Edge of Infinity: The Clash between Edge Effect and Infinity Assumption for the Distribution of Charge on a Conducting Plate

TL;DR

This paper challenges the common assumption that large conducting plates have uniform charge distribution by demonstrating that edge effects influence the entire plate, even at large sizes, through exact solutions of Poisson's equation.

Contribution

It provides a detailed analysis and exact solutions showing that edge effects persist regardless of plate size, invalidating the infinity assumption in charge distribution models.

Findings

Edge effects influence the entire charge distribution on large plates.

The infinity assumption fails due to persistent edge effects.

Exact solutions reveal non-uniform charge distributions across the plate.

Abstract

We re-examine a familiar problem given in introductory physics courses, about determining the induced charge distribution on an uncharged ``infinitely-large'' conducting plate when placing parallel to it a uniform charged dielectric plate of the same size. We show that, no matter how large the plates are, the edge effect will always be strong enough to influence the charge distribution deep in the central region, which totally destroyed the infinity assumption (that the surface charge densities on the two sides are uniform and of opposite magnitudes). For a more detailed analysis, we solve Poisson's equation for a similar setting in two-dimensional space and obtain the exact charge distribution, helping us to understand what happens how charge distributes at the central, the asymptotic, and the edge regions.