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We use the method of field decomposition, a technique widely used inrelativistic magnetohydrodynamics, to study the small velocity approximation(SVA) of the Lorentz transformation in Maxwell equations for slowly movingmedia. The "deformed" Maxwell equations derived under the SVA in the lab framecan be put into the conventional form of Maxwell equations in the medium'scomoving frame. Our results show that the Lorentz transformation in the SVA upto $O(v/c)$ ($v$ is the speed of the medium and $c$ is the speed of light invacuum) is essential to derive these equations: the time and charge densitymust also change when transforming to a different frame even in the SVA, notjust the position and current density as in the Galilean transformation. Thismarks the essential difference of the Lorentz transformation from the Galileanone. We show that the integral forms of Faraday and Ampere equations for slowlymoving surfaces are consistent with Maxwell equations. We also present Faradayequation the covariant integral form in which the electromotive force can bedefined as a Lorentz scalar independent of the observer's frame. No evidencesexist to support an extension or modification of Maxwell equations.

Lorentz transformation in Maxwell equations for slowly moving media