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We demonstrate the ability of a simple algorithm based on the venerable method of images (MOIs), to accurately model the detailed frequency response of a multidimensional, rectangular, lossy resonant cavity. The convergence properties of the model's infinite series solution are shown to be determined by the cavity's quality factor Q. A 1D example demonstrates that the MOI series converges to the exact solution. Next, a comparison to precisely measure 2D cavity data confirms that a straightforward extension of the 1D algorithm to multiple dimensions provides accurate results. The algorithm is short, easily understandable by undergraduate students and relatively undemanding to code. An example using ®mathematica is provided.

Two-dimensional coupled nonlinear equations of projectile motion with air resistance in the form of quadratic drag are often treated as inseparable and solvable only numerically. However, when they are recast in terms of the angle between the projectile velocity and the horizontal, they become completely uncoupled and possess analytic solutions for projectile velocities as a function of that angle. The equations relating the time and position coordinates to this angle are not integrable in terms of elementary functions but are easy to integrate numerically. Additionally, energy equations explicitly including dissipation terms can be developed as integrals of the equations of motion. One-dimensional numerical integrations can be treated in a pedagogically straightforward way using numerical analysis software or even within a spreadsheet, making this topic accessible to undergraduates. We present this approach with sample numerical results for velocity components, trajectories, and energy-balance of a baseball-sized projectile.

[Media: see text] A smoke chamber and light-sheet video microscope setup is relatively easy to construct and provides opportunities for undergraduates to participate in a variety of advanced experiments, including the demonstration of Brownian motion and the interaction of induced electrostatic dipoles in aerosol particle agglomeration. We present results of these experiments along with information to allow replication of the setup in undergraduate physics laboratories. A theoretical model of the rate of aerosol agglomeration of long dipole chains as a function of electric field agrees with experiments at field strengths up to 200 kV m−1.

A lossless beam-splitter has certain (complex-valued) probability amplitudes for sending an incoming photon into one of two possible directions. We use elementary laws of classical and quantum optics to obtain general relations among the magnitudes and phases of these probability amplitudes. Proceeding to examine a pair of (nearly) single-mode wavepackets in the number-states |n1> and |n2> that simultaneously arrive at the splitter's input ports, we find the distribution of photon-number states at the output ports using an argument inspired by Feynman's scattering analysis of indistinguishable Bose particles. The result thus obtained coincides with that of the standard quantum-optical treatment of beam-splitters via annihilation and creation operators a and a†. A simple application of the Feynman method provides a form of justification for the Bose enhancement implicit in the well-known formulas a|n>=sqrt(n)|n-1> and a†|n>=sqrt(n+1)|n+1>., 13 pages, 2 figures, 22 equations, 18 references and endnotes

The brain is an electric organ. As such, for well over a hundred years, physicists, engineers, biologists, and physicians have used electromagnetic theory to try to understand how the brain works and to diagnose and treat disease. The field of electro-neuroscience is immense with thousands of papers being published each year. In this paper, we provide physics students with an introduction to the field using a conventional model of the head that employs four concentric spheres having different conductivities to represent the scalp, skull, cerebrospinal fluid (CSF), and the brain itself. To ground the discussion, we use the specific case of non-invasive DC electrical stimulation of the brain through electrodes applied to the surface of the scalp, a procedure known as transcranial DC stimulation or tDCS. Currently, tDCS is under clinical investigation for such diseases as depression, anxiety, and chronic pain as well as to enhance the performance of athletes and the training of fighter pilots. We solve the corresponding physics problem from a charge perspective and explain why the charge distributions look the way they do using what we call the “sensing pixel” technique. This paper can introduce students to the ways in which electromagnetic theory is being applied to problems in neuroscience; in this case, the problem of how to non-invasively stimulate the brain to treat disease or improve performance.

The speed of light is an important fundamental constant in physics, and so determining its value is a common undergraduate laboratory experiment. Methods to measure the light speed can help students practice their experimental skills and become familiar with the concepts of modern precision measurement techniques. In this paper, we demonstrate that a tabletop optical setup, comprised of an affordable Red Pitaya STEMlab board and a low-cost laser diode module, can be used to accurately determine the speed of light by measuring the frequency response of the phase shift between intensity-modulated light beams reflected by two end mirrors separated by 50 cm. By using the STEMlab built-in Bode analyzer to automatically scan the modulation frequency over the range from 10 to 40 MHz, the frequency response of phase is measured and recorded. These phase shift data are then used to calculate the speed of light with an uncertainty of less than 0.5%. With the help of the Red Pitaya board, the number of required electronic instruments for our setup is reduced. All of the required components are commercially available, and no electronic construction work is necessary so that teachers and students can implement the experiment in a plug-and-play manner.

The standard method to evaluate many definite integrals that are encountered in physics is contour integration. Here, we show how these can be evaluated by other means, enlarging the toolbox available to students and enabling the discussion of physical problems where these integrals arise before contour integration is introduced.

Gravitational microlensing is one of the methods to detect exoplanets–planets outside our solar system. Here, we focus on the theoretical modeling of systems with three lensing objects and in particular circumbinary systems. Circumbinary systems include two stars and at least one planet and are estimated to represent a sizeable portion of all exoplanets. Extending a method developed for binary lenses to the three lens case, we explore the parameter space of circumbinary systems, producing exact magnification maps and light curves.

We present undergraduate-friendly instructions on how to produce light pulses propagating through warm Rubidium vapor with speeds less than 400 m/s, i.e., nearly a million times slower than c. We elucidate the role played by electromagnetically induced transparency (EIT) in producing slow light pulses and discuss how to achieve the required experimental conditions. The optical setup is presented, and details provided for preparation of pump, probe, and reference pulses of the required size, frequency, intensity, temporal width, and polarization purity. EIT-based slow light pulses provide the most widely studied architecture for creating quantum memories. Therefore, the basic concepts presented here are useful for physics and engineering majors who wish to get involved in the development of cutting-edge quantum technologies.

A continuous fractional component (CFC) approach increases the probability of particle swaps in the context of vapor-liquid equilibrium simulations using the Gibbs ensemble Monte Carlo algorithm. Two variants of the CFC approach are compared for simulations of pure Lennard-Jones (LJ) fluids and binary LJ mixtures as examples. The details of an exemplary CFC implementation are presented. Recommendations are provided to reduce the effort required for the suggested problems.

We discuss the particle-in-cell (PIC) method, which is one of the most widely used approaches for the kinetic description of plasmas. The positions and velocities of the charged particles take continuous values in phase space, and spatial macroscopic quantities, such as the charge density and self-generated electric fields, are calculated at discrete spatial points of a grid. We discuss the computer implementation of the PIC method for one-dimensional plasmas in the electrostatic regime and discuss a desktop application (PlasmAPP), which includes the implementation of different numerical and interpolation methods and diagnostics in a graphical user interface. To illustrate its functionality, the electron-electron two-stream instability is discussed. Readers can use PlasmAPP to explore advanced numerical methods and simulate different phenomena of interest.

This paper shows how to apply Leibniz's integral rule to calculate the action variables for the Kepler problem. This method offers an attractive alternative to the usual technique of complex contour integration. The method presented here to calculate definite integrals has a broad scope and is especially suitable for undergraduates who are unfamiliar with complex analysis.

A hydrogen atom is supposed to be described by a generalization of the Schrödinger equation, in which the Hamiltonian depends on an iterated Laplacian and a Coulomb-like potential [Formula: see text]. Starting from previously obtained solutions for this equation using the [Formula: see text] expansion method, it is shown that new light can be shed on the problem of understanding the dimensionality of the world as proposed by Paul Ehrenfest. A surprising new result is obtained. Indeed, for the first time, we can understand that not only the sign of the energy but also the value of the ground state energy of hydrogen atoms is related to the threefold nature of space.

A few years ago, one of the former Editors of this journal launched “a call to action” (E. F. Taylor, Am. J. Phys. 71, 423–425 (2003)) for a revision of teaching methods in physics in order to emphasize the importance of the principle of least action. In response, we suggest the use of Hamilton's principle of stationary action to introduce the Schrödinger equation. When considering the geometric interpretation of the Hamilton–Jacobi theory, the real part of the action [Formula: see text] defines the phase of the wave function [Formula: see text], and requiring the Hamilton–Jacobi wave function to obey wave-front propagation (i.e., [Formula: see text] is a constant of the motion) yields the Schrödinger equation.

Modulated strain displacements were measured with a quadrature Michelson–Morley interferometer employing polarization optics and two lock-in amplifiers to filter noise and thermal drift. The advantages of the technique, its limitations, and estimates on the accuracy are discussed, including an algorithm to correct for non-ideal components and non-linear effects. Instructions for the construction and setup of the quadrature interferometer are provided. To test the interferometer, the dynamic converse piezoelectric effect was used, and by modulating the electric field across the sample, the [Formula: see text] strain coefficient for x-cut quartz was determined to be [Formula: see text], which is within 1.5 standard deviations of the accepted standard. The measurements had a standard deviation of 4.1 pm, resulting in standard errors as low as 5 fm/V after fitting.

Resonant transmission occurs when constructive interference results in the complete passage of an incoming wave through an array of barriers. In this paper we explore such a scenario with one dimensional models. We adopt wave packets with finite width to illustrate the deterioration of resonance with decreasing wave packet width, and suggest an approximate wave function for the transmitted and reflected components, derived from aspects of both the wave packet and plane wave approaches. A comparison with exact numerical calculations shows excellent agreement, and provides insight into the scattering process., Comment: 15 pages, 6 figures

We show that infrared telescopes can be constructed at low cost using consumer-grade thermal infrared imagers and commercially available germanium lenses. Using these telescopes in the laboratory, introductory astronomy students can image nearby celestial objects to observe properties that are not seen in the visible region, in particular, variations in temperature across the surface.

We share the design for a simple apparatus that, when paired with an Arduino processor and a computer, can be used in a wide range of laboratory measurements: observing linear kinematics, confirming Faraday's and Lenz's laws, measuring magnetic moments, and observing the effects of eddy currents. The setup is simple, inexpensive, easy to replicate, and can even be fabricated and used by students working at home.

Increasing interest in wave propagation in phononic systems and metamaterials motivates the development of experimental designs, measurement techniques, and fabrication methods for use in basic research and classroom demonstrations. The simplest phononic system, the monatomic chain, exhibits rich physics such as dispersion and frequency-domain filtering. However, a limited number of experimental studies showcase monatomic chains for macroscale observation of phonons. Herein, we discuss the design, fabrication, and testing of monatomic lattices as enabled by three-dimensional (3D) printing. Using this widely available technology, we provide design guidelines for realization of a monatomic chain composed of 3D printed serpentine springs and press-fitted cylindrical masses. We also present measurement techniques that record propagating waves and algorithms for the experimental determination of dispersion behavior.

We present and implement a design for an automated system that fabricates multilayer photonic crystal structures. The device is constructed with low-cost materials. A polystyrene/cellulose acetate multilayer Bragg reflector was fabricated to confirm the device's capability. A distributed feedback laser was also fabricated and characterized. The system has also been used to fabricate microlasers for a Modern Physics laboratory assignment in which students measure fluorescence, amplified spontaneous emission, lasing from one-dimensional Bragg reflectors, and lasing from scattering media., 20 pages

A ball that rolls on carpet while also spinning around a vertical axis will experience a drift force that acts perpendicular to its velocity, opposite to the tangential velocity component of the front point of the ball. Here, we present a model of this motion based on three assumptions: the ball rolls without slipping around the point of contact directly below its center of mass; the ball experiences rolling resistance due to a forward-shifted normal force; and the ball experiences a forward-shifted kinetic friction (drift) force. This model produces a simple analytic solution that is consistent with experimental data. Our measurements suggest that the kinetic friction force acts near the front of the contact patch between the ball and carpet.

This Resource Letter provides information and guidance for those looking to incorporate computation into their courses or to refine their own computational practice. We begin with general resources, including policy documents and supportive organizations. We then survey efforts to integrate computation across the curriculum as well as provide information for instructors looking to teach a computational physics course specifically. An overview of education research into computation in physics, including materials from beyond Physics Education Research, is then provided, followed by suggestions for tools, languages, and environments. We conclude with some emerging topics for which only preliminary resources exist but represent important topics for future innovation.

The matrix method is used to determine the mass spectrum (energy levels) of quarkonium, a composite particle comprising a quark and an anti-quark. This two-body system is similar to the hydrogen atom but at a reduced length scale. The results obtained by solving the Schrödinger equation for this system are in agreement with experimental and theoretical results obtained via other techniques, showing that problems with complicated potentials can be tackled by undergraduates.

We have developed a complete collection of freely available instructional materials to assist faculty in creating a student-centered quantum mechanics (QM) class that engages students while supporting them in developing both sense-making and calculational skills. Our materials are grounded in research on students' understanding of quantum mechanics and are intended to be adaptable to a variety of instructional settings and faculty styles or preferences. They were designed for a spins-first instructional paradigm and include a set of learning goals, concept (“clicker”) questions, pre-lecture surveys, and homework and exam questions, along with example lecture notes from three instructors at three different institutions. In this work, we describe what active learning can look like in the upper-division as well as describe each of the instructional tools and provide a few representative examples. We also discuss how these materials are used at each of our institutions, illustrating how they may be adapted for use at different institutions.

A straightforward method of determining the eccentricity of Earth's orbit and the position of aphelion and perihelion relative to the vernal equinox from solstice and equinox times is described. The only assumption made is that the orbital eccentricity is small. Using dates for these phenomena adopted from a desk calendar gives the eccentricity to an accuracy of about 10%.

We have designed a low-cost radio telescope system named the Bose Horn Antenna Radio Telescope (BHARAT) to detect the 21 cm hydrogen line emission from our Galaxy. The system is being used at the Radio Physics Laboratory (RPL) (Radio Physics Lab, IUCAA NCRA-TIFR, < http://www.iucaa.in/∼rpl/ >, < http://www.ncra.tifr.res.in/ncra/rpl/ >), Inter-University Centre for Astronomy and Astrophysics (IUCAA), India, for laboratory sessions and training students and teachers. It is also a part of the laboratory curriculum at several universities and colleges. Here, we present the design of a highly efficient, easy to build, and cost-effective dual-mode conical horn used as a radio telescope and describe the calibration procedure. We also present some model observation data acquired using the telescope for facilitating easy incorporation of this experiment in the laboratory curriculum of undergraduate or post-graduate programs. We have named the antenna after Acharya (teacher or an influential mentor) Jagadish Chandra Bose, honoring a pioneer in radio-wave science and an outstanding teacher, who inspired several world renowned scientists.

International audience; Why does the Sun have a radius around 696 000 km? We will see in this article that dimensional arguments can be used to understand the size of the Sun and of a few other things along the way. These arguments are not new and can be found scattered in textbooks. They are presented here in a succinct way in order to better confront the kinematic and mechanical viewpoints on size. We derive and compare a number of expressions for the size of the Sun and relate large and small scales. We hope that such presentation will be useful to students, instructors, and researchers alike.

We present an intuitive, conceptual, and semi-rigorous introduction to the Markov Chain Monte Carlo method using a simple model of population dynamics and focusing on a few elementary distributions. We start from two states, then three states, and finally generalize to many states with both discrete and continuous distributions. Despite the mathematical simplicity, our examples include the essential concepts of the Markov Chain Monte Carlo method, including ergodicity, global balance and detailed balance, proposal or selection probability, acceptance probability, the underlying stochastic matrix, and error analysis. Our experience suggests that most senior undergraduate students in physics can follow these materials without much difficulty.

An important property of oscillating systems like RLC circuits is the Q factor, which quantifies the strength of damping in the system. The Q factor is inversely proportional to the resistance for a series RLC circuit but increases with the resistance in a parallel RLC circuit. The surprising behavior of the parallel RLC circuit makes building and modeling this circuit an interesting project for a student laboratory. We describe an experiment that has been performed to explore this topic, share an example of the results that can be obtained, and suggest analyses that students might perform.

A circuit involving a charging supercapacitor in series with a non-Ohmic tungsten lamp displays a wealth of interesting behavior. Most notably, the current through the lamp decreases in time according to a power-law function as opposed to the exponential time dependence observed in RC circuits with Ohmic resistors. We use a combination of computational and analytical techniques to model this power-law behavior as well as the behavior of the filament's temperature and resistance as the supercapacitor charges. Our results agree well with experiment, and the experiment described here can be modified to be appropriate for physics courses at a wide range of levels.

A nonlinear electronic circuit comprising of three nodes with a feedback loop is analyzed. The system has two stable states, a uniform state and a sinusoidal oscillating state, and it transitions from one to another by means of a Hopf bifurcation. The stability of this system is analyzed with nonlinear equations derived from a repressilator-like transistor circuit. The apparatus is simple and inexpensive, and the experiment demonstrates aspects of nonlinear dynamical systems in an advanced undergraduate laboratory setting.

This work describes an experimental method for measuring the speed of light in air. It uses optical feedback from a visible picosecond diode laser operated below the threshold and a voltage-controlled oscillator to determine the time required for a pulse to travel a known distance. The experimental setup is compact, fitting into a space of 1 × 0.5 m2, and at the same time, can determine the speed of light with an uncertainty of 0.03%. The method does not require fast detectors or oscilloscopes.

In digital data transmission, single mode optical fibers are commonly used since they can carry very short optical pulses without any significant distortions. In contrast, multimode fibers support many propagation modes that travel with different speeds; thus, they cannot maintain the shape of a light pulse. This feature of multiple propagation modes can be a benefit since it makes possible the transmission of data through several channels simultaneously. We demonstrate how multimode fibers can be used to transmit images. Because of the different propagation constants of the modes, the transmitted image is scrambled to apparently random speckle patterns. A simple neural network can be used to model the transmission through the multimode fiber. We show how the neural network can be trained to recognize a set of patterns with high accuracy.

We treat a horizontal oscillator damped by constant-magnitude sliding friction by extending the analogy between the simple harmonic motion of a mass on a spring and the uniform circular motion of a mass attached to the end of a string. In the presence of sliding friction, the motion of the mass on a spring becomes the horizontal projection of the path of a mass attached to a string winding around two nails separated by a well-defined distance; this path is a spiral consisting of connected semi-circles of diminishing radii. This graphical analysis is very simple and pedagogically useful. It can also be generalized to any oscillation affected by other forces of constant magnitude but not necessarily constant direction.

We present a set of experiments and computations suitable for introducing upper-level undergraduate physics and engineering students to the interdisciplinary field of nanoplasmonics for periods ranging from a week-long advanced laboratory session to a summer research project. The end product is a tunable optofluidic device capable of detecting changes in a fluid medium as low as 0.002 refractive index units. The sensing element—a thin gold film on a glass prism coupled to a microfluidic cell—owes its sensitivity to the bound nature of the surface plasmon–polariton waves that are resonantly excited by evanescently coupled light at the gold–fluid interface. Pedagogically, surface plasmon resonance (SPR) sensing immerses students in the rich physics of nanoscale optics and evanescent waves in constructing and operating a precision apparatus and in developing theoretical, analytical, and numerical models to aid both in the physical understanding and engineering optimization of the SPR sensor.

[Media: see text] We describe a pandemic-inspired, modern physics distance lab course, focused both on engaging undergraduate physics majors in scientific research from their homes and on building skills in scientific paper reading and writing. To introduce the experimental and analytic tools, students are first asked to complete a traditional lab assignment in which collections of Hexbugs™, randomly moving toy automatons, are used to model gas molecules and to confirm the ideal gas law. Subsequently, after consulting the literature, students propose and implement semester-long experiments using Hexbugs™, smartphones, and materials commonly found at home to model various concepts in statistical mechanics and electrical conduction. A sample project focused on the Drude model, in which Hexbugs™ on a tilted plane are used to model electrical conduction, is described in detail. Alongside the research projects, students write formal, peer-reviewed scientific papers on their work, modeling the professional publication process as closely as possible. Somewhat paradoxically, we found that the pandemic-inspired exigency of reliance on simple, home-built experiments enabled an increased focus on developing experimental research skills and achieving the laboratory learning objectives recommended by the American Association of Physics Teachers.

The pioneering work of G.I. Taylor on the turbulent dispersion of aerosols is exactly one century old and provides an original way of introducing both diffusive processes and turbulence at an undergraduate level. Light enough particles transported by a turbulent flow exhibit a Brownian-like motion over time scales larger than the velocity correlation time. Aerosols are therefore subjected to an effective turbulent diffusion at large length scales. However, the case of a source of pollutant much smaller than the integral scale is not completely understood. Here, we present experimental results obtained by undergraduate students in the context of the COVID-19 pandemic. The dispersion of a smoke of oil droplets by a turbulent flow is studied in a wind tunnel designed for pedagogical purposes. It shows a ballistic-like regime at short distance, followed by Taylor's diffusive-like regime, suggesting that the continuous cascade is bypassed. Accordingly, measurements show that the CO$_2$ concentration emitted when breathing typically decays as the inverse squared distance to the mouth, which is not the law expected for a diffusion. The experiment offers the possibility for students to understand the role of fluctuations in diffusive processes and in turbulence. The non-linear diffusive equations governing aerosol dispersion, based on a single correlation time, allow us to model the airborne transmission risk of pathogens, indoors and outdoors. We discuss the pedagogical interest of making students work on applied scientific problems, as the results obtained in this study have been used to provide public health policy recommendations to prevent transmission in French shopping malls., 23 pages, 7 figures; submitted to the American Journal of Physics

Although physical laws or theories are often invoked in debates over “causality” and “determinism,” our best current understanding of physics assigns only a limited (though still very broad) validity to these concepts. It may be, thus, helpful (particularly when having to deal with the challenges posed by quantum mechanics) to think of them as prejudices, extrapolated from our experience with a limited (essentially classical) set of phenomena and/or theoretical models. This paper discusses how, over time, different physical theories have either reinforced or challenged these prejudices, focusing specifically on conservative “Laplacian” mechanics, dissipative mechanics (thermal physics), and quantum mechanics.

Long-term changes in the tilt of the Earth's axis, relative to the plane of its orbit, are of great significance to long-term climate change, because they control the size of the arctic and Antarctic circles. These “Milankovitch cycles” have hitherto been calculated by classical perturbation methods or by direct numerical integration of Newton's equations of motion. This paper presents an approximate calculation from simple considerations of angular momentum using similar methods to those used to study the precession of a spinning top. It is an instructive exercise in classical mechanics and gives a simple explanation of the phenomenon in terms of angular momentum. It is shown that the main component of “Milankovitch cycles” has a period of 41,000 yr and is due to one of the modes of precession of the Earth-Venus system. The other mode of this system produces a component of period 29,500 yr, and a third component of period 54,000 yr results from the influence of the precession of the orbits of Jupiter and Saturn. These results agree closely with several of the numerical simulations in the literature and strongly suggest that some other results in the literature are incorrect.

We present the design and construction of a laboratory apparatus that provides advanced undergraduates with hands-on observations of electron quantum tunneling and the electronic density of states of various materials. The instrument is inspired by the scanning tunneling microscope (STM), but its implementation is simplified by limiting the tip motion to the single dimension along the tip-sample separation ( z-axis); we refer to the device as the z-axis tunneling microscope (ZTM). Students are able to use the ZTM to measure electron tunneling probability as a function of barrier width, estimate relative material work functions, and observe differences in local electronic structure among metals, semimetals, and semiconductors. We share results obtained by third-year undergraduate physics students using the instrument for their final projects in an advanced instructional lab course.

The Efimov effect (in a broad sense) refers to the onset of a geometric sequence of many-body bound states as a consequence of the breakdown of continuous scale invariance to discrete scale invariance. While originally discovered in three-body problems in three dimensions, the Efimov effect has now been known to appear in a wide spectrum of many-body problems in various dimensions. Here we introduce a simple, exactly solvable toy model of two identical bosons in one dimension that exhibits the Efimov effect. We consider the situation where the bosons reside on a semi-infinite line and interact with each other through a pairwise $\delta$-function potential with a particular position-dependent coupling strength that makes the system scale invariant. We show that, for sufficiently attractive interaction, the bosons are bound together and a new energy scale emerges. This energy scale breaks continuous scale invariance to discrete scale invariance and leads to the onset of a geometric sequence of two-body bound states. We also study the two-body scattering off the boundary and derive the exact reflection amplitude that exhibits a log-periodicity. This article is intended for students and non-specialists interested in discrete scale invariance., Comment: 14 pages, 4 eepic figures; title changed, typos corrected, references and an appendix added

We revisit the classical pursuit curve problem solved by Pierre Bouguer in the 18th century, taking into account that information propagates at a finite speed. To a certain extent, this could be seen as a relativistic correction to that problem, though one does not need Einstein's theory of relativity in order to derive or understand its solution. The discussion of this generalized problem of pursuit constitutes an excellent opportunity to introduce the concept of retarded time without the complications inherent to the study of electromagnetic radiation (where it is usually seen for the first time), which endows the problem with a clear pedagogical motivation. We find the differential equation which describes the problem, solve it numerically, compare the solution to Bouguer's for different values of the parameters, and deduce a necessary and sufficient condition for the pursuer to catch the pursued, complementing previous work by Hoenselaers., 15 pages, 7 figures, submitted to the American Journal of Physics

Kinesthetic (or embodied) representations help students build intuition and deep understanding of concepts. This paper presents a series of kinesthetic activities for a spins-first undergraduate quantum mechanics course that supports students in reasoning and developing intuition about the complex-valued vectors of spin states. The arms representation, used in these activities, was developed as a tangible representation of complex numbers: Students act as an Argand diagram, using their left arm to represent numbers in the complex plane. The arms representation is versatile and can be expanded to depict complex-valued vectors with groups of students. This expansion enables groups of students to represent quantum mechanical state vectors with their arms. We have developed activities using the arms representation that parallel the progression of a spins-first approach by starting with complex numbers, then representing two- and three-state systems, considering time-dependence, and, eventually, extending to approximate wavefunctions. Each activity illustrates the complex nature of quantum states and provides a tangible manipulative from which students can build intuition about quantum phenomena.

Armed with an astrolabe and Kepler's laws one can arrive at accurate estimates of the orbits of planets., Comment: 28 pages, 19 figures

Stereographic projection mapping is typically introduced to explain the point at infinity in the complex plane. After this brief exposure in the context of complex analysis, students rarely get an opportunity to fully appreciate stereographic projection mapping as an elegant and powerful technique on its own with many fruitful applications in the physical sciences. Here, using a classical description of nuclear magnetic resonance in the rotating frame, I show how stereographic projection mapping to and from the Bloch sphere can be used for visualizing solutions to Bloch's equation and the Bloch–Riccati equation, respectively. After developing the fundamentals of stereographic projection mapping using examples drawn from nuclear spin precession in the rotating frame, the method is then applied to visualizations of composite pulse excitation of a spin-1/2 system and to radiation damping in a system of isolated spins-1/2. In the case of the radiation-damped system, these visualizations provide particularly vivid illustrations of loxodromic Möbius transformation dynamics.

We revisit the golfer's curse, which is the possibility that a golf ball can emerge from the cylindrical hole into which it has entered. Our analysis focuses on three constants of the motion. One of these is the energy, because we assume that the ball rolls without slipping on the inner wall of the hole, losing only a small amount of energy to rolling resistance; the other two are related to the angular momentum about the contact point of the ball with the inner wall of the hole. We develop an analysis of the motion of the ball and report measurements of the moment of inertia of a real golf ball. Solving the equation of motion along the vertical direction, we address the question of whether or not the ball could complete a vertical oscillation without reaching the bottom of the hole. We also present measurements of the dynamical friction for a golf ball and discuss dissipation in slip conditions. We conclude by proposing a challenge to golf players: to find a way to send a ball into a hole in order to make it emerge.

Shielding involves much more than just putting a conductive screen in between an emitting source and a susceptible device. Starting from Maxwell's equations, the concept of electromagnetic shielding is formally explained. The physical working mechanisms behind the two basic forms of shielding, electric field and magnetic field shielding, are given, and the link between them at higher frequencies is clarified. Several aspects, like the effect of gridding or weaving a shield, the effect of the finite size of a shield, and the penetration through the metal of a shield, are discussed based on very simple canonical shielding topologies that can be solved analytically. Although the classical paradigm to explain shielding based on the notions of skin depth and eddy current is not followed, conceptual links with this classical paradigm are explained.

In recent years, more courses in electromagnetism are using the “Système International” (SI) units as opposed to Gaussian-cgs. The confusing notation used to formulate SI with origins in the early 19th century still persists in instruction. This work shows that electromagnetism may be taught relatively painlessly in the units that virtually everyone uses by employing a new presentation that makes the equations nearly as simple as those in the Heaviside–Lorentz system commonly used by theoretical physicists. Introducing a new coupling constant κ and some new notation for the fields, it is possible to dispense with ϵ0 and μ0 and the conceptual framework from which they come. As a result, it is possible achieve much greater clarity, while using all the same symbols and relations as in the extant literature.