Mathematically Deriving Novel Field Equations of Electromagnetics and Gravity for Sources with Velocity, Acceleration, and Spin, Respectively

Abstract

In this article, we show that the vector calculus identities are powerful tool in both rederiving the existing physical laws, such as Maxwell equations, and discovering new physical laws. First, we show that the following physical laws can be rederived from the vector calculus identities: (1) Poynting's Theorem; (2) Gauss's Law for Magnetism; (3) Charge density Continuity equation; (4) Wave equation of potential A, electric field E, and magnetic field B; (5) Faraday's Law of Induction; (6) energy density of the magnetic field; (7) time change of energy density of the electric field; (8) time change of energy density of the magnetic field. To rederive the above physical laws (1) to (8) shows the validation and power of the vector calculus identities in discovering physical laws. Then, applying the vector calculus identities, we derive NEW physical field equations: (1) for electric charges moving with either uniform velocity v, or constant acceleration a, or continuous spin S; (2) for gravitational charges (mass) moving with either uniform velocity v, or constant acceleration a, or continuous spin S. Electromagnetics and gravitoelectromagnetics are derived from the vector calculus identities and have the same form, and thus can be unified. The new physical field equations need to be tested experimentally.