Single Gauge-Invariant Electric Field-Impulse Embedding All Fields and Potentials for Current and Charge Distributions

This article presents the single gauge-invariant electric field-impulse $\boldsymbol {I} {}_{E}$ as the retarded solution of its wave equation, which embeds all fields and potentials for current and charge distributions. Many traditional wave equations for electromagnetic (EM) fields, Lorenz- and Coulomb-gauge potentials, can be combined into or simplified via a single wave equation for $\boldsymbol {I} {}_{E}$ . This wave equation is universal and gauge-invariant, which bypasses all gauge conditions for potentials and includes Poisson equations as its special cases for static charge-integrator and direct current. The retarded wave solution $\boldsymbol {I} {}_{E}$ is provided in complete and convenient expressions involving current and charge-integrator (instead of charge). It is shown that the single $\boldsymbol {I} {}_{E}$ actually carries the complete information of all electrostatic, magnetostatic, and electrodynamic fields. Thus, it can aptly describe all EMs simultaneously and provide their ultimate unification, by embedding not only electrodynamics but also electrostatics and magnetostatics. In particular, for the traditional wave equations for EM fields, the solutions derived by Jefimenko and their variants can be unified in the single quantity $\boldsymbol {I} {}_{E}$ . This unification also applies to the classical electrostatic and magnetostatic solutions named after Coulomb, Poisson, and Biot-Savart, since they all can be embedded simultaneously as parts of the same single quantity $\boldsymbol {I} {}_{E}$ . For the solutions of traditional wave equations for potentials, including those for Lorenz and Coulomb gauges, they can also be merged uniquely in the same gauge-invariant $\boldsymbol {I} {}_{E}$ . Applications of electric field-impulse are demonstrated for embedding and unifying all the electric and magnetic fields for Hertzian electric and magnetic dipoles. Numerical results further highlight the completeness of $\boldsymbol {I} {}_{E}$ in contrast to the inadequacy of fields to describe all EM phenomena.