Table of contents

An introduction to quantum mechanics based on the sum-over-paths (SOP) method originated by Richard P Feynman and developed by E F Taylor and coworkers is presented. The Einstein–Brillouin–Keller (EBK) semiclassical quantization rules are obtained following the SOP approach for bounded systems, and a general approach to the calculation of propagation amplitude is discussed for unbounded systems. These semiclassical results are obtained when the SOP is limited to the trajectories classically allowed. EBK semiclassical quantization and the topological Maslov index are used to deduce the correct quantum mechanical results for systems which live in a two-dimensional world as quantum dots and quantum rings. In the latter systems, the semiclassical propagation amplitude is used to discuss the Aharonov–Bohm effect. The development involves only elementary calculus and also provides a theoretical introduction to the quantum nature of low-dimensional nanostructures.

This paper deals with subtleties and misunderstandings regarding the Clausius relation. We start by demonstrating the relation in a new and simple way, explaining clearly the assumptions made and the extent of its validity. Then follows a detailed discussion of some confusions and mistakes often found in the literature. The addressed points include the issue of temperature in the Clausius relation and closely related concepts, such as heat, reversibility and reservoir. The ideas presented in this study are primarily intended for graduate students and teachers, and may also be of interest to undergraduate students with a solid background in thermodynamics.

The generalized second-order partial derivatives of 1/r, where r is the radial distance in three dimensions (3D), are obtained using a result of the potential theory of classical analysis. Some non-spherical-regularization alternatives to the standard spherical-regularization expression for the derivatives are derived. The utility of a spheroidal-regularization expression is illustrated on an example from classical electrodynamics.

When two gravitating bodies reside in a material medium, Newton's law of universal gravitation must be modified to account for the presence of the medium. A modified expression of Newton's law is known in the literature, but lacks a clear connection with existing gravitational theory. Newton's law in the presence of a homogeneous material medium is herein derived on the basis of classical, Newtonian gravitational theory and by a general relativistic use of Archimedes' principle. It is envisioned that the techniques presented herein will be most useful to graduate students and those undergraduate students having prior experience with vector analysis and potential theory.

Thermoacoustic instability can appear in any thermal device when the unsteady heat transfer is favourably coupled with the fluctuations of acoustic pressure. In this paper, we present a project type of physical measuring and modelling task; the aim of our project is to help our students increase their knowledge of thermoacoustics. Our paper proposes several experiments and describes some tools' setups that are easy to obtain and work with. Free software is offered to analyse the signals with a personal computer. In this paper, the basis of standing wave theory and the tie between thermodynamics and acoustical oscillations are also discussed; some devices and technical applications of thermoacoustic oscillations are presented. The objective of this paper is to present the theory of frequency shifting of thermoacoustic oscillations as well. The frequencies of the acoustic modes in the excited state are of interest for practical purposes; the differences between the calculated and the measured values of these frequencies are shown. The behaviour of the properties of the exited modes shows the complexity of the real thermoacoustic systems; the mathematical modelling intended to simulate the effect of frequency shifting is observed in tests. We think that these experiments can be implemented in physics courses on thermodynamics for graduates or specialized courses for undergraduates.

GEANT4 simulations of the processes affecting the transport and collection of optical photons generated inside a scintillation detector were carried out, with the aim to complement the educational material offered by textbooks to third-year physics undergraduates. Two typical situations were considered: a long scintillator strip with and without a wavelength shifter fibre embedded. Various aspects were studied, including the reflection of photons on realistic coating interfaces, their absorption in the bulk material and the effect of the wavelength shifter fibres. Examples of experimental results obtained with such a detector setup, as actually obtained within a third-year laboratory activity, are also included.

The research on the cooling law began with an article by Newton published in 1701. Later, many studies were performed by other scientists confirming or confuting Newton's law. This paper presents a description and an interpretation of Newton's article, provides a short overview of the research conducted on the topic during the 18th century, and discusses the relationships between the research on cooling laws and the definition of a temperature scale, as it was treated in Newton's article and in the work of Dalton, including Dalton's search for the absolute zero of temperature. It is shown that these scientists considered the exponential cooling law as a fundamental principle rather than a conjecture to be tested by means of experiments. The faith in the simplicity of natural laws and the spontaneous idea of proportionality between cause and effect seem to have strongly influenced Newton and Dalton. The topic is developed in a way that can be suitable for both undergraduate students and general physicists.

The question whether to walk slowly or to run when it starts raining in order to stay as dry as possible has been considered for many years—and with different results, depending on the assumptions made and the mathematical descriptions for the situation. Because of the practical meaning for real life and the inconsistent results depending on the chosen parameters, this problem is well suited to undergraduate students learning to decide which parameters are important and choosing reasonable values to describe a physical problem. Dealing with physical parameters is still useful at university level, as students do not always recognize the connection between pure numbers and their qualitative and quantitative influence on a physical problem. This paper presents an intuitive approach which offers the additional advantage of being more detailed, allowing for more parameters to be tested than the simple models proposed in most other publications.

In this paper a simplified two-parameter model of the greenhouse effect on the Earth is developed, starting from the well known two-layer model. It allows both the analysis of the temperatures of the inner planets, by focusing on the role of the greenhouse effect, and a comparison between the temperatures the planets should have in the absence of greenhouse effect and their actual ones. It may also be used to predict the average temperature of the Earth surface in the future, depending on the variations of the concentration of greenhouse gases in the atmosphere due to human activities. This model can promote an elementary understanding of global warming since it allows a simple formalization of the energy balance for the Earth in the stationary condition, in the presence of greenhouse gases. For these reasons it can be introduced in courses for undergraduate physics students and for teacher preparation.

In this paper we give a general solution to the problem of the damped harmonic oscillator under the influence of an arbitrary time-dependent external force. We employ simple methods accessible for beginners and useful for undergraduate students and professors in an introductory course of mechanics.

The dynamics of a cylinder rolling on a horizontal plane acted on by an external force applied at an arbitrary angle is studied with emphasis on the directions of the acceleration of the centre-of-mass and the angular acceleration of the body. If rolling occurs without slipping, there is a relationship between the directions of these accelerations. If the linear acceleration points to the right, then the angular acceleration is clockwise. On the other hand, if it points to the left, then the angular acceleration is counterclockwise. In contrast, if rolling and slipping occurs, the direction of the linear acceleration does not determine the direction of the angular acceleration. For example, the linear acceleration may point to the right and the angular acceleration clockwise or counterclockwise depending on the external force orientation and point of application.

It is a widespread misunderstanding in introductory physics courses that the motion of rolling bodies in general can be calculated using the point of contact as a reference point when equating the rate of change of angular momentum to the torque. In this paper I discuss in general two correct rules to be used instead, in order to derive the equation of motion of rolling bodies, taking moments about the point of contact. I also clarify that the point of contact either can be reckoned the fixed point on the rolling body instantly at the point of contact or the geometrical point defined as the point of contact at any time. Altogether this gives four different ways of deriving the equation of motion with some of them being easier than others depending on the case under consideration. The four different methods are as an illustration applied to a case presented recently by Turner and Turner (2010 Am. J. Phys.78 905–7).

The first-order equation relating object and image location for a mirror of arbitrary conic-sectional shape is derived. It is also shown that the parabolic reflecting surface is the only one free of aberration and only in the limiting case of distant sources.

An anharmonic solution to the differential equation describing the oscillations of a simple pendulum at large angles is discussed. The solution is expressed in terms of functions not involving the Jacobi elliptic functions. In the derivation, a sinusoidal expression, including a linear and a Fourier sine series in the argument, has been applied. The coefficients of the Fourier series are found, and it turns out that they are rapidly decreasing. The solution is found to be very close to the exact solution. In the small-angle regime the solution formula for the displacement becomes equivalent to the solution describing the linear pendulum. During the analysis a new formula for the period of the simple pendulum is found, which is more accurate as compared to most previously published results. The formula gives the period with an accuracy better than 0.0004% for angles up to , and within 0.025% for angles up to radians. In the small-angle regime the formula becomes equivalent to the result for the period of the linear pendulum. The present derivation of an anharmonic solution to the equation of motion describing a simple pendulum, as well as the derivation of a new expression for the pendulum period, is obtained in terms of elementary functions. It is believed to give valuable insight into a better understanding of the behaviour of the simple pendulum in undergraduate courses on classical mechanics.

In this paper we present an experimental strategy to measure the micro power dissipation due to Foucault 'eddy' currents in a copper cylinder rolling on two parallel conductive rails in the presence of a magnetic field. Foucault power dissipation is obtained from kinematical measurements carried out by using a common PC webcam and video analysis done by means of software tools freely available within Windows operating system (Paint and Movie Maker). The proposed method allows us to experimentally discern the contribution to dissipation due to the velocity-independent rolling friction from that owed to the viscous-like friction emerging from complex electrodynamic interactions among eddy currents and the external magnetic field. In this way a microdissipation of some tens of µW is measured. The easily reproducible experimental setup, the simple implementation of data analysis and the discussion on various experimental approaches and strategies make the proposed activity highly significant for university undergraduates, since involved crucial skills can be efficiently strengthened.

In recent years there has been a considerable increase in the publishing of textbooks and monographs covering what was formerly known as random or irregular deterministic motion, now referred to as deterministic chaos. There is still substantial interest in a matter that is included in many graduate and even undergraduate courses on classical mechanics. Based on the Hamiltonian formalism, the main objective of this paper is to provide, from the physicist's point of view, an overall and intuitive review of this broad subject (with some emphasis on the Kolmogorov–Arnold–Moser theorem and the stability of planetary motions) which may be useful to both students and instructors.

Problems involving polarized plane waves and currents on sheets perpendicular to the wavevector involve only one component of the fields, so it is possible to discuss electrodynamics in one dimension. Taking for simplicity linearly polarized sinusoidal waves, we can derive the field emitted by currents (analogous to dipole radiation in three dimensions) and reflection and transmission (analogous to Thomson scattering). Some aspects of the results are not intuitive, and for a physical understanding we need to see the problem in three dimensions. Eventually we apply these results to a linear model of the sheet, and we discuss the limit from a thick sheet.

We described a simple idea for experimental verification of the uncertainty principle for light waves. We used a single-slit diffraction of a laser beam for measuring the angular width of zero-order diffraction maximum and obtained the corresponding wave number uncertainty. We will assume that the uncertainty in position is the slit width. For the acquisition of the experimental data and their further analysis, we used a computer. Because of its simplicity this experiment is very suitable for demonstration, as well as for a quantitative exercise at universities and final year of high school studies.

In this paper, an analytical solution for the differential equation of the simple but nonlinear pendulum is derived. This solution is valid for any time and is not limited to any special initial instance or initial values. Moreover, this solution holds if the pendulum swings over or not. The method of approach is based on Jacobi elliptic functions and starts with the solution of a pendulum that swings over. Due to a meticulous sign correction term, this solution is also valid if the pendulum does not swing over.

We present Edaq530, a low-cost, compact and easy-to-use digital measurement solution consisting of a thumb-sized USB-to-sensor interface and measurement software. The solution is fully open-source, our aim being to provide a viable alternative to professional solutions. Our main focus in designing Edaq530 has been versatility and transparency. In this paper, we shall introduce the capabilities of Edaq530, complement it by showing a few sample experiments, and discuss the feedback we have received in the course of a teacher training workshop in which the participants received personal copies of Edaq530 and later made reports on how they could utilize Edaq530 in their teaching.

We present a laboratory experiment that allows undergraduate or graduate students to get introduced to colloidal crystal research concepts in an interesting way. Moreover, such experiments and studies can also be useful in the field of crystallography or solid-state physics. The work concerns the growth of colloidal crystal thin films obtained from the crystallization of a latex colloidal solution in a wedge cell. Depending on the thickness of the sample, microcrystals with different structures and orientation are obtained. Colloidal arrangements are studied by scanning electronic microscopy images of the top and edge views of several areas of the crystals.

Unlike other standard equations in introductory classical mechanics, the Bernoulli equation is not Galilean invariant. The explanation is that, in a reference frame moving with respect to constrictions or obstacles, those surfaces do work on the fluid, constituting an extra term that needs to be included in the work–energy calculation. A quantitative example is presented here for a horizontal tapered pipe. A frame-independent expression for the pressure drop in the pipe is obtained. The concepts discussed in this paper are accessible to introductory undergraduate physics majors.

This study focuses on students' conceptions of electric and magnetic fields at university level and of the interrelations between them. A total of 33 students participated in a paper and pencil test after the completion of first-year electricity and second-year electromagnetism courses. The conceptions were investigated in the contexts of a charged particle, a charging capacitor, electromagnetic induction, and an electromagnetic plane wave. The results show that students do not include the vector nature of field quantities in their reasoning. In addition, learning the interactions between electric and magnetic fields is very challenging for university students. Even after completing relevant studies, the students had difficulty in recognizing the symmetry relations between the electric and magnetic fields in all of the areas under study. Consequently, it appears that the concepts of electric and magnetic fields and their interrelations should be emphasized not only in the context of electromagnetism but also in other relevant areas such as electricity and optics. Explicit recommendations are presented in the discussion section at the end of this paper.

By using the symmetry and time-independence properties of Schwarzschild spacetime it is demonstrated that an energy conservation law may be expressed in terms of local velocity. From this form three important results may be derived very concisely. This highlights analogies and differences between relativistic and classical approaches to mechanics and serves as an illustration of the power that the use of particular expressions may provide in solving physical problems.

Single and double electromagnetically induced transparencies (EIT) in a medium, consisting of four-level atoms in the inverted-Y configuration, are discussed using mechanical and electrical analogies. A three-coupled spring–mass system subject to damping and driven by an external force is used to represent the four-level atom mechanically. The equations of motion of this system are solved analytically, which revealed single and double EIT. On the other hand, three coupled RLC circuits are used, as the electrical analogue, to explore and experimentally demonstrate single and double EIT. The simplicity of these two models makes this experiment appropriate for undergraduate students and easy to incorporate into a college physics laboratory.

We present a consistent series of activities, including experiments and basic computational studies, investigating the shape and optical properties of water drops in connection with novel technological devices. Most of the work can be carried out with simple teaching equipment and is well suited to undergraduate students. Firstly, we show how the mass variations of a sessile drop can be used to control its curvature and hence to produce lenses with tunable focal distance. Alternatively, the shape of the drop can be varied using electrowetting on dielectric (EWOD). We propose a simple pedagogical approach to this phenomenon in connection with historical electrostatic apparatus. A detailed process for the preparation of an EWOD device is given, together with a focimetric method allowing the analysis of electrowetting effects in practical exercises. Finally, the manipulations of a commercialized variable focus lens illustrate that EWOD is at the heart of most recent technological developments, making practical work in optics more attractive than traditional exercises using conventional lenses.

FisL@bs is a network of remote and virtual laboratories for physics university education via the Internet that offers students the possibility of performing hands-on experiments in different fields of physics in two ways: simulation and real remote operation. This paper gives a detailed account of a novel way in physics in which distance learning students can gain practical experience autonomously. FisL@bs uses the same structure as AutomatL@bs, a network of virtual and remote laboratories for learning/teaching of control engineering, which has been in operation for four years. Students can experiment with the laboratories offered using an Internet connection and a Java-compatible web browser. This paper, specially intended for university educators but easily comprehensible even for undergraduate students, explains how the portal works and the hardware and software tools used to create it. In addition, it also describes two physics experiments already available: spring elasticity and the laws of reflection and refraction.

The surface plasmon wave is a surface wave confined at the interface between a dielectric and a metal. The excitation of the surface plasmon resonance (SPR) on a gold thin film is discussed within the Kretschmann configuration, where the coupling with the excitation light is achieved by means of a prism in total reflection. The electromagnetic principles are detailed and a simple experimental setup is described that can be used for laboratory experiments for senior students in the third or fourth year of university. This experiment allows accurate determination of the angle of plasmon extinction and discussion of the principles of biosensors based on the SPR. A slight modification of the setup allows the investigation of the dependence of SPR on wavelength and illustrates the damping of SPR due to its coupling with the interband transitions of the gold thin film.

Transparent prism foil is part of a backlight system in LCD monitors that are widely used today. This paper describes the optical properties of the prism foil and several pedagogical applications suitable for undergraduate introductory physics level. Examples include experiments that employ refraction, total internal reflection, diffraction and image formation in a nontrivial way and are therefore particularly useful for active learning strategies.

A sticky capture thread from the spiral element of spider orb-webs is formed of almost regularly spaced droplets that surround a supporting axial fibre. From the perspective of physical optics it represents a periodic linear array of scattering elements that acts as a diffraction grating. This is a novel aspect, which is of vital importance for the understanding of the overall scattering pattern. To demonstrate its significance, we present our experimental findings and compare them with results of a simplified model.

In this paper the method of converting mechanical work into electrical energy with the participation of a preliminarily charged condenser while the electrodes are sliding in it is presented. Using this method, we can obtain a considerable increase of converted electrical power, depending on the initial energy of the charged condenser, distance between the electrodes and frequency of conversion. Exemplary time–voltage, time–current and instantaneous electrical power (for determined system work conditions) dependences are described in the paper. A recommended resistor resistance value is given to obtain the maximal system efficiency at the determined condenser capacity and assumed conversion frequency. The method can be used to convert renewable wind energy into electric energy in an ecologically pure manner.

We give a lower bound for the energy of a quantum particle in the infinite square well. We show that the bound is exact and identify the well-known element that fulfils the equality. Our approach is not directly dependent on the Schrödinger equation and illustrates an example where the wavefunction is obtained directly by energy minimization. The derivation presented can serve as an example of a variational method in an undergraduate level university course in quantum mechanics.

We show that a remark made by Hernández-Saldaña in 2010 (Eur. J. Phys.31 1319) concerning the validity of an expression first presented by us in 2006 (Proc. 17th Biennial Congress of the Australian Institute of Physics Paper 27) for the optimal angle of projection for greatest forward skew in the trajectory of a projectile launched in a linear resisting medium is in error. We also draw attention to an earlier treatment (2006 Int. J. Math. Educ. Sci. Technol.37 411) of the locus of apexes for such a projectile. When expressed in Cartesian form, the locus can be written in terms of the now familiar, though less common, secondary real branch of the Lambert W function.

The author replies to a comment made on a previous paper (Hernández-Saldaña 2010 Eur. J. Phys.31 1319).

See PDF file for details.