  Utilizing the integral expression of Coulomb's Law to determine the electric
potential from a continuous charge distribution is a canonical exercise in
Electricity and Magnetism (E&M). In this study, we use both think-aloud
interviews and responses to traditional exam questions to investigate student
difficulties with this topic at the upper-division level. Leveraging a
theoretical framework for the use of mathematics in physics, we discuss how
students activate, construct, execute and reflect on the integral form of
Coulomb's Law when solving problems with continuous charge distributions. We
present evidence that junior-level E&M students have difficulty mapping
physical systems onto the mathematical expression for the Coulomb potential.
Common challenges include difficulty expressing the difference vector in
appropriate coordinates as well as determining expressions for the differential
charge element and limits of integration for a specific charge distribution. We
discuss possible implications of these findings for future research directions
and instructional strategies.
