The theory of distributions in teaching electrostatics: Multipoles, forces, and boundary conditions

This paper presents a way of teaching the standard curriculum of upper-undergraduate electrostatics, the electrostatic potential, the multipole expansion, and the boundary conditions at material interfaces, organized around the theory of distributions. Our aim is practical rather than foundational. In standard treatments the student meets a different technique for each situation, with singular objects introduced informally, multipole terms named but not constructed, and boundary conditions derived by geometric arguments that violate the hypotheses of the theorems they invoke. The difficulties this produces are as much mathematical as physical. We show how a single distributional procedure replaces that patchwork, so that each manipulation the standard course asks the student to accept on the authority of the textbook becomes a result they can derive. We also report on six semesters of classroom experience teaching the material this way in an Electromagnetism II course, including the diagnostic discussions through which students come to question arguments they had previously accepted. The framework also yields results we did not encounter in our review of the literature, namely an explicit charge representation of the 2ⁿ-pole point distribution, a physical interpretation of the dipole and quadrupole moments in terms of centers of charge and moments of inertia, and closed-form expressions for the force and torque on multipoles of arbitrary order.