In a bounded domain O⊂ R ³ of class C ^{1 , 1}, we consider a stationary Maxwell system with the perfect conductivity boundary conditions. It is assumed that the dielectric permittivity and the magnetic permeability are given by η(x/ ε) and μ(x/ ε) , where η(x) and μ(x) are symmetric (3 × 3) -matrix-valued functions; they are periodic with respect to some lattice, bounded and positive definite. Here ε> 0 is the small parameter. We use the following notation for the solutions of the Maxwell system: u _ε and v _ε are the electric and magnetic field intensities, w _ε and z _ε are the electric and magnetic displacement vectors. It is known that u _ε, v _ε, w _ε, and z _ε weakly converge in L ₂(O) to the corresponding homogenized fields u, v, w, and z (the solutions of the homogenized Maxwell system with the effective coefficients), as ε→ 0. We improve the classical results and find approximations for u _ε, v _ε, w _ε, and z _ε in the L ₂(O) -norm. The error terms do not exceed Cε(‖q‖L2+‖r‖L2), where the divergence free vector-valued functions q and r are the right-hand sides of the Maxwell equations.