Upper-division student difficulties with Separation of Variables

TL;DR

This study investigates upper-division physics students' difficulties with the separation of variables technique in solving Laplace's equation, highlighting common challenges and implications for teaching.

Contribution

It provides a detailed analysis of student difficulties with separation of variables in electrostatics, using multiple assessment methods and an analytical framework.

Findings

Students struggle to recognize when to use separation of variables.

Common errors include recalling the correct form and boundary conditions.

Differences in errors between Cartesian and spherical geometries.

Abstract

Separation of variables can be a powerful technique for solving many of the partial differential equations that arise in physics contexts. Upper-division physics students encounter this technique in multiple topical areas including electrostatics and quantum mechanics. To better understand the difficulties students encounter when utilizing the separation of variables technique, we examined students' responses to midterm exam questions and a standardized conceptual assessment, and conducted think-aloud, problem-solving interviews. Our analysis was guided by an analytical framework that focuses on how students activate, construct, execute, and reflect on the separation of variables technique when solving physics problems. Here we focus on student difficulties with separation of variables as a technique to solve Laplace's equation in both Cartesian and spherical coordinates in the context of junior-level electrostatics. Challenges include: recognizing when separation of variables is the appropriate tool; recalling/justifying the separated form of the potential and the need for the infinite sum; identifying implicit boundary conditions; and spontaneously reflecting on their solutions. Moreover, the type and frequency of errors was often different for SoV problems in Cartesian and spherical geometries. We also briefly discuss implication of these our findings for instruction.