Maxwell equations provide a complete description of the electromagnetic (EM) phenomena, which have been one of the key fundamental-theories of modern physics, such as electromagnetism, optics, quantum theories, etc. The vacuum permittivity and permeability (P&P) in the constitutive relation were regarded as constant, resulting into that the vacuum lightspeed is constant with time. However, neither the Maxwell equations nor any experiments demonstrated that the P&P must be invariable with time. Here, the P&P are assumed as time-varied, achieving a mathematical result exhibits that the P&P are increased with time exponentially. Then, the Maxwell equations of static and covariant EM fields are presented, which include the different constitutive relations. The EM characters in the time-varied case are investigated that, the lightspeed, level EM intensity, and magnetic effect of charge-motion decreases with time. The EM energy transforms into a new energy form caused by the P&P variation, and a new momentum-tensor form caused by propagation is found as well. At last, the Maxwell equations based on the observation results at our time are presented. Meanwhile, since the level EM intensity decreases with time, the energy per frequency is time-decreased, resulting into that the Planck constant decreases exponentially, and correspondingly, the Rydberg constant increases with time. A red-shift of absorption spectrum is predicted based on this finding, which is coincident with the Hubble observation results.

Manuscript and Supplemental document, 27 pages, 11 figures