[Paper Review] Electric field lines of an arbitrarily moving charged particle

S.G. Arutunian, М. А. Агинян+3 more|arXiv (Cornell University)|Sep 22, 2021

Laser-Plasma Interactions and Diagnostics11 references4 citations

TL;DR

This paper derives the electric field lines of a charged particle undergoing arbitrary motion by reducing their equations to homogeneous linear differential equations with variable coefficients. For trajectories where the ratio of torsion to curvature times the Lorentz factor is constant—such as planar motion—these equations simplify to constant-coefficient forms, enabling analytical solutions and visualizations for a charge in a monochromatic linearly polarized wave.

ABSTRACT

In this paper it is shown that the equations of electric field lines of an arbitrarily moving charged particle in the general case are reduced to homogeneous, linear differential equations with variable coefficients. For trajectories where the expression b=Zeta/(Gamma*Kappa) is a constant (Zeta - orbit torsion, Kappa - orbit curvature, Gamma - Lorentz factor of a particle) these equations are reduced to homogeneous, linear differential equations with constant coefficients. This case, in particular, includes all planar trajectories. This paper presents solutions of the equations of electric field lines and corresponding illustrations both in the orbital plane and outside it for a charge moving in a flat monochromatic linearly polarized wave.

Motivation & Objective

• To derive the general equations governing electric field lines of a charged particle in arbitrary relativistic motion.
• To identify conditions under which the field line equations reduce to linear differential equations with constant coefficients.
• To provide analytical solutions and visual illustrations of electric field lines in both the orbital plane and transverse directions.
• To examine the field structure for a charge oscillating in a monochromatic linearly polarized electromagnetic wave.
• To extend classical field line analysis to non-uniform, relativistic trajectories beyond the standard dipole or uniform motion cases.

Proposed method

• Formulating the electric field lines as solutions to a system of ordinary differential equations derived from the Liénard–Wiechert potentials.
• Expressing the field line equations in terms of the particle's trajectory, curvature (κ), torsion (ζ), and Lorentz factor (Γ).
• Introducing the invariant b = ζ/(Γκ) to classify trajectories: when b is constant, the field line equations become linear with constant coefficients.
• Solving the resulting differential equations analytically for the case of planar motion and for motion in a linearly polarized wave.
• Generating 2D and 3D visualizations of field lines in the orbital plane and perpendicular to it using numerical and analytical solutions.
• Validating the solutions through consistency checks with known results for uniform motion and dipole radiation.

Experimental results

Research questions

• RQ1 Under what conditions do the electric field line equations for an arbitrarily moving charged particle reduce to linear differential equations with constant coefficients?
• RQ2 How do electric field lines behave for a relativistic charged particle undergoing planar motion?
• RQ3 What is the structure of electric field lines when the particle moves in a monochromatic linearly polarized electromagnetic wave?
• RQ4 Can analytical solutions be derived for field lines in non-uniform, relativistic trajectories beyond the dipole approximation?
• RQ5 How does the geometric invariant b = ζ/(Γκ) influence the topology and solvability of electric field line configurations?

Key findings

• The electric field lines of an arbitrarily moving charged particle are governed by a system of homogeneous linear differential equations with variable coefficients.
• When the ratio b = ζ/(Γκ) is constant—such as in all planar trajectories—the field line equations reduce to linear differential equations with constant coefficients, enabling exact analytical solutions.
• For a charge oscillating in a monochromatic linearly polarized wave, the paper derives and illustrates the electric field lines both in the orbital plane and in transverse planes.
• The solutions reveal complex, non-symmetric field line patterns that reflect the relativistic and time-dependent nature of the source motion.
• The method successfully captures the field structure in regions of high curvature and rapid acceleration, where standard approximations fail.
• The derived equations and solutions are consistent with known limits, such as uniform motion and dipole radiation, validating the approach.