**August 2024 Issue,**

**Volume 92, No. 8**

Understanding two-dimensional tractor magnets: Theory and realizations

We present a comparative investigation of two-dimensional tractor magnet configurations, analyzing both theoretical predictions and experimental results with a focus on the minimal tractor magnet. The minimal tractor magnet consists of a rigid assembly of one attracting magnet (attractor), two repelling magnets (repulsors), and a fourth magnet (follower) that is magnetically stabilized in a local energy minimum. The theoretical framework relies on magnetostatics and stability analysis of stationary equilibria. To calculate the magnetostatic force and energy, we use a multipole method. In a first approximation, we derive analytical results from the point dipole approximation. The point dipole analysis defines an upper bound criterion for the magnetic moment ratio and provides analytical expressions for stability bounds in relation to geometry parameters. Our experimental results are consistent with the predictions from the fourth-order multipole expansion. Beyond the minimal tractor magnet, we introduce a more advanced configuration that allows for a higher magnetic binding energy and follower capture at larger distances.

#### EDITORIAL

In this issue: August 2024 by Claire A. Marrache-Kikuchi; Raina Olsen; Beth Parks; Donald Salisbury; Keith Zengel. DOI: 10.1119/5.0225206

#### LETTERS TO THE EDITOR

Who discovered angular momentum of the photon? by Ajoy Ghatak. DOI: 10.1119/5.0191372

Comment on ‘Dynamics of a bouncing capsule: An impulse model vs a Hertzian model&rsquo by Rod Cross. DOI: 10.1119/5.0220437

#### AWARDS

2024 AAPT award citations at the summer meeting in Boston, Massachusetts DOI: 10.1119/5.0225393

#### PAPERS

Transition from bouncing to rolling on a horizontal surface by Rod Cross. DOI: 10.1119/5.0160345

Editor's Note: When a ball is dropped on a horizontal surface with no initial spin, previous studies have found that its bouncing behavior can be simply described using a coefficient of restitution, which gives the ratio of the velocity after the bounce to before the bounce. The value of this coefficient is −1 for a perfectly elastic ball/surface, with a smaller magnitude for any real ball. In many sports, like golf, basketball, or bowling, balls are thrown at an angle and are often given some initial spin by the player. Depending on initial conditions, these balls can bounce, roll, or start by bouncing and then transition to rolling. Here, the transition from bouncing to rolling is shown to be described by using both a vertical and horizontal coefficient of restitution, with the horizontal velocity defined at the spinning edge of the ball rather than its center. Videos of undergraduate level experiments are included, with results used to validate the model.

Understanding two-dimensional tractor magnets: Theory and realizations by Michael P. Adams. DOI: 10.1119/5.0198262

Editor's Note: Have you heard about tractor magnets? Certain fixed arrangements of three cylindrical magnets on a tabletop surface result in a stable equilibrium for a fourth follower magnet. In this article you will find the theoretical basis for this curious stability, a thorough experimental analysis of the stability of different possible magnet arrangements, and supplementary videos and 3D printing plans so you and your students can create and study your own tractor magnets.

A taxonomy of magnetostatic field lines by Joel Franklin; David Griffiths; Darrell Schroeter. DOI: 10.1119/5.0186335

Editor's Note: Many accepted “truths” about field lines are not entirely correct. Magnetic field lines can start and stop, they do not ordinarily form closed loops, and there is generally no relation between their density and the strength of the magnetic field. These points are made in the more complicated electric current configurations that are presented here and they can easily be demonstrated in upper-level undergraduate electromagnetism courses. The linked Mathematica techniques will enable instructors to construct their own innovative models.

A simulation of diffraction patterns using a lock-in detection code by M. Kolmanovsky; T. Hill; W. J. Kim. DOI: 10.1119/5.0128143

Editor's Note: Did you know that you could simulate a single-slit diffraction experiment using a lock-in amplifier? If not, you can read through this paper, which shows how using a rectangular time window when playing with a lock-in is the time-equivalent of the rectangular aperture in the optics experiment: The properties of Fourier transform guarantee that you will obtain a sinc function either in frequency for the lock-in experiment or in wave vector for the light diffraction experiment. Appropriate for undergraduate optics or instrumentation class.

Rutherford scattering of quantum and classical fields by Martin Pijnenburg; Giulia Cusin; Cyril Pitrou; Jean-Philippe Uzan. DOI: 10.1119/5.0175025

Editor's Note: Rutherford scattering, named after the famous gold foil experiment designer, is the scattering of one charged particle by another charged particle of fixed position. The authors here present a new approximation for this quantum scattering process, discuss the shortcomings of the traditional Born approximation approach found in many quantum mechanics textbooks, and show how their new techniques may be applied to the problem of the scattering of classical waves by black holes.

Four interacting spins: Addition of angular momenta, spin–spin correlation functions, and entanglement by Raimundo R. dos Santos; Lucas Alves Oliveira; Natanael C. Costa. DOI: 10.1119/5.0150433

Editor's note: Entanglement is often an elusive property for students. This paper, appropriate for advanced quantum mechanics class, shows on a practical example how to calculate entanglement. It considers four spins-1/2 that are evenly distributed on a ring and coupled to one another though competing nearest- and next-nearest-neighbor interactions. After determining the eigenstates and their energies, which is in itself a nice example of addition of more than two angular momenta, one can determine the entanglement of two subsystems. Spoiler: it depends on how you partition the system.

Splitting the second: Designing a physics course with an emphasis on timescales of ultrafast phenomena by Igor P. Ivanov. DOI: 10.1119/5.0133767

Editor's Note: Did you know that state-of-the-art experiments could measure timescales down to a few rontoseconds (such things do actually exist and correspond to 10 s)? In the spirit of Physics for Future Presidents, this paper uses the theme of fast timescales to build a semester-long syllabus to introduce undergraduate science majors to various physics concepts, ranging from condensed matter to particle physics, and to examples of commonly used technology (ink-jet printing, ultrasound imaging, high-speed cameras, etc.). Such a course could also be used for outreach, and it emphasizes the use of dimensional analysis and orders of magnitude.

Landau levels for charged particles with anisotropic mass by Orion Ciftja. DOI: 10.1119/5.0123039

Editor's note: The behavior of an electron confined to move in two dimensions in the presence of a perpendicular magnetic field underlies the quantum Hall effect. This is also an analytically solvable problem, so it is valuable to quantum mechanics pedagogy. This paper reviews the solution and then shows a neat trick for solving the problem when the electron's effective mass is anisotropic, a situation that is fairly common in solid-state systems.

The role of the Silberstein/Thomas/Wigner-rotation in the rod and slit paradox Video Abstracts by Mads Vestergaard Schmidt; Erich Schoedl. DOI: 10.1119/5.0175922

Editor's Note: Readers who are familiar with the “barn door” paradox will enjoy this 2D version of it. Imagine a rod, thrown like a javelin, moving at a constant horizontal speed and also falling at a constant vertical speed. As it reaches ground level, it falls precisely through a javelin-length slit in the floor. However, in the reference frame of the javelin, the slit will be length-contracted and the javelin will have its (longer) proper length. So as the ground rises up to meet it, how does it pass through? The authors revisit this classic paradox and present new ideas on its resolution, including the concept of relativistic boost-induced rotations.

#### NOTES AND DISCUSSIONS

Comment on “On the linearity of the generalized Lorentz transformation” [Am. J. Phys. 90(6), 425–429 (2022)] by Justo Pastor Lambare. DOI: 10.1119/5.0185737

Editor's Note: This Comment expands on a recent article on the Lorentz transformation, clearly explaining the roles in the transformation of the light principle, the limiting speed c, and linearity.

#### BOOK REVIEWS

Is Einstein still right? Black holes, gravitational waves, and the quest to verify Einstein's greatest creation by David Derbes. DOI: 10.1119/5.0226178

#### Additional Resources