Point Charge Dynamics Near a Grounded Conducting Plane

TL;DR

This paper clarifies and correctly solves the classic image charge problem involving a grounded conducting plane and a point charge, including calculating the time for the charge to reach the plane, and discusses related electromagnetism force scenarios.

Contribution

The paper provides a corrected solution to the classic image problem and clarifies the calculation of the charge's impact time, addressing previous inconsistencies in the literature.

Findings

Corrected the calculation of the impact time τ for a point charge near a grounded plane.

Presented a consistent solution to the classic image problem in electromagnetism.

Discussed various 1/r^n force scenarios between different charge configurations.

Abstract

The classic image problem in electromagnetism involves a grounded infinite conducting plane and a point charge. The force of attraction between the point charge and the plane is identified using an equivalent-field picture of an image charge with opposite sign equidistant behind the plane resulting in a 1/{\it r}$^{\rm 2}$ force of attraction between the original charge and the plane. If the point charge is released from rest it will reach the plane in a time $\tau$. This time $\tau$ has not been calculated correctly up to now. Clarification of the inconsistency is presented along with a correct solution to the classic image problem. Other electromagnetism problems are mentioned with attractive 1/{\it r}$^{\it n}$ forces (where {\it n} $\ge$ 1). Such situations arise between a point charge and a line charge (1/{\it r}), between a line charge and a point dipole (1/{\it r}$^{2}$), between a point charge and a point dipole (1/{\it r}$^{3})$, and between a point dipole and a second point dipole (1/{\it r}$^{4}$).