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

Full two-scale asymptotic expansion and higher-order constitutive laws in the homogenisation of the system of Maxwell equations