The higher derivatives of motion are rarely discussed in the teaching of classical mechanics of rigid bodies; nevertheless, we experience the effect not only of acceleration, but also of jerk and snap. In this paper we will discuss the third and higher order derivatives of displacement with respect to time, using the trampolines and theme park roller coasters to illustrate this concept. We will also discuss the effects on the human body of different types of acceleration, jerk, snap and higher derivatives, and how they can be used in physics education to further enhance the learning and thus the understanding of classical mechanics concepts.
The European Physical Society (EPS) is a not for profit association whose members include 41 National Physical Societies in Europe, individuals from all fields of physics, and European research institutions.
As a learned society, the EPS engages in activities that strengthen ties among the physicists in Europe. As a federation of National Physical Societies, the EPS studies issues of concern to all European countries relating to physics research, science policy and education.
ISSN: 1361-6404
With a world-wide readership and authors from every continent, European Journal of Physics is a truly international journal dedicated to maintaining and improving the standard of taught physics in universities and other higher education institutes.
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David Eager et al 2016 Eur. J. Phys. 37 065008
Giulia Polverini and Bor Gregorcic 2024 Eur. J. Phys. 45 025701
The paper aims to fulfil three main functions: (1) to serve as an introduction for the physics education community to the functioning of large language models (LLMs), (2) to present a series of illustrative examples demonstrating how prompt-engineering techniques can impact LLMs performance on conceptual physics tasks and (3) to discuss potential implications of the understanding of LLMs and prompt engineering for physics teaching and learning. We first summarise existing research on the performance of a popular LLM-based chatbot (ChatGPT) on physics tasks. We then give a basic account of how LLMs work, illustrate essential features of their functioning, and discuss their strengths and limitations. Equipped with this knowledge, we discuss some challenges with generating useful output with ChatGPT-4 in the context of introductory physics, paying special attention to conceptual questions and problems. We then provide a condensed overview of relevant literature on prompt engineering and demonstrate through illustrative examples how selected prompt-engineering techniques can be employed to improve ChatGPT-4's output on conceptual introductory physics problems. Qualitatively studying these examples provides additional insights into ChatGPT's functioning and its utility in physics problem-solving. Finally, we consider how insights from the paper can inform the use of LLMs in the teaching and learning of physics.
Joseph Ivin Thomas 2020 Eur. J. Phys. 41 055305
The double slit experiment was the first demonstrative proof of the wave nature of light. It was expounded by the English physician-physicist Thomas Young in 1801 and it soon helped lay to rest the then raging Newton–Huygens debate on whether light consisted of a fast-moving stream of particles or a train of progressive waves in the ether medium. In the experiment, light is made to pass through two very narrow slits spaced closely apart. A screen placed on the other side captures a pattern of alternating bright and dark bands called fringes which are formed as a result of the phenomenon of interference. In prior work by the same author, it was shown that the conventional analysis of Young's experiment that is used in many introductory physics textbooks, suffers from a number of limitations in regards to its ability to accurately predict the positions of these fringes on the distant screen. This was owing to the adoption of some needless and paradoxical assumptions to help simplify the geometry of the slit barrier-screen arrangement. In the new analysis however, all such approximations were discarded and a hyperbola theorem was forwarded which was then suitably applied to determine the exact fringe positions on screens of varied shapes (linear, semi-circular, semi-elliptical). This paper further builds on that work by laying down the mathematical framework necessary for counting fringes and then comparing their distributions on differently shaped screens, using MATLAB software package for numerical–graphical simulation. In addition, a pair of equivalent laws of proportionality are predicted that govern the distribution of fringes independent of the shape of the detection screen employed.
Thomas Quick and Johannes Grebe-Ellis 2024 Eur. J. Phys. 45 045301
Lens phenomena, such as caustics, image distortions, and the formation of multiple images, are commonly observed in various refracting geometries, including raindrops, drinking glasses, and transparent vases. In this study, we investigate the ball lens as a representative example to showcase the capabilities of Berry's eye caustic as an optical tool. Unlike the conventional paraxial approximation, the eye caustic enables a comprehensive understanding of image transformations throughout the entire optical space. Through experimental exploration, we establish the relationship between the eye caustic and traditional light caustics. Furthermore, we provide mathematical expressions to describe both the caustic and the image transformations that occur when viewing objects through the ball lens. This approach could be of interest for optics education, as it addresses two fundamental challenges in image formation: overcoming the limitations of the paraxial approximation and recognizing the essential role of the observer in comprehending lens phenomena.
P Berg 2023 Eur. J. Phys. 44 025002
A typical undergraduate course in mechanics does not cover the fascinating and important gravity-assist manoeuvre that allows satellites or other spacecrafts to navigate through our solar system on efficient and desired paths. Instead, it usually remains a mystery to students how energy is conserved when a spacecraft gains speed as it flies past a planet. Indeed, one might be led to believe that the curved path of the planet is the root cause for the gain in speed, requiring consideration of gravity-assist within the framework of the restricted three-body problem. This contribution will emphasize that this extension is not required to explain the gain in kinetic energy. Instead, a simple, scaffolded analysis of the planet-satellite system alone, using elementary physics, two reference frames and analytical methods, provides a sufficient explanation. Our simplified analysis is successfully validated against mission data from Voyager 2's gravity-assist manoeuvre around Jupiter.
Peter Wulff 2024 Eur. J. Phys. 45 023001
Language is an important resource for physicists and learners of physics to construe physical phenomena and processes, and communicate ideas. Moreover, any physics-related instructional setting is inherently language-bound, and physics literacy is fundamentally related to comprehending and producing both physics-specific and general language. Consequently, characterizing physics language and understanding language use in physics are important goals for research on physics learning and instructional design. Qualitative physics education research offers a variety of insights into the characteristics of language and language use in physics such as the differences between everyday language and scientific language, or metaphors used to convey concepts. However, qualitative language analysis fails to capture distributional (i.e. quantitative) aspects of language use and is resource-intensive to apply in practice. Integrating quantitative and qualitative language analysis in physics education research might be enhanced by recently advanced artificial intelligence-based technologies such as large language models, as these models were found to be capable to systematically process and analyse language data. Large language models offer new potentials in some language-related tasks in physics education research and instruction, yet they are constrained in various ways. In this scoping review, we seek to demonstrate the multifaceted nature of language and language use in physics and answer the question what potentials and limitations artificial intelligence-based methods such as large language models can have in physics education research and instruction on language and language use.
Rod Cross 2024 Eur. J. Phys. 45 035008
A simple experiment is described where a metal ring was rotated in a vertical plane on a horizontal rod. The ring rotated about 100 times before coming to a stop, so the friction force on the ring remained very small. However, measurements of the rotation frequencies of the ring around the rod and around its centre of mass indicated that the ring was sliding rather than rolling, with an unusually low coefficient of sliding friction.
Jan Stegemann et al 2023 Eur. J. Phys. 44 035402
With the advent of quantum technology, the need for affordable, flexible and robust laboratory experiments not only for students, but also at high school level is increasing. Here, for the first time, we report on a simple modular 3D printed low-cost (<250 €) setup which fulfils these needs for quantum sensing experiments based on nitrogen-vacancy centers in diamonds. Commercially available setups for optically detected magnetic resonance in microdiamonds used as quantum sensor for magnetic fields are not only beyond the reach of any high school (>10 000 €), but also have shortcomings from a didactical point of view, as all the components of the setup are hidden within a 'black box', doomed to be successful 'plug and play'. In contrast, our open-source experimental kit consists of optical components that are placed inside 3D printed open-framed cubes, that can be arranged freely on a grid. This modular and flexible design can provide an inquiry-based learning experience both at undergraduate and high school level.
Friedrich Herrmann and Tobias Vorbach 2020 Eur. J. Phys. 41 045803
The geodynamo usually appears as a somewhat intimidating subject. Its understanding seems to require knowledge of the intricate theory of magnetohydrodynamics. The solution of the corresponding equations can only be achieved numerically. It seems to be a subject for the specialist. We show that one can understand the basics of the functioning of the geodynamo solely by using the well-known laws of electrodynamics. The topic is not only important for geophysicists. The same physics is the cause for the magnetic fields of Sun-like stars, of the very strong fields of neutron stars, and also of the cosmic magnetic fields.
Jed Brody and Kristen Gram 2024 Eur. J. Phys. 45 035403
We present, for the non-specialist, an experimental implementation of Shor's factoring algorithm. We are unaware of any other single reference that explains, in a beginner-friendly way, complete circuits that implement Shor's algorithm. We perform the experiment with IBM quantum processors, which are remotely accessible online for free. We reproducibly factor 15 with one 7-qubit processor (ibm_perth), while four other quantum processors exhibit excessive error.
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Nilüfer Didiş Körhasan and Ceylan Çıtak 2024 Eur. J. Phys. 45 045701
A body of beliefs based on previous knowledge, attitudes, and experiences creates physics expectations of students about what they will learn and which skills they will use in physics lessons. Previous research indicated these views about knowledge and learning influenced physics learning by affecting their learning approaches. This research aims to investigate the physics expectations of undergraduate engineering and education students, who take various and different numbers of physics courses in Türkiye. The physics expectations of students were examined due to certain variables and they were compared with experts' ideas. In the study, the Turkish version of the Maryland Physics Expectations Survey II (MPEX II) was given to 831 engineering and education students in different universities. The results indicated the statistical differences in the physics expectations scores according to the department that was registered, the number of physics courses taken, and the reading of popular physics books. However, no significant difference was observed in the physics expectations scores according to gender, being an engineering student or education student, the year of the program, and the instructional approach of physics courses. The mean of the percentages of students who thought like experts in item by item was around 2/5; however, when expert-like thinking was considered for the overall scores, almost 12% of students answered at least half of the items like experts.
Justo Pastor Lambare 2024 Eur. J. Phys. 45 045601
The Sagnac effect is an interferometric phenomenon produced by rotation. It has a rich history and presently has numerous technological applications. Despite some persistent claims to the contrary, we explain why the Sagnac effect does not prove relativity either incorrect or inconsistent. Analyzing such misunderstandings has didactic value because it allows us to review some subtle relativity concepts. It also reveals the importance of basing scientific reasoning on rigorous logical thinking to avoid confusion derived from prejudices based on our limited everyday human experience.
Thomas Quick and Johannes Grebe-Ellis 2024 Eur. J. Phys. 45 045301
Lens phenomena, such as caustics, image distortions, and the formation of multiple images, are commonly observed in various refracting geometries, including raindrops, drinking glasses, and transparent vases. In this study, we investigate the ball lens as a representative example to showcase the capabilities of Berry's eye caustic as an optical tool. Unlike the conventional paraxial approximation, the eye caustic enables a comprehensive understanding of image transformations throughout the entire optical space. Through experimental exploration, we establish the relationship between the eye caustic and traditional light caustics. Furthermore, we provide mathematical expressions to describe both the caustic and the image transformations that occur when viewing objects through the ball lens. This approach could be of interest for optics education, as it addresses two fundamental challenges in image formation: overcoming the limitations of the paraxial approximation and recognizing the essential role of the observer in comprehending lens phenomena.
Xabier Prado et al 2024 Eur. J. Phys. 45 035009
Throughout history, scales have served as instrumental tools for quantifying the weight of objects, relying on a comparative assessment against a specified reference weight. Scales featuring uneven arms, such as the bismar scale, have proven particularly adept at gauging masses within a specific range relative to a predetermined reference mass. On the other hand, the kinematics of elastic collisions hinge on the inertial masses of the colliding entities. By observing the aftermath of a collision between a known reference mass and an object of unknown mass, one can deduce the latter's mass. In this contribution, we highlight a fascinating and clear analogy between these two methodologies. We do so by adapting a geometric approach, initially applicable to the bismar scale, to both non-relativistic and relativistic elastic collisions, encompassing phenomena such as Compton scattering.
J Barata et al 2024 Eur. J. Phys. 45 035306
In this work we develop an experiment wherein the variation of the fringe visibilities with wavelength provides insights into the geometry of various stellar sources or components. The experiment is based on the Michelson stellar interferometer in which a filter wheel has been inserted between a detecting camera and a telescope obscured by a double-aperture lid. The spatial and temporal incoherent light emitting from stellar sources has been simulated using polymer optical fibres and a broadband light-emitting diode. By measuring the visibilities of the central fringe at different wavelengths we are able to determine the morphology of the analysed light sources, provided that the baseline used is sufficiently long. The experiment is suitable for postgraduate students seeking to delve deeper into light coherence theory and to gain practical experience in optical stellar interferometry.
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Manfred Euler 2024 Eur. J. Phys. 45 023003
Although synchronization effects play an important role in many areas of basic and applied science, their treatment in undergraduate physics courses requires more attention. Based on acoustic experiments with a driven organ pipe, the article proposes analytical, numerical and qualitative approaches to this universal phenomenon, suitable for introductory teaching. The Adler equation is developed, a first-order nonlinear differential equation describing the phase dynamics of driven self-sustained oscillations in the weak coupling limit. Analytical solutions, intuitive mechanical analogues and properties of the resulting comb spectra are discussed. The underlying phase model is paradigmatic for synchronization-based self-organization phenomena in a wide range of fields, from physics and engineering to life and social sciences.
Andrej Vidak et al 2024 Eur. J. Phys. 45 023002
The use of augmented reality (AR) allows for the integration of digital information onto our perception of the physical world. In this article, we present a comprehensive review of previously published literature on the implementation of AR in physics education, at the school and the university level. Our review includes an analysis of 96 papers from the Scopus and Eric databases, all of which were published between 1st January 2012 and 1st January 2023. We evaluated how AR has been used for facilitating learning about physics. Potential AR-based learning activities for different physics topics have been summarized and opportunities, as well as challenges associated with AR-based learning of physics have been reported. It has been shown that AR technologies may facilitate physics learning by providing complementary visualizations, optimizing cognitive load, allowing for haptic learning, reducing task completion time and promoting collaborative inquiry. The potential disadvantages of using AR in physics teaching are mainly related to the shortcomings of software and hardware technologies (e.g. camera freeze, visualization delay) and extraneous cognitive load (e.g. paying more attention to secondary details than to constructing target knowledge).
Peter Wulff 2024 Eur. J. Phys. 45 023001
Language is an important resource for physicists and learners of physics to construe physical phenomena and processes, and communicate ideas. Moreover, any physics-related instructional setting is inherently language-bound, and physics literacy is fundamentally related to comprehending and producing both physics-specific and general language. Consequently, characterizing physics language and understanding language use in physics are important goals for research on physics learning and instructional design. Qualitative physics education research offers a variety of insights into the characteristics of language and language use in physics such as the differences between everyday language and scientific language, or metaphors used to convey concepts. However, qualitative language analysis fails to capture distributional (i.e. quantitative) aspects of language use and is resource-intensive to apply in practice. Integrating quantitative and qualitative language analysis in physics education research might be enhanced by recently advanced artificial intelligence-based technologies such as large language models, as these models were found to be capable to systematically process and analyse language data. Large language models offer new potentials in some language-related tasks in physics education research and instruction, yet they are constrained in various ways. In this scoping review, we seek to demonstrate the multifaceted nature of language and language use in physics and answer the question what potentials and limitations artificial intelligence-based methods such as large language models can have in physics education research and instruction on language and language use.
Francesco Bernardini et al 2024 Eur. J. Phys. 45 013001
This pedagogical article elucidates the fundamentals of trapped-ion quantum computing, which is one of the potential platforms for constructing a scalable quantum computer. The evaluation of a trapped-ion system's viability for quantum computing is conducted in accordance with DiVincenzo's criteria.
Jongheon Baek 2023 Eur. J. Phys. 44 063001
Mass–energy equivalence (MEE) has become a basis of modern physics. In spite of the current educational trends highlighting modern physics education, it has been pointed out that interpretations of MEE are still not in general agreement. In addition, the derivations of MEE found in textbooks gloss over some logical oversights. MEE is often introduced with only a declarative knowledge that mc2 represents the rest energy of a particle, making the learning process difficult for students. To resolve the instructional challenges, distinguished papers on MEE are analyzed. By specifying common features of derivations in each paper, it is found that there are at least three types of MEE. By identifying the entire hierarchical structure of each one, a type of MEE is suggested that can potentially be useful in the establishment of the connection between the particle and field.
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Blanco
The ALE-3 mission plans to create artificial "shooting stars" in the upper atmosphere by launching pellets sequentially from a satellite in low Earth orbit. How does one arrange for all pellets to re-enter over the intended location simultaneously? Starting with conservation of energy and angular momentum, I derive a version of Kepler's equation that gives time as a function of radial distance in an orbit, which can be used to find the transfer time from the pellet's ejection to its re-entry as a function of its launch velocity. I show that for a given pellet ejection speed, there is a launch angle from the satellite that results in the fastest transfer time. I use these results to determine the pellets' launch times and velocities for simultaneous arrival at a desired re-entry point. These results can be applied to de-orbiting any set of objects launched from a satellite, and use concepts that can be covered in an advanced undergraduate course in physics or aerospace engineering. Supplementary materials are provided online.
Tufino et al
In this article we describe how we successfully incorporated data analysis in Python in a first-year laboratory course without significantly altering the course structure and without overburdening students. We show how we created and used carefully designed Jupyter Notebooks with exercises and physics application examples that allow students to master data analysis programming in the laboratory course. These Notebooks guide students through the fundamentals of data handling and analysis in Python as they conduct simple experiments. We present our teaching approach and the developed materials. The effectiveness of this intervention is assessed through results from pre- and post-course questionnaires and students' group work. The results presented give insights about advantages and challenges of introducing computation at the early stage of the curriculum in a laboratory course setting and are informative for other instructors and the physics education research community.
Li et al
Research based active learning strategies have been shown to have positive effects on students' learning in physics class. However, the exploration of active learning's impact on students' sense of belonging in physics classrooms remains limited. In this paper, we present our findings pertaining to students' sense of belonging in an introductory physics course at a large public research university. This course was taught by three different instructors, one of whom integrated research-based active learning strategies, while the other two employed traditional lecture-based approaches. Our results show a statistically significant decrease in the sense of belonging among students in the traditionally taught classes, in contrast to a modest increase in students' sense of belonging in the active learning class. Furthermore, we observed a statistically significant gender difference in students' sense of belonging at the beginning of the course, which was closed by the end of the active learning class but persisted in the traditional classes. In addition, the active learning class exhibited a positive impact on students' academic outcomes, and this effect was partially mediated by the sense of belonging. Our findings indicate that active learning classes, particularly those thoughtfully designed to foster inclusive environments, may benefit students' sense of belonging and enhance their academic performance without relying on the use of specific belonging interventions.
Berry
A Hamiltonian in two space dimensions whose kinetic-energy contributions have opposite signs is studied in detail. Solutions of the time-independent Schrödinger equation for fixed energy are superpositions of plane waves, with wavevectors on hyperbolas rather than circles. The local velocity (e.g. in the Madelung representation) is proportional to the kinetic momentum, i.e. local particle velocity, not the more familiar canonical momentum (phase gradient). The patterns of the associated streamlines are different, especially near phase singularities and phase saddles where the kinetic and canonical streamline patterns have opposite indices. Contrasting with the superficially analogous circular smooth solutions of kinetically isotropic Hamiltonians are wave modes that are anisotropic in position and also discontinuous. Pictures illustrating these phenomena are included. The occurrence of familiar concepts in unfamiliar guises could be useful for teaching quantum or wave physics at graduate level.
Briggs
The standard description of the transition from quantum to classical mechanics presented in most text books is the proof that the quantum expectation values of position and momentum obey equations of Newtonian form. This is the Ehrenfest Theorem. It is combined with the requirement that wave packets remain localised to describe a single particle moving according to classical mechanics. Hence, the natural spreading of wave packets is viewed as a quantum effect. In contradiction to this view, here it is argued that the spreading, where different momentum components separate, is the signature of the quantum to classical transition. The asymptotic spatial wave function becomes proportional to the initial momentum space wave function, which mirrors exactly the well-known far-field diffraction pattern in optics. Trajectories, defined as the locus of the normals to the expanding wave front, are used to illustrate the transition from quantum to classical motion. Again this is the analogue of the wave to beam transition in optics. It is suggested that this analysis of the quantum to classical transition should be incorporated routinely into introductory quantum mechanics courses.mechanics courses.
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Philip Rodriguez Blanco 2024 Eur. J. Phys.
The ALE-3 mission plans to create artificial "shooting stars" in the upper atmosphere by launching pellets sequentially from a satellite in low Earth orbit. How does one arrange for all pellets to re-enter over the intended location simultaneously? Starting with conservation of energy and angular momentum, I derive a version of Kepler's equation that gives time as a function of radial distance in an orbit, which can be used to find the transfer time from the pellet's ejection to its re-entry as a function of its launch velocity. I show that for a given pellet ejection speed, there is a launch angle from the satellite that results in the fastest transfer time. I use these results to determine the pellets' launch times and velocities for simultaneous arrival at a desired re-entry point. These results can be applied to de-orbiting any set of objects launched from a satellite, and use concepts that can be covered in an advanced undergraduate course in physics or aerospace engineering. Supplementary materials are provided online.
Eugenio Tufino et al 2024 Eur. J. Phys.
In this article we describe how we successfully incorporated data analysis in Python in a first-year laboratory course without significantly altering the course structure and without overburdening students. We show how we created and used carefully designed Jupyter Notebooks with exercises and physics application examples that allow students to master data analysis programming in the laboratory course. These Notebooks guide students through the fundamentals of data handling and analysis in Python as they conduct simple experiments. We present our teaching approach and the developed materials. The effectiveness of this intervention is assessed through results from pre- and post-course questionnaires and students' group work. The results presented give insights about advantages and challenges of introducing computation at the early stage of the curriculum in a laboratory course setting and are informative for other instructors and the physics education research community.
Yangquiting Li et al 2024 Eur. J. Phys.
Research based active learning strategies have been shown to have positive effects on students' learning in physics class. However, the exploration of active learning's impact on students' sense of belonging in physics classrooms remains limited. In this paper, we present our findings pertaining to students' sense of belonging in an introductory physics course at a large public research university. This course was taught by three different instructors, one of whom integrated research-based active learning strategies, while the other two employed traditional lecture-based approaches. Our results show a statistically significant decrease in the sense of belonging among students in the traditionally taught classes, in contrast to a modest increase in students' sense of belonging in the active learning class. Furthermore, we observed a statistically significant gender difference in students' sense of belonging at the beginning of the course, which was closed by the end of the active learning class but persisted in the traditional classes. In addition, the active learning class exhibited a positive impact on students' academic outcomes, and this effect was partially mediated by the sense of belonging. Our findings indicate that active learning classes, particularly those thoughtfully designed to foster inclusive environments, may benefit students' sense of belonging and enhance their academic performance without relying on the use of specific belonging interventions.
Michael V Berry 2024 Eur. J. Phys.
A Hamiltonian in two space dimensions whose kinetic-energy contributions have opposite signs is studied in detail. Solutions of the time-independent Schrödinger equation for fixed energy are superpositions of plane waves, with wavevectors on hyperbolas rather than circles. The local velocity (e.g. in the Madelung representation) is proportional to the kinetic momentum, i.e. local particle velocity, not the more familiar canonical momentum (phase gradient). The patterns of the associated streamlines are different, especially near phase singularities and phase saddles where the kinetic and canonical streamline patterns have opposite indices. Contrasting with the superficially analogous circular smooth solutions of kinetically isotropic Hamiltonians are wave modes that are anisotropic in position and also discontinuous. Pictures illustrating these phenomena are included. The occurrence of familiar concepts in unfamiliar guises could be useful for teaching quantum or wave physics at graduate level.
Pietro Campana et al 2024 Eur. J. Phys.
This paper presents a new laboratory activity aimed at developing knowledge and expertise in microwave applications at cryogenic temperatures. The experience focuses on the detection of infrared photons through Microwave Kinetic Inductance Detectors (MKIDs). The experimental setup, theoretical concepts, and activities involved are detailed, highlighting the skills and knowledge gained through the experience. This experiment is designed for graduate students in the field of quantum technologies.
J J Teixeira et al 2024 Eur. J. Phys.
This work presents and analyzes the acceleration data inside an aircraft during a parabolic flight. The data used was obtained during flight from the aircraft's automatic recordings and from a portable data-logger with a built-in 3-axis accelerometer connected to a graphic calculator. The comparison of the accelerations obtained by the two methods shows a good agreement; however, during the higher accelerations about -18 m/s2, the discrepancies reach values of 10%. From the fit curves to the altitude data during each parabolic manoeuver performed by the aircraft, it was possible to estimate martian and lunar gravity, as well as the values of the acceleration of gravity during the microgravity moments. The analysis presented can also help improve understanding of the concepts of inertial forces and the equivalence between gravity and acceleration.
Alejandro Morales et al 2024 Eur. J. Phys.
Normally, the properties of evanescent optical waves are obtained by developing the Fresnel equations that are expressed in the complex numbers field when the incident angle exceeds the critical angle. Instead of using complex numbers, here we use real functions and the mathematical concept of linear independence to obtain all the properties of the evanescent wave.
Thomas Quick and Johannes Grebe-Ellis 2024 Eur. J. Phys. 45 045301
Lens phenomena, such as caustics, image distortions, and the formation of multiple images, are commonly observed in various refracting geometries, including raindrops, drinking glasses, and transparent vases. In this study, we investigate the ball lens as a representative example to showcase the capabilities of Berry's eye caustic as an optical tool. Unlike the conventional paraxial approximation, the eye caustic enables a comprehensive understanding of image transformations throughout the entire optical space. Through experimental exploration, we establish the relationship between the eye caustic and traditional light caustics. Furthermore, we provide mathematical expressions to describe both the caustic and the image transformations that occur when viewing objects through the ball lens. This approach could be of interest for optics education, as it addresses two fundamental challenges in image formation: overcoming the limitations of the paraxial approximation and recognizing the essential role of the observer in comprehending lens phenomena.
R G Dias et al 2024 Eur. J. Phys.
In this paper, we address the topological characterization of the
wave equation solutions in continuous two-dimensional (2D) periodic geometries with
Neumann or Dirichlet boundary conditions. This characterization is
relevant in the context of 2D vibrating membranes and our approach
allows one to understand the topological behavior recently observed
in acoustic three-dimensional (3D) artificial lattices. In particular, the dependence
of the topological behavior on the experimental positioning of the
coupling channels is explained using simple arguments and a simple
method of construction of an equivalent effective tight-binding Hamiltonian
is presented.
Rod Cross 2024 Eur. J. Phys. 45 035008
A simple experiment is described where a metal ring was rotated in a vertical plane on a horizontal rod. The ring rotated about 100 times before coming to a stop, so the friction force on the ring remained very small. However, measurements of the rotation frequencies of the ring around the rod and around its centre of mass indicated that the ring was sliding rather than rolling, with an unusually low coefficient of sliding friction.