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For the system of Maxwell equations of electromagnetism in an l-periodic composite medium of overall size L (0 < l < L < ∞), in the low-frequency quasi-static 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 [V. P. Smyshlyaev and K. D. Cherednichenko, J. Mech. Phys. Solids, 48 (2000), pp. 1325{1357], we propose 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 standard homogenized operator of second order. [ABSTRACT FROM AUTHOR]