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Equations of a Moving Mirror and the Electromagnetic Field

We consider a system composed of a mobile slab and the electromagnetic field. We assume that the slab is made of a material that has the following properties when it is at rest: it is linear, isotropic, non-magnetizable, and ohmic with zero free charge density. Using instantaneous Lorentz transformations, we deduce the set of self-consistent equations governing the dynamics of the system and we obtain approximate equations to first order in the velocity and the acceleration of the slab. As a consequence of the motion of the slab, the field must satisfy a wave equation with damping and slowly varying coefficients plus terms that are small when the time-scale of the evolution of the mirror is much smaller than that of the field. Also, the motion of the slab and its interaction with the field introduce two effects in the slab's equation of motion. The first one is a position- and time-dependent mass related to the $\textit{effective mass}$ taken in phenomenological treatments of this type of systems. The second one is a velocity-dependent force that can give rise to friction and that is related to the much sought $\textit{cooling}$ of mechanical objects.