September 2025
Volume 93, Issue No. 9
We present an experimental analysis of the magnetic field generated by a square coil. In contrast to the usual circular coil, an analytical expression is easily derived over the whole space. This allows quantitative comparisons not only along the coil axis but also in the coil plane, where we demonstrate a smooth transition from the infinite-wire model to the dipole model. Unlike the 1D Hall probes usually found in teaching laboratories, we use a smartphone whose 3D sensor allows precise alignment with respect to the coil's symmetry elements.
EDITORIAL
In this issue: September 2025 by Beth Parks; Cameron Reed; Todd Springer; Keith Zengel. DOI: 10.1119/5.0293275
PAPERS
Shallow- and deep-water ocean waves: Deconstructing the dispersion relation by Tamara Beitzel Barriquand; Jessica M. Kleiss. DOI: 10.1119/5.0223337
Editor's Note: When designing lessons about waves, physicists should learn from the experts: oceanographers! This paper shares teaching modules that can be used independently or together at a variety of instructional levels, from introductory lessons on wavelength and period to more in-depth explorations of phase and group velocities for sophomore-level or advanced courses. The online supplementary material shares carefully developed instructional materials designed to address student misconceptions.
Strategies for human-powered water ferries by Christophe Odin. DOI: 10.1119/5.0226840
Editor's Note: To spark student interest, Snell's law is often introduced through a classic optimization problem of a dog traveling over both land and water to retrieve a stick in the shortest time. In this article, we encounter a few less-familiar but highly engaging optimization problems in the context of kayaking and seafaring. What is the most efficient way to cross a river when the paddler is quickly tiring and losing power? How can a paddler leverage eddy currents near the opposite shore to arrive more quickly at the destination than by paddling straight across? Analyzing these situations involves a variety of analytical, approximate, and numerical methods, all of which can be used to develop students' skill at mathematical modeling of real-world situations.
On the action for a charged particle moving in a magnetic field by Manoj K. Harbola. DOI: 10.1119/5.0238984
Editor's Note: Students usually first encounter the concept of action in classical dynamics, and then again with the Bohr–Wilson–Sommerfeld quantization condition in quantum mechanics. If a particle is moving in a vector potential as opposed to a scalar one, however, calculating the action becomes more complicated. This is the case for a charged particle moving in a magnetic field, where knowledge of the canonical momentum is usually required. This paper shows how the action in this case can be more easily constructed by the addition of a term involving the flux passing through the area enclosed by the particle's trajectory. Appropriate for students in upper-level dynamics and electromagnetism classes.
A new simple approach to the problem of a metallic sphere encased by a dielectric shell in a uniform electric field by Trinh Duc Thien. DOI: 10.1119/5.0263990
Editor's Note: This manuscript shows how a problem usually solved by separation of variables can be solved by students at an earlier stage of their physics education, including possibly in a university-level course on an introduction to electromagnetism.
Interference of interference effects by Kevin J. Randles; S. J. van Enk. DOI: 10.1119/5.0256745
Editor's Note: As instructors know, it is easy for intuition to go astray when analyzing quantum mechanical puzzles. This paper analyzes an interferometer that combines aspects of the Mach–Zehnder interferometer with the Hong–Ou–Mandel effect, showing how students might believe an incorrect argument and how the correct analysis can be presented. The problem posed in this paper will be an excellent addition to courses that teach these two interference effects, and supplementary online material presents an expanded version of the paper with additional background, references, and solutions to sample problems.
Incorporating a research project and coding exercises into existing undergraduate physics courses by Mara D. Casebeer; Alex Frano. DOI: 10.1119/5.0227376
Editor's Note: At times, early undergraduate students have difficulty seeing how their introductory courses are relevant to their broader career goals, leading to lack of engagement and decreased retention rates. To address these issues, the authors of this article recommend a two-pronged approach. First, as part of their introductory course sequence, students were tasked with writing a paper and delivering a presentation on a recent publication authored by researchers in the local community. Second, in order to develop pertinent and marketable skills, the authors incorporated a series of coding exercises into their introductory physics courses. The article includes extensive supplementary material, including a complete set of coding exercises which can be adapted by interested instructors for use in their own classrooms. Readers seeking to enhance the relevance of their introductory physics courses will find the ideas and resources presented in this article fruitful.
Computational physics in the advanced lab: Experiment and simulation of thermal diffusion in metal rods by Yash Mohod; M. C. Sullivan. DOI: 10.1119/5.0269525
Editor's Note: The authors introduce a simple setup of practical interest: a metal rod heated near its center. The temperature everywhere in the rod at all times is governed by a modified thermal diffusion equation that has no apparent analytical solution. The system will therefore be of interest to teachers of advanced physics lab classes who wish to introduce their students to applications of numerical techniques, experimental design skills, and conceptual approaches to the interplay between simulations and measurements.
ADVANCED TOPICS
Escape from a one-dimensional metastable potential well revisited by Alex Edison; Anupam Garg. DOI: 10.1119/5.0236109
Editor's Note The authors present pedagogically useful calculations of the escape rate for particles from a metastable cubic parabola quantum well. This presentation will be of interest to teachers preparing to discuss decay, tunneling, the WKB approximation, and ways to connect their course material to advanced approaches involving path integrals.
INSTRUCTIONAL LABORATORIES AND DEMONSTRATIONS
Experimental investigation of the axial and in-plane magnetic fields of a square coil using a smartphone by R. Mathevet; C. M. Fabre; N. Lamrani; P. Marchou. DOI: 10.1119/5.0276359
Editor's Note: Readers will value two contributions of this paper: It points out that the 3D nature of a smartphone’s magnetometer is very convenient for aligning the magnetometer with the field, and it reminds readers that, unlike a circular loop, the field of a rectangular loop is easily calculated. The combination of these two contributions will allow instructors to create laboratory experiences that help their students better understand magnetic fields.
NOTES AND DISCUSSIONS
Alternative apparent depth method for measuring the refractive index in solid materials by M. Isaias Rodríguez-Rodríguez; Itzel Muñoz-Juárez; Evelyn Hernández-Ramos; Daniel Aguirre-Aguirre; José Rufino Díaz-Uribe. DOI: Am. J. Phys. 93, 755–758 (2025) https://doi.org/10.1119/5.0264709
Editor's Note: The index of refraction, as used in Snell's law, explains fascinating optical effects such as how an object placed at the bottom of a glass beaker appears to float when viewed from the side. This paper shows how this “apparent depth” effect can be exploited to create a simple laboratory exercise that lets students measure the index of refraction while also gaining a better understanding of this effect. The measurements require only a small rectangular slab of the transparent material, so they can easily be performed in “at home” laboratories as well as in the regular teaching lab.
A compact derivation of the arc-length-weighted average Sun–planet separation in a Kepler orbit by B. Cameron Reed. DOI: 10.1119/5.0274956
Editor's Note A standard problem in astronomy and classical mechanics courses is the calculation of the average radius of a planet's elliptical orbit around the Sun. Though the statement of the problem is often the same in textbooks, articles, and lecture notes, there are at least four different approaches in common use for actually calculating the average radius, only two of which produce the same result. In this note, readers will find a handy summary of the typical approaches, a discussion of the conceptual differences between the various averages, as well as some new insights on useful symmetries in ellipses.
BOOK REVIEWS
Dark Matter: Evidence, Theory, and Constraints by Ethan O. Nadler. DOI: 10.1119/5.0294201
ppropriate for introductory-level students familiar with fluid pressure.
Additional Resources
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