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We consider two different asymptotic limits of the Vlasov-Maxwell system describing a quasineutral plasma with a uniform ionic background. In the first case, as the magnetic field is preserved in the limiting process, we obtain the so-called electron magnetohydrodynamics equations. In the second case, we obtain the incompressible Euler equations with no more magnetic field left.

Incompressible Euler and E-MHD as scaling limits of the Vlasov-Maxwell system