Observations on student difficulties with mathematics in upper-division electricity and magnetism

We discuss common difficulties in upper-division electricity and magnetism (E&M) in the areas of Gauss’s law, vector calculus, and electric potential using both quantitative and qualitative evidence. We also show that many of these topical difficulties may be tied to student difficulties with mathematics. At the junior level, some students struggle to combine mathematical calculations and physics ideas, to account for the underlying spatial situation when doing a mathematical calculation, and to access appropriate mathematical tools. We discuss the implications of our findings for E&M instruction at the junior level.

Figure 1

Illustrations from CUE diagnostic questions. Students are asked not to solve the question, but to give “the easiest method you would use to solve the problem” (half credit) and “why you chose that method” (half credit). (a) “A solid non-conducting sphere, centered on the origin, with a non-uniform charge density that depends on the distance from the origin, $\rho (r)={\rho }_0{e}^{-r^2/a^2}$. Find $E$ (or $V$) at point $P$.” (b) “A charged insulating solid sphere of radius $R$ with a uniform volume charge ${\rho }_0$, with an off-center spherical cavity carved out of it. Find $E$ (or $V$) at point $P$, at a distance $4R$ from the sphere.”

Figure 2

Student drawing from Gauss’s law interview. After drawing axes and the nonuniform blob of charge, the student drew the left-hand Gaussian surface, (A), and explained as detailed in the text. The student next drew the isolated sphere of charge, (B), to help explain her point, followed by the right-hand Gaussian surface, (C).

Figure 3

Student drawing from interview. Portions of the picture in red are from the question statement which is reprinted with permission from Singh [41]. Dark gray is the student’s drawing, and explanation of the student’s drawing are black. After drawing electric field lines radiating outwards from each charge, the student drew three vectors of equal magnitude at each measurement point—one for the contribution to the electric field from each charge. At this point he does not attend to the magnitude of the vectors. To the right of that the student then drew the sum of those vectors as the total electric field. In this stage he attends to the magnitude of the vectors.

Figure 4

Student response to a midterm question about a thin shell of uniform charge. $N=51$.

Figure 5

Left: Student responses to a midterm question about a thin shell of uniform charge. Not included in this graph is one student whose answer was unclear. $N=51$. Right: Student responses to the CUE question about potential described in the text. $N=324$.

Figure 6

Student drawing from interview question about potential. The student’s sketch is to the left and her calculations to the right. She originally drew the charged shell as hollow, and explicitly mentioned that $E=0$ there. The scribbles visible in the shell are part of a later explanation.