**August 2023 Issue,**

**Volume 91, No. 8**

Infinite-fold energy degeneracy in 2D square lattices of magnetic spheres

We show that a two-dimensional square lattice of magnets can be studied by placing small cylindrical neodymium magnets inside plastic spherical shells and floating them on water, leaving their magnetic moments free to re-orient within the plane. Experimentally, anti-correlated dipole orientations between nearest neighbors appear to be favored energetically. This motivates the construction of a simplified single-variable energy function for a 2D square lattice of magnetic dipoles. For odd numbers of spheres, this ansatz yields a continuum of dipole configurations with the same energies, matching the observed behavior that the orientation of the dipoles in these lattices can be rotated freely. The behavior of square lattices with even numbers of spheres is strikingly different, showing strongly preferred orientations. While the energy calculated in this simplified model is larger than that of the actual ground state for finite size clusters, its asymptotic value in the limit where the number of spheres goes to infinity is in good agreement with the literature value. Additionally, rectangular arrangements of magnetic spheres with and without a defect are analyzed within the class of the single variable energy function. Simple experimental demonstrations qualitatively reproduce several interesting results obtained from all these analyses.

#### EDITORIAL

In this issue: August 2023 by Claire A. Marrache-Kikuchi; Raina Olsen; Beth Parks; B. Cameron Reed; Donald Salisbury; Todd Springer. DOI: 10.1119/5.0164053

#### LETTERS TO THE EDITOR

Physics writing in the era of artificial intelligence by M. A. Antonacci; M. A. Maize. DOI: American Journal of Physics 91, 575 (2023) doi: https://doi.org/10.1119/5.0159871

Atom-emitting-a-photon solved on the back of an envelope by Ramandeep S. Johal. DOI: American Journal of Physics 91, 576 (2023) doi: https://doi.org/10.1119/5.0162000

#### AWARDS

2023 AAPT award citations at the summer meeting in Sacramento, California by Beth Parks. DOI: American Journal of Physics 91, 577–578 (2023) doi: https://doi.org/10.1119/5.0161894

#### PAPERS

An alternative way to solve the small oscillations problem by Sergio A. Hojman. DOI: 10.1119/5.0106530

An algebraic approach employing new constants of the motion for dealing with coupled small oscillations is presented here. It offers new insight into undergraduate mechanics and can be most profitably applied in higher dimensional interacting systems using algebraic manipulation software.

An experimental demonstration of level attraction with coupled pendulums by Chenyang Lu; Bentley Turner; Yongsheng Gui; Jacob Burgess; Jiang Xiao; Can-Ming Hu. DOI: 10.1119/5.0081906

As quantum technologies have developed, it has become more important to teach students about how the effects of dissipation can be understood, controlled, and even used. When two pendulums with the same natural frequency are spring coupled, the eigenmodes break degeneracy to form an avoided crossing, also called level repulsion. This creates a beating pattern in the coupled system. But dissipative coupling, in which energy dissipation is proportional to the relative velocity of the pendulums, causes level attraction. Pendulums with slightly different natural frequencies oscillate in tandem when dissipatively coupled. This paper shows how to build a dissipative coupling apparatus and analyzes the system behavior in comparison to the more commonly used spring coupled system.

Rendezvous revisited: The search for fast-intercept solutions by Eric M. Edlund. DOI: 10.1119/5.0095559

Imagine you're on a spacecraft in a circular orbit around a planet or star. You'd like to catch another object at a different point on the same orbit. Your engine thrusters can only provide a brief impulse; in which direction should you aim your thrust so as to quickly intercept the target? Such questions may show up in science fiction as a spacecraft attempts to catch a rival ship, or when a torpedo needs to be fired to quickly neutralize an approaching asteroid. However, there are real-world aerospace engineering applications as well! A systematic study of this question was initiated by Edlund and published in AJP in 2021. In the current paper, the author concludes the analysis by exploring the “fast-intercept solutions” which are mathematically more complicated, but lead to a quicker intercept of the target. Explorations in this direction would make excellent projects for undergraduate students studying advanced mechanics.

Infinite-fold energy degeneracy in 2D square lattices of magnetic spheres Video Abstracts by Kyongwan Kim. DOI: /10.1119/5.0121937

Anyone who has played with magnets knows that when they are grouped together, their dipole interactions fix their orientations relative to each other. This strong interaction makes it difficult to use permanent magnets to study the dipole-dipole interaction. This paper presents a new way to study this interaction in two dimensions by mounting dipole magnets inside plastic shells and floating them on a water surface to create a compelling demonstration or student laboratory. The paper also reveals and explains an unexpected result: When dipoles are organized into squares, arrays with odd numbers of dipoles can have their orientation direction rotated continuously, while those with even numbers of dipoles have a fixed orientation. A video abstract accompanies the online version of this paper.

Helping students apply superposition principle in problems involving spherical, cylindrical, and planar charge distributions by Jing Li; Alexandru Maries; Chandralekha Singh. DOI: 10.1119/5.0127127

As we add complications to physics problems, we often find that students have forgotten concepts that had earlier been mastered. This paper demonstrates that students have difficulty applying the superposition problem in problems that also require the use of Gauss's law. It provides a set of tested exercises that will help students overcome this difficulty.

Modeling the induced voltage of a disk magnet in free fall by Nolan Samboy. DOI: 10.1119/5.0141862

Electromagnetic induction is often demonstrated in classroom and laboratory settings, but quantitative tests of Faraday's law tend to be less frequent. This paper describes an undergraduate-level experiment where the induced voltage created by dropping a disk magnet through a coil wrapped around a plastic pipe is recorded and compared to an analytic model based on treating the magnet as a dipole. Results are in excellent agreement with the theory; measurements are sensitive enough to detect the difference in the peak voltages induced as the magnet enters and exits the coil as a result of its gravitational acceleration. The paper shows how the experiments can be adapted to make them appropriate for students at a variety of levels.

Self-inductance and magnetic flux by Diego J. Castaño; Teresa M. Castaño. DOI: 10.1119/5.0098417

Did you ever get a headache from wondering what integration contour or surface you should use when dealing with an E&M exercise? If yes, this paper is for you. It deals with determining the inductance L=Φ/I of a circuit. In practical situations, determining which I to use (total current? local current?), or what surface to consider to determine the magnetic flux Φ, might take some head-scratching. This paper solves your troubles by proposing a more tractable way of calculating a circuit's self-inductance, using what is known as a weighted flux. You can use this article to prepare your undergraduate E&M classes on induction, or as some reading material for students.

Introducing corrugated surfaces in electrostatic problems via a perturbative approach by Alexandre P. Costa; Lucas Queiroz; Edson C. M. Nogueira; Danilo T. Alves. DOI: 10.1119/5.0099928

Certain electrostatics problems will be familiar to all students who have studied electromagnetism. Examples of these classic problems include calculating the electric potential from a point charge near an infinite conducting plane or calculating the resulting electric field when an infinitely long conducting cylinder is placed in a uniform electric field. Physical situations such as these become both more realistic and more challenging if the conducting surface is not symmetrical, or if it is irregular in some way. In this article, examples of these more difficult problems are analyzed perturbatively under the assumption that the conducting surfaces possess small corrugations (as any realistic surface would). Readers will enjoy this stimulating article as they gain exposure to some challenging electrostatics problems as well as the mathematical technique of solving differential equations perturbatively using Green's functions.

The nuclear optical model and its optical-scattering analog: Mie scattering by F. D. Becchetti. DOI: 10.1119/5.0152813

It's delightful when what we've learned in one field of physics gives insight into an entirely different field. This paper points out parallels between optical Mie scattering and the nuclear optical model of scattering and may help those who are familiar with one of these topics to better understand the other.

Cosmological horizons by Michele Re Fiorentin; Stefano Re Fiorentin. DOI: 10.1119/5.0127840

To support instructors who wish to teach about cosmological horizons, this paper presents a pedagogical treatment of our understanding of the topic for the case of a flat universe. Suitable for introductory courses in relativity or cosmology.

Conditional probability as found in nature: Facilitated diffusion by Ori Hachmo; Ariel Amir. DOI: 10.1119/5.0123866

Diffusion is remarkably slow, but nature finds fascinating ways to overcome that speed limit. Transcription factors regulate gene expression in cells, and to find their target, these proteins must traverse the entire DNA molecule, a process that would take thousands of hours if it proceeded purely by diffusion. Instead, a facilitated diffusion process is used, in which the protein regularly falls off the DNA and reattaches in a random location, speeding the search process to the timescale of tens of minutes that is required for cells to live. This paper would form the basis of an interesting lecture or problem set in a course on biophysics or mathematical methods.

#### NOTES AND DISCUSSIONS

Perfect and imperfect pitchfork bifurcations in a V-shaped spring-mass system: Comment on “Hysteresis in a simple V-shaped spring-mass system” [Am. J. Phys. 89, 663–665 (2021)] by J. J. Bissell; A. M. Nagaitis. DOI: 10.1119/5.0079561

Although students already have an intuitive understanding of common thermodynamic phases, phase transitions can still be a difficult concept to grasp. This comment on a previous paper by C. Ong shows how varying a length scale in a simple mechanical mass-spring system causes a transition from one equilibrium point to three equilibria (a pitchfork bifurcation) as the spring forces go from tension to compression. The authors have also analyzed oscillations, which slow as the system approaches the critical point. Though the difficulty of finding springs that work in both compression and tension likely precludes a laboratory demonstration in which the full range of behavior can be observed in a single system, a two-system demonstration could be used.