Notes on Gauss law applied for time varying electric field in vacuum

Gauss's law in integral form states that closed surface integral of electric field is proportional to net charge present within the volume bounded by this closed surface. Gauss's law in differential form states that divergence of electric field is proportional to charge's volume density. From Gauss's law it is concluded that in vacuum, where no electric charges are present, divergence of electric field and closed surface integral of electric field are zero. In this paper we make an attempt to theoretically prove that for time-varying electric fields Gauss's law is not valid. In other words, for time-varying electric field even in vacuum divergence of electric field and closed surface integral of electric field are non-zero. We do this by considering simple sinusoidal motion of the charge and then by calculating closed surface integral at specific time of time-varying electric field due to charge motion. Results show that for charges at still and at motion with constant velocity Gauss's law is valid. However for charges moving with acceleration, Gauss's law is no longer valid.

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