Quadrupolarization as a Teaching Tool

In the standard undergraduate Electricity and Magnetism curriculum, the topic of continuously-polarized matter is treated extensively. However, the related topic of continuously-quadrupolarized matter is not discussed, despite its potential relevance for quadrupolar fluids. Here, we extend the usual treatment of surface and bound charge to the quadrupolar case. Using only undergraduate-level techniques, we discuss how to convert quadrupolarization (quadrupole moment per unit volume) into an effective surface charge density, volume charge density, and normal surface polarization. This method allows one to efficiently solve for electrostatic potential due to quadrupolarization and serves as a pedagogical tool to 1) introduce students to normal surface dipole moment and the related boundary condition, 2) present a generally-applicable method to treat non-normally oriented surface polarization, 3) familiarize students with component notation, and 4) sharpen student's boundary value problem solving skills. We illustrate the method through four worked examples: 1) a uniformly quadrupolarized sphere, 2) a concentrically quadrupolarized sphere, 3) an infinite cylinder uniformly quadrupolarized parallel to axis, and 4) an infinite cylinder uniformly quadrupolarized perpendicular to axis.