  Two fundamental models in plasma physics are given by the
Vlasov-Maxwell-Boltzmann system and the compressible Euler-Maxwell system which
both capture the complex dynamics of plasmas under the self-consistent
electromagnetic interactions at the kinetic and fluid levels, respectively. It
has remained a long-standing open problem to rigorously justify the
hydrodynamic limit from the former to the latter as the Knudsen number
$\varepsilon$ tends to zero. In this paper we give an affirmative answer to the
problem for smooth solutions to both systems near constant equilibrium in the
whole space in case when only the dynamics of electrons is taken into account.
The explicit rate of convergence in $\varepsilon$ over an almost global time
interval is also obtained for well-prepared data. For the proof, one of main
difficulties occurs to the cubic growth of large velocities due to the action
of the classical transport operator on local Maxwellians and we develop new
$\varepsilon$-dependent energy estimates basing on the macro-micro
decomposition to characterize the asymptotic limit in the compressible setting.
