[Paper Review] The Electric Field of a Uniformly Charged Non-Conducting Cubic Surface

Kaitlin McCreery, Henry Greenside|arXiv (Cornell University)|Jul 26, 2016

Experimental and Theoretical Physics Studies3 citations

TL;DR

This paper presents a comprehensive analysis of the electric field produced by a uniformly charged non-conducting cubic surface, using symmetry, superposition, and analytical methods to show that the internal field is non-zero and exhibits complex spatial structure—pointing inward near face centers and outward near edges and corners. The key contribution is an explicit analytical expression for the electric field derived via potential integration and verified numerically, offering deep pedagogical insight for advanced undergraduates.

ABSTRACT

As an integrative and insightful example for undergraduates learning about electrostatics, we discuss how to use symmetry, Coulomb's Law, superposition, Gauss's law, and visualization to understand the electric field produced by a non-conducting cubic surface that is covered with a uniform surface charge density. We first discuss how to deduce qualitatively, using only elementary physics, the surprising fact that the electric field inside the cubic surface is nonzero and has a complex structure, pointing inwards towards the cube center from the midface of each cube and pointing outwards towards each edge and corner. We then discuss how to understand the quantitative features of the electric field by plotting an analytical expression for E along symmetry lines and on symmetry surfaces. This example would be a good choice for group problem solving in a recitation or flipped classroom.

Motivation & Objective

• To provide an integrative, multi-concept example for teaching electrostatics that combines symmetry, superposition, Coulomb’s law, Gauss’s law, and visualization.
• To demonstrate that the electric field inside a uniformly charged non-conducting cubic surface is non-zero and has a complex spatial structure, contrary to common intuition from spherical or conducting cases.
• To derive and validate an explicit analytical expression for the electric field E(x,y,z) using symbolic computation and numerical discretization.
• To enable deep conceptual understanding through qualitative reasoning, analytical calculation, and visualization on symmetry lines and surfaces.
• To support pedagogical use in recitations or flipped classrooms via structured, group-based problem-solving activities.

Proposed method

• Use of symmetry and superposition to qualitatively deduce the direction and structure of the electric field inside the cubic surface.
• Derivation of the electric potential V(x,y,z) by summing contributions from six rectangular charged faces using the formula for the potential of a finite rectangle.
• Symbolic computation in Mathematica to compute the electric field as the negative gradient of the potential, E = -∇V.
• Numerical validation via a discretized model: approximating each face with a grid of N×N point charges and summing their individual contributions to E and V.
• Plotting the analytical E-field along symmetry lines and surfaces to visualize field behavior and infer full 3D structure via continuity.
• Use of series expansion (e.g., Taylor series near vertices) to analytically confirm logarithmic divergence of the field magnitude near edges and corners.

Experimental results

Research questions

• RQ1 Why is the electric field inside a uniformly charged non-conducting cubic surface non-zero, despite its high symmetry?
• RQ2 How does the direction of the electric field vary across different regions of the cube’s interior—face centers, edges, and corners?
• RQ3 What is the analytical form of the electric field E(x,y,z) for a uniformly charged cubic shell, and how can it be computed symbolically?
• RQ4 How does the electric field magnitude behave near the edges and vertices of the cube, and does it diverge?
• RQ5 Can a simple numerical method based on point charge discretization reproduce the analytical results with sufficient accuracy?

Key findings

• The electric field inside the uniformly charged cubic surface is non-zero and exhibits a complex spatial structure: pointing inward near the center of each face and outward near edges and corners.
• The electric field diverges logarithmically near edges and vertices, with the magnitude scaling as |log(d)| for small distance d to the singularity, as confirmed by series expansion near a vertex.
• An explicit analytical expression for the electric field E(x,y,z) was derived using symbolic differentiation of the potential, which is several pages long but computationally evaluable.
• Numerical validation using a discretized grid of point charges (N ≥ 10 per face) confirmed the analytical results to high accuracy, except near singularities and on the grid itself.
• Plotting the field along symmetry lines and surfaces allows full reconstruction of the 3D field structure due to continuity and symmetry.
• The field at the vertex (1/2,1/2,1/2) along the line (1/2−x,1/2−x,1/2−x) behaves as E ≈ −2log(x)(x̂+ŷ+ẑ), confirming logarithmic divergence.