Electric field of a point charge in truncated hyperbolic motion

TL;DR

This paper derives the electric field of a point charge undergoing truncated hyperbolic motion, showing the continuity of field lines and the emergence of a delta function contribution in the limit of infinite past acceleration.

Contribution

It extends the Lienard-Wiechert formula to truncated hyperbolic motion, clarifying the behavior of electric fields during transition phases and the role of delta functions.

Findings

Electric field remains continuous across transition time.

Lienard-Wiechert formula applies to both motion phases with modified retarded time.

Delta function appears as transition time approaches negative infinity.

Abstract

We find the electric field of a point charge in `truncated hyperbolic motion', in which the charge moves at a constant velocity followed by motion with a constant acceleration in its instantaneous rest frame. The same Lienard-Wiechert formula holds for the acceleration phase and the constant velocity phase of the charge's motion. The only modification is that the formula giving the retarded time is different for the two motions, and the acceleration is zero for the constant velocity motion. The electric field lines are continuous as the retarded time increases through the transition time between constant velocity and accelerated motion. As the transition time approaches negative infinity the electric field develops a delta function contribution that has been introduced by others as necessary to preserve Gauss's law for the electric field.