[Paper Review] Double (implicit and explicit) dependence of the electromagnetic field of an accelerated charge on time: Mathematical and physical analysis of the problem

Andrew Chubykalo, S.J. Vlaev|arXiv (Cornell University)|Mar 25, 1998

Quantum and Classical Electrodynamics10 references7 citations

TL;DR

This paper demonstrates that the electromagnetic field of a uniformly accelerated charge, derived from the Liénard-Wiechert potentials, fails to satisfy Maxwell’s equations when only retarded (implicit) time dependence is considered. The authors show that only by including both retarded interaction and direct (action-at-a-distance) interaction—i.e., explicit time dependence—does the field fully comply with Maxwell’s equations, resolving a long-standing inconsistency in classical electrodynamics.

ABSTRACT

We considered the electromagnetic field of a charge moving with a constant acceleration along an axis. We found that this field obtained from the Liénard-Wiechert potentials does not satisfy Maxwell equations if one considers exclusively a retarded interaction (i.e. pure implicit dependence this field on time). We show that if and only if one takes into account both retarded interaction and direct interaction (so called "action-at-a-distance") the field produced by an accelerated charge satisfies Maxwell equations.

Motivation & Objective

• To investigate why the electromagnetic field of a uniformly accelerated charge derived from Liénard-Wiechert potentials fails to satisfy Maxwell’s equations.
• To identify the mathematical and physical origin of this inconsistency in the context of retarded potentials.
• To demonstrate that the inclusion of both retarded (implicit) and direct (explicit) time dependence—via action-at-a-distance—restores consistency with Maxwell’s equations.
• To resolve a longstanding issue in classical electrodynamics concerning the field of an accelerated charge.

Proposed method

• The authors analyze the electromagnetic field of a charge undergoing constant acceleration along a straight line using the Liénard-Wiechert potentials.
• They examine the time dependence of the field, distinguishing between explicit time dependence (from the field point and source position) and implicit time dependence (through the retarded time).
• The field expressions are derived and substituted into Maxwell’s equations to test their validity.
• A comparison is made between solutions based solely on retarded interaction and those incorporating both retarded and direct interaction terms.
• The analysis employs differential equations and tensor calculus to verify whether the field satisfies the full set of Maxwell’s equations.
• The study uses a relativistic formulation to ensure consistency with special relativity and classical field theory.

Experimental results

Research questions

• RQ1 Why does the electromagnetic field of a uniformly accelerated charge, as derived from the Liénard-Wiechert potentials, fail to satisfy Maxwell’s equations when only retarded time dependence is considered?
• RQ2 What mathematical and physical conditions are necessary for the field of an accelerated charge to be consistent with Maxwell’s equations?
• RQ3 How does the inclusion of direct (action-at-a-distance) interaction affect the time dependence and consistency of the electromagnetic field?
• RQ4 Can a field derived from retarded potentials alone account for the full dynamics of an accelerated charge in vacuum?
• RQ5 What role does explicit time dependence play in resolving inconsistencies in the field equations for accelerated charges?

Key findings

• The electromagnetic field of a uniformly accelerated charge derived from the Liénard-Wiechert potentials does not satisfy Maxwell’s equations when only retarded (implicit) time dependence is considered.
• The inconsistency arises because the retarded potential formulation alone fails to account for the full time evolution of the field, particularly in the presence of acceleration.
• Only when both retarded interaction and direct (action-at-a-distance) interaction are included does the field satisfy all four Maxwell equations.
• The inclusion of explicit time dependence—beyond the retarded time—restores consistency with the source-free Maxwell equations.
• The solution requires a non-local, non-retarded contribution to the field, indicating that the standard retarded potential approach is insufficient for accelerated charges.
• The field derived with both implicit and explicit time dependence is shown to be a valid solution of Maxwell’s equations in the context of a uniformly accelerated point charge.