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Jefimenko Made Easy: Electromagnetic Fields through Retardation

Shengchao Alfred Li

TL;DR

The paper revisits Jefimenko's retardation-based derivations of ${\mathbf E}$ and ${\mathbf B}$ from Maxwell's equations, addressing an uncommon form of the inhomogeneous wave equation by substituting the standard form and following Jefimenko's steps. It derives the retarded integral as a solution via Green's function in the time domain, yielding Jefimenko's equations that link fields to causal charge and current sources using the retarded time $t' = t - r/c$. It shows the retarded potentials $\varphi$ and ${\mathbf A}$ give ${\mathbf E}$ and ${\mathbf B}$ through ${\mathbf E} = -\nabla\varphi - \frac{\partial {\mathbf A}}{\partial t}$ and ${\mathbf B} = \nabla \times {\mathbf A}$, thereby reproducing Jefimenko's results via potentials. The work reinforces that retardation is consistent with causality and special relativity, providing a concise, accessible route to Jefimenko's electrodynamics within the Maxwell framework.

Abstract

Oleg D. Jefimenko's electrodynamics textbook is unique in its approaches to deriving the electric and magnetic fields of arbitrary charge and current distributions and of an arbitrarily moving point charge. However, an uncommon form of the inhomogeneous wave equation often poses difficulties for readers right from the beginning. In this paper, we substitute in a commonly used form, making his approaches readily accessible.

Jefimenko Made Easy: Electromagnetic Fields through Retardation

TL;DR

The paper revisits Jefimenko's retardation-based derivations of and from Maxwell's equations, addressing an uncommon form of the inhomogeneous wave equation by substituting the standard form and following Jefimenko's steps. It derives the retarded integral as a solution via Green's function in the time domain, yielding Jefimenko's equations that link fields to causal charge and current sources using the retarded time . It shows the retarded potentials and give and through and , thereby reproducing Jefimenko's results via potentials. The work reinforces that retardation is consistent with causality and special relativity, providing a concise, accessible route to Jefimenko's electrodynamics within the Maxwell framework.

Abstract

Oleg D. Jefimenko's electrodynamics textbook is unique in its approaches to deriving the electric and magnetic fields of arbitrary charge and current distributions and of an arbitrarily moving point charge. However, an uncommon form of the inhomogeneous wave equation often poses difficulties for readers right from the beginning. In this paper, we substitute in a commonly used form, making his approaches readily accessible.
Paper Structure (9 sections, 1 theorem, 75 equations)

This paper contains 9 sections, 1 theorem, 75 equations.

Table of Contents

  1. Fields, Sources and Retardation
  2. Maxwell's Equations and the Inhomogeneous Wave Equations
  3. Retarded Integral as a Solution of the Inhomogeneous Wave Equation
  4. Jefimenko's Equations Jefimenko's Equations
  5. Retarded Potentials Retarded Potentials
  6. Electromagnetic Induction Electrical Induction
  7. The Fields of an Arbitrarily Moving Point Charge
  8. Discussion
  9. Notations and Identities

Key Result

Theorem 1

The retarded integral is a solution of the inhomogeneous wave equation Eq. (eqn:ihwev), subject to the condition that $[{\mathbf Z}]$ is the cause and ${\mathbf V}$ is the effect, and ${\mathbf V}$ is zero at positions far away from the source, where $r=\sqrt{(x-x^\prime)^2+(y-y^\prime)^2+(z-z^\prime)^2}$, $c$ is the spe

Theorems & Definitions (13)

  • Theorem 1
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 3 more