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

This paper presents the single gauge-invariant electric field-impulse <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I_E</i> 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, etc., can be combined into or simplified via a single wave equation for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I_E</i>. 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I_E</i> is provided in complete and convenient expressions involving current and charge-integrator (instead of charge). It is shown that the single <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I_E</i> actually carries the complete information of all electrostatic, magnetostatic and electrodynamic fields. Thus, it can aptly describe all electromagnetics 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I_E</i>. 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I_E</i>. 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I_E</i>. 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I_E</i> in contrast to the inadequacy of fields to describe all EM phenomena.