall size $L$ ($0<l<L<\infty$), in the low-frequency
quasistatic approximation, we develop an electromagnetic version of
strain-gradient theories, where the magnetic field is not a function of the
magnetic induction alone but also of its spatial gradients, and the electric
field depends not only on the displacement but also on displacement gradients.
Following the work (Smyshlyaev, V.P., Cherednichenko, K.D., 2000. On rigorous
derivation of strain gradient effects in the overall behaviour of periodic
heterogeneous media, ${\mathit J.\ Mech.\ Phys.\ Solids\ }{\mathbf{48}},$
$1325-1357$), we develop a combination of variational and asymptotic approaches
to the multiscale analysis of the Maxwell system. We provide rigorous
convergence estimates of higher order of smallness with respect to the inverse
of the "scale separation parameter" $L/l.$ Using a special "ensemble averaging"
procedure for a family of periodic problems, we derive an infinite-order
version of the classical homogenised system of Maxwell equations.
