The electric flux through a square of a point charge near one corner

Abstract

This paper presents two geometric models that can be used by students studying electrostatics to explore the electric flux of a point charge through a square in the limit of infinitesimal displacement from one corner. We first use qualitative reasoning to show that the limit is not a single value, but depends on the direction of displacement, and has significant discontinuities at the plane inside the square and along its sides. In the first model, the square is treated as one face of a cube. With this model, we discuss how to use Gauss’s law to determine the limiting flux through the square for displacements along symmetry lines of the cube, and their reflections through the planes of the faces. The second model treats the square as the base of a square pyramid and enables derivation of the limiting flux for displacements in an arbitrary direction. The result is verified against that obtained analytically, by integrating the definition of electric flux over the area of the square, in the limit of infinitesimal displacement from the corner. Both models require and facilitate an intuitive understanding of Gauss’s law and the limit concept, and the quantitative reasoning involves a minimum of algebra and no calculus. The first model, especially, is suitable for use in a guided-inquiry tutorial or problem-solving workshop for undergraduate physics students.