Upper-division student difficulties with separation of variables

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 or 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 separation of variables problems in Cartesian and spherical geometries. We also briefly discuss implication of these findings for instruction.

Figure 1

An example of a canonical exam problem targeting Cartesian SOV. Variations on this question in our data include providing a nonconstant potential on the fourth side [e.g., $V_o(x,y=a)=V_o\sin \pi x/a$], placing the nongrounded side at $y=0$ rather than $y=a$, or placing one of the grounded sides at infinity.

Figure 2

An example of a canonical exam problem targeting spherical SOV. Variations on this question include providing a simpler or more complex expression for the boundary [e.g., $V(R,\theta )=V_o\cos \theta$], giving the boundary condition in terms of Legendre polynomials [e.g., $V(R,\theta )=P_9(\cos \theta )$], or only asking for the potential inside or outside the sphere.

Figure 3

The multiple-response CUE question related to the Cartesian SOV. The prompt has been shortened and paraphrased; see Ref. [13] for full prompt.