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The problem of electromagnetic wave propagation in the linearly accelerated dielectric medium is studied by using the extended Maxwell‐Minkowski theory. First, from the viewpoint of the extended Maxwell‐Minkowski theory, the well‐known problem of reflection and transmission of electromagnetic waves by the moving dielectric with constant velocity is studied by using the Galilean transformations instead of the Lorentz transformations. It is shown that the problem can be solved and the same results can be obtained as those obtained by using the Lorentz transformations. Then, with the aid of the covariant properties of Maxwell’s equations, a rigorous solution of the electromagnetic wave in a linearly accelerated dielectric is founded. The phenomena of bending of light rays in the accelerated system and the relativistic energy velocity and phase velocity addition law for the accelerated motion are derived from the rigorous solution obtained. The problem of reflection and transmission of the electromagnetic wave at normal incidence by a linearly accelerated dielectric interface is solved using the solution obtained. The reflected wave and the transmitted wave are obtained, and their interesting properties concerning the Doppler effect and drag effect are discussed in comparison with the case of constant‐velocity motion.