Upper-division students’ difficulties with Ampère’s law

This study presents and interprets some conceptual difficulties junior-level physics students experience with Ampère’s law. We present both quantitative data, based on students’ written responses to conceptual questions, and qualitative data, based on interviews of students solving Ampère’s law problems. We find that some students struggle to connect the current enclosed by an Ampèrian loop to the properties of the magnetic field while some students do not use information about the magnetic field to help them solve Ampère’s law problems. In this paper, we show how these observations may be interpreted as evidence that some students do not see the integral in Ampère’s law as representing a sum and that some students do not use accessible information about the magnetic field as they attempt to solve Ampère’s law problems. This work extends previous studies into students’ difficulties with Ampère’s law and provides possible guidance for instruction.

Figure 1

Question 17 from the CUE [5]: Consider an infinite nonmagnetizeable cylinder with a uniform volume current density $J$. Where is the $\mathit{B}$-field maximum? Explain how to determine this. The detailed grading rubric for this question assigned two points for correctness and five points for reasoning.

Figure 2

Average percentage correct on CUE question 17. The number of students in each group who took the CUE postinstruction are as follows: $\text{CU}1=20$, $\text{CU}2=42$, $\text{CU}3=27$, $\text{CU}4=35$, $\text{nonCU}=31$, and $\text{Trad}=27$.

Figure 3

Average score for both correctness (gray bars) and reasoning (white bars) on CUE question 17.

Figure 4

The post-Ampère’s law tutorial question: We have a large (infinite) sheet with a uniform current density $J$ flowing down it. The current runs in the $+y$ direction, as shown. (The sheet is infinite in the $y$ and $z$ directions.) Consider several small Ampèrian loop choices, labeled a-d. List ALL of these which might prove useful in learning something quantitative about $\mathit{B}\left(x,y,z\right)$ somewhere. For each loop, explain briefly why you did NOT choose it if you didn’t, or what it’s useful for if you did.

Figure 5

The solenoid problem: Find the magnetic field of a very long solenoid with $n$ closely spaced turns per unit length on a cylinder of radius $R$ and carrying a steady current $I$. Adapted from example 5.9 in Griffiths [11]. The Ampèrian loops shown above were not given to students.

Figure 6

(Color) The toroid problem: Find the magnetic field of a toroid with a long wire wrapped around it with $n$ closely spaced turns per unit length, a steady current $I$, and radii as indicated in the figure. Adapted from example 5.10 in Griffiths [11]. The Ampèrian loop $a$ shown above was not given to students.