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.Comment: 13 pages, 3 figures, submitted to Phys. Rev. ST-PE