Table of contents

Computers are now so common in our everyday life that it is difficult to imagine the computer-free scientific life of the years before the 1980s. And yet, in spite of an unquestionable rise, the use of computers in the realm of education is still in its infancy. This is not a problem with students: for the new generation, the pre-computer age seems as far in the past as the the age of the dinosaurs. It may instead be more a question of teacher attitude. Traditional education is based on centuries of polished concepts and equations, while computers require us to think differently about our method of teaching, and to revise the content accordingly.

Our brains do not work in terms of numbers, but use abstract and visual concepts; hence, communication between computer and man boomed when computers escaped the world of numbers to reach a visual interface. From this time on, computers have generated new knowledge and, more importantly for teaching, new ways to grasp concepts. Therefore, just as real experiments were the starting point for theory, virtual experiments can be used to understand theoretical concepts. But there are important differences. Some of them are fundamental: a virtual experiment may allow for the exploration of length and time scales together with a level of microscopic complexity not directly accessible to conventional experiments. Others are practical: numerical experiments are completely safe, unlike some dangerous but essential laboratory experiments, and are often less expensive. Finally, some numerical approaches are suited only to teaching, as the concept necessary for the physical problem, or its solution, lies beyond the scope of traditional methods. For all these reasons, computers open physics courses to novel concepts, bringing education and research closer. In addition, and this is not a minor point, they respond naturally to the basic pedagogical needs of interactivity, feedback, and individualization of instruction. This is why one can foresee the rapid emergence of computer-assisted education as the legitimate third standard for physics teaching, along with the traditional use of theory and experiment.

The following papers give practical examples of physics (or physics-related) concepts which are, or which could be, used in present student courses. We hope that they will exemplify the use of computers for physics teaching (personal computers in particular), and help to illustrate that 'e-science' is becoming a powerful and indispensable new tool for scientific education.

The pioneering Fermi–Pasta–Ulam (FPU) numerical experiment played a major role in the history of computer simulation because it introduced this concept for the first time. Moreover, it raised a puzzling question which was answered more than 10 years later. After an introduction to this problem, we briefly review its history and then suggest some simple numerical experiments, with the Matlab© code provided, to study various aspects of the 'FPU' problem.

Integration of information and communication technologies (ICTs) in special relativity teaching may bring multiple and complementary methods for introducing either difficult or abstract counterintuitive concepts. This paper describes multimedia content developed at Lyon University to enhance the learning process of undergraduate students. Two categories of animated scenarios have been identified: real experiments and thought experiments. Both typical examples of these scenarios and their impacts on the teaching process are discussed.

Molecular simulations are very convenient to get insight into the structure and thermal behaviour of small atomic clusters. In this paper, we show how the classical Monte Carlo method can be used to tackle both the global optimization problem of molecular structure and the finite temperature properties. Smarter schemes to compute the caloric curves are also illustrated in the example of the histogram reweighting method.

We present known and new applications of pseudo random numbers and of the Metropolis algorithm to phenomena of physical and mechanical interest, such as the search of simple clusters isomers with interactive visualization, or vehicle motion planning. The progression towards complicated problems was used with first-year graduate students who wrote most of the programs presented here. We argue that the use of pseudo random numbers in simulation and extrema research programs in teaching numerical methods in physics allows one to get quick programs and physically meaningful and demonstrative results without recurring to the advanced numerical analysis methods.

Although mainly devoted to the study of equilibrium systems, Monte Carlo simulations are also a helpful tool for investigating out-of-equilibrium dynamics. By studying simple adsorption models that may illustrate a numerical master's degree course, we show how computer simulations can handle some general features of non-equilibrium dynamics, in particular, the sensitivity to initial conditions and the emergence of long relaxation times.

Phase-field models are very attractive in view of their numerical simplicity. With only a few lines of code, one can model complex physical situations such as dendritic growth. From this point of view, they constitute very interesting tools for teaching purposes at graduate level. The main difficulty with these models is in their formulation, which incorporates the physical ingredients in a subtle way. We discuss these approaches on the basis of two examples: dendritic growth and multiphase flows.

The density-dependent distribution function of the orientation of the long axis of rod-like colloidal particles in the liquid crystalline nematic phase is numerically calculated as the solution of a nonlinear integral equation. From this solution, which is obtained by an iterative method that is explained in detail, the free energy, the pressure and the chemical potential can be computed, as well as the nematic order parameter. Moreover, the first-order phase transition from the disordered isotropic fluid phase to the nematic phase is analysed numerically. This study, presented at the level of advanced undergraduates or starting graduate students, not only illustrates the existence of liquid crystalline ordering for nonspherical particles, but also shows explicitly how thermodynamics, structure and symmetry are related.

Physics concepts have often been borrowed and independently developed by other fields of science. In this perspective, a significant example is that of the entropy in information theory. The aim of this paper is to provide a short and pedagogical introduction to the use of data compression techniques for the estimate of the entropy and other relevant quantities in information theory and algorithmic information theory. We consider in particular the LZ77 algorithm as a case study and discuss how a zipper can be used for information extraction.

Some examples are given on how methods of computational statistical physics are applied outside physics.

We developed a highly interactive, multi-windows Java applet which made it possible to simulate and visualize within any platform and internet the Carnot cycle (or engine) in a real-time computer experiment. We extended our previous model and algorithm (Galant et al 2003 Heat Transfer, Newton's Law of Cooling and the Law of Entropy Increase Simulated by the Real-Time Computer Experiments in Java (Lecture Notes in Computer Science vol 2657) pp 45–53, Gall and Kutner 2005 Molecular mechanisms of heat transfer: Debye relaxation versus power-law Physica A 352 347–78) to simulate not only the heat flow but also the macroscopic movement of the piston. Since in reality it is impossible to construct a reversible Carnot engine, the question arises whether it is possible to simulate it at least in a numerical experiment? The positive answer to this question which we found is related to our model and algorithm which make it possible to omit the many-body problem arising when many gas particles simultaneously interact with the mobile piston. As usual, the considerations of phenomenological thermodynamics began with a study of the basic properties of heat engines, hence our approach, besides intrinsic physical significance, is also important from the educational, technological and even environmental points of view.

The double cone ascending an inclined V-rail is a common exhibit used for demonstrating concepts related to centre of mass in introductory physics courses (see http://www.physics.ncsu.edu/pira/). While the conceptual explanation is well known—the widening of the ramp allows the centre of mass of the cone to drop, overbalancing the increase in altitude due to the inclination of the ramp—there remains rich physical content waiting to be extracted through deeper exploration. Such an investigation seems to be absent from the literature. This paper seeks to remedy the omission.

Based upon previous discussions on the structure of compact stars geared towards undergraduate physics students, a real experiment involving two upper-level undergraduate physics students, a beginning physics graduate and two advanced graduate students was conducted. A recent addition to the physics curriculum at Florida State University, The Physics of Stars, sparked quite a few students' interests in the subject matter involving stellar structure. This, coupled with Stars and statistical physics by Balian and Blaizot (1999 Am. J. Phys.67 1189) and Neutron stars for undergraduates by Silbar and Reddy (2004 Am. J. Phys.72 892), is the cornerstone of this small research group who tackled solving the structure equations for compact objects in the summer of 2004. Through the use of a simple finite-difference algorithm coupled to Microsoft Excel and Maple, solutions to the equations for stellar structure are presented in the Newtonian regime appropriate to the physics of white dwarf stars.

Friction is, in general, a very complicated function of velocity. The treatment in textbooks of mechanical systems with friction is limited to the simplest cases, and there are few examples of a canonical approach to these systems. In this work, we discuss some general aspects of systems with friction of the form f = −mλ_nvⁿ and add some examples of their canonical treatment, which we hope will be useful in advanced courses on mechanics. We also discuss a difference in the motion accordingly if the exponent in the friction force is n < 2, or n ⩾ 2.

It is a fact that one of the main roles of the university is to present the core concepts in every discipline. However it is often difficult for the students to understand and comprehend the several concepts taught, because of previous lack in relevant knowledge or due to today's information overload. For example, previous research in the physics department in the Aristotle University of Thessaloniki (Greece) indicated that the graduate students are often not familiar with those factors that determine the Earth's climate, which is a common everyday subject. In this project we describe a multimedia application appropriate for climate simulation experiments (http://dmod.physics.auth.gr/EBMC.html). The application is based on the energy balance model and is appropriate not only for undergraduates, but also for people with a basic scientific knowledge. We have used this model as part of an undergraduate teaching course. The results indicate that it can be a useful educational tool for understanding the factors that determine the Earth's climate.

Thermodynamics is a very general theory, based on fundamental symmetries. It generalizes classical mechanics and incorporates theoretical concepts such as field and field equations. Although all these ingredients are of the highest importance for a scientist, they are not given the attention they perhaps deserve in most undergraduate courses. Nowadays, powerful computers in conjunction with equally powerful software can ease the exploration of the crucial ideas of thermodynamics. The purpose of the present work is to show how the utilization of symbolic computation software can lead to a complementary understanding of thermodynamics. The method was applied to first and second year physics students in the Aristotle University of Thessaloniki (Greece) during the 2002–2003 academic year. The results indicate that symbolic computation software is appropriate not only for enhancing the teaching of the fundamental principles in thermodynamics and their applications, but also for increasing students' motivation for learning.

When the concept of scattering differential cross-section is introduced in classical mechanics textbooks, usually it is first supposed that the target is a fixed, hard sphere. In this paper we calculate the scattering differential cross-section in the case of the hard target being a fixed figure of revolution of any shape. When the target is a paraboloid of revolution, we find the well-known formula corresponding to Rutherford's scattering. In addition, we analyse the inverse problem, i.e. given a differential cross-section, what is the profile of the corresponding hard target?

We analyse the human electrocardiogram with simple nonlinear time series analysis methods that are appropriate for graduate as well as undergraduate courses. In particular, attention is devoted to the notions of determinism and stationarity in physiological data. We emphasize that methods of nonlinear time series analysis can be successfully applied only if the studied data set originates from a deterministic stationary system. After positively establishing the presence of determinism and stationarity in the studied electrocardiogram, we calculate the maximal Lyapunov exponent, thus providing interesting insights into the dynamics of the human heart. Moreover, to facilitate interest and enable the integration of nonlinear time series analysis methods into the curriculum at an early stage of the educational process, we also provide user-friendly programs for each implemented method.

The foundation of non-equilibrium thermodynamics is discussed. The behaviour of thermodynamic systems is described with the help of internal variables. It is shown that the requirement that a thermodynamic system cannot fulfil any work via internal variables is equivalent to the conventional formulation of the second law of thermodynamics. This work has to be included into the balance energy equation (the first law of thermodynamics). These statements, in line with axioms introducing internal variables, can be considered as basic principles of non-equilibrium thermodynamics. The statements do not affect equilibrium thermodynamic functions, but allow one to obtain consistent definitions of non-equilibrium functions. On some examples, it is demonstrated that the developed formalism provides the known results for non-equilibrium functions in a direct way, which encourages us to investigate and apply the formalism further.

We compare the magnetic field at the centre and the self-magnetic flux through a current-carrying circular loop, with those obtained for current-carrying polygons with the same perimeter. As the magnetic field diverges at the position of the wires, we compare the self-fluxes utilizing several regularization procedures. The calculation is best performed utilizing the vector potential, thus highlighting its usefulness in practical applications. Our analysis answers some of the intuition challenges students face when they encounter a related simple textbook example. These results can be applied directly to the determination of mutual inductances in a variety of situations.

We analyse the motion of a swing which is an interesting physical system. Due to the limitation of the conventional method using the Taylor series, an alternative method called the 'two-timing method' is used in order to explore its stability nature (i.e., long-term behaviour) properly. It is shown by the numerical work that the two-timing method supplies a good analytic approximation whose amplitude has an exponential dependence on time. By the two-timing method, the exponent is expressed in terms of the pumping and the frictional force. And we discuss the dependence on the pumping manner of the swing's motion.

We again consider the analysis of a cyclist on a bicycle in motion around a circular track. An expression derived earlier (Horsfield 1998 Eur. J. Phys.19 347–50) for the motion is corrected, by inclusion of two neglected features of a more complete description: (1) the distinction between the velocity of the axles of the wheels, and that of the centre of mass of the system, which is usually located at some height above the axles; (2) the application of the parallel-axes theorem to the 'spin term'. The corresponding two correction terms turn out to be significant, being of the same order in the relevant parameters as the correction term introduced in Horsfield (1998) with respect to the 'lowest-order' formula.

Recent studies of the rotational properties of magic cubes are extended to a corresponding electrical problem. While expecting the dipole moments of magic cubes to vanish, it was a surprise to find their quadrupole moments also vanished. These properties are shown to follow from the RCP (row, column, pillar) symmetry of the orthogonal line sums, namely the semi-magic property of the cubes, which opens up the conclusions to a much wider class of charge distributions.

We extend the standard treatment of the asymmetric infinite square well to include solutions that have zero curvature over part of the well. This type of solution, both within the specific context of the asymmetric infinite square well and within the broader context of bound states of arbitrary piecewise-constant potential energy functions, is not often discussed as part of quantum mechanics texts at any level. We begin by outlining the general mathematical condition in one-dimensional time-independent quantum mechanics for a bound-state wavefunction to have zero curvature over an extended region of space and still be a valid wavefunction. We then briefly review the standard asymmetric infinite square well solutions, focusing on zero-curvature solutions as represented by energy eigenstates in position and momentum space.

Brownian motion appears to be a good subject for investigation at advanced students' laboratory [1]. The paper presents such an investigation carried out in Physics Laboratory II at the Institute of Experimental Physics of Wroclaw University. The experiment has been designed to find viscosity of liquids from Brownian motion phenomenon. Authors use modern technology that helps to proceed with measurements and makes the procedure less time and effort consuming. Discussion of the process of setting up the experiment and the results obtained for three different solutions of glycerin in water are presented. Advantages and disadvantages of the apparatus are pointed out along with descriptions of possible future uses.

A simple and low-cost digital correlation method is proposed to investigate weak photoelectrical signals, using a high-speed photodiode as detector, which is directly connected to a programmably triggered sound card analogue-to-digital converter and a personal computer. Two testing experiments, autocorrelation detection of weak flickering signals from a computer monitor under background of noisy outdoor stray light and cross-correlation measurement of the surface velocity of a motional tape, are performed, showing that the results are reliable and the method is easy to implement.

In earlier studies, it has been demonstrated that cooperative work has a positive influence on physics distance learning at university entrance level. In this paper, it is shown clearly that this teaching method in physics is advantageous for both genders. In fact, the collaboration seems to help male students more than female students to manage the course and to complete it. No difference was seen between the genders either in performance, measured in throughput (productivity) for the course, or in intrinsic motivation. Without cooperative work, the productivity for male students is far lower than for female students.

We generalize a recent derivation of the relativistic expressions for momentum and kinetic energy from the one-dimensional to the three-dimensional case.

The importance of variational calculus is usually not reflected in the basic, undergraduate engineering curriculum. In this paper, we present the result of a pedagogical project aimed at introducing variational calculus at an early stage in the engineering education. This introduction is based on the classical brachistochrone problem and combines both theoretical and experimental analysis.

The discretized Schrödinger equation is most often used to solve one-dimensional quantum mechanics problems numerically. While it has been recognized for some time that this equation is equivalent to a simple tight-binding model and that the discretization imposes an underlying bandstructure unlike free-space quantum mechanics on the problem, the physical implications of this equivalence largely have been unappreciated and the pedagogical advantages accruing from presenting the problem as one of solid-state physics (and not numerics) remain generally unexplored. This is especially true for the analytically solvable discretized finite square well presented here. There are profound differences in the physics of this model and its continuous-space counterpart which are direct consequences of the imposed bandstructure. For example, in the discrete model the number of bound states plus transmission resonances equals the number of atoms in the quantum well.

The motion of a charged particle in a magnetic mirror device is compared to the motion of a bead spiralling into a smooth hollow cone. The complete solution of the motion of the bead is derived and physical aspects of the solution obtained are discussed. Similarities between the motion of the bead and that of a charged particle in a magnetic mirror device are pointed out. The effort is primarily aimed at enhancing the physical understanding of the mechanics of a charged particle in a magnetic mirror device and secondarily at proposing an equivalent mechanics problem, devoid of any electro-dynamical aspect but still possessing all the interesting features of the original problem. The proposed problem can be taught at upper-division undergraduate level.

This paper discloses two important discoveries. These are: (i) discovery of ambiguity in the well-established laws of reflection and refraction of light which have been in regular use for many years, and (ii) discovery of generalized vectorial laws of reflection and refraction of light. The existing definitions of angle of incidence, angle of reflection and angle of refraction are considered first. Each of these definitions is found to be ambiguous, not in compliance with the fundamental definition of angle in geometry. Two typical questions (one in the case of reflection and the other for refraction) have been addressed, which cannot be dealt with by using the existing laws of reflection and refraction of light. Thus, the existing laws of reflection and refraction of light seem to be ambiguous in respect of generality and their validity in a broad sense is questionable. With a view to removing the ambiguities, proper definitions of the above three angles are given first and then the statement of the generalized vectorial law of reflection (as well as that of refraction) has been offered.

Two alternative expressions exist for the diffusive flux in inhomogeneous systems: Fick's law and the Fokker–Planck law. Here we re-examine the origin of these expressions and perform numerical and physical experiments to shed light on this duality. We conclude that in general the Fokker–Planck expression should be conceded preference, in spite of the fact that Fick's law seems to be more popular.

Here, first, the static formulation of tidal forces with completely spherical symmetry is reviewed (a differential calculus-based method is additionally introduced for readers with more familiarity with/interest in mathematical tools). Then, the result is generalized to the tidal force of the Moon acting on the Earth by considering the rotational dynamics and the (real) oblate spheroidal shape of the Earth.

The empirical laws of chemical kinetics are studied from a microscopic point of view. An analysis based on elementary probability theory and combinatorics is enough to explain the kinetics law observed in experiments. Thus, an out-of-equilibrium system may be examined with tools available to a student beginning a study of statistical physics.

CCD (charged coupling device) detectors are made of a large number of pixels, forming a two-dimensional lattice. The dimensions of pixels are in the µm range. So, when irradiated by visible light, the lattice gives rise to diffraction. This self-diffraction may be studied experimentally. The CCD camera is illuminated by a HeNe laser beam. Diffraction patterns are either observed by looking at the light reflected by the camera or measured when looking at the signal given by the camera itself. This may serve as an introduction to usual diffraction by a lattice or as an analogy with two-dimensional electron diffraction (LEED).

We make remarks on the paper by Margaritondo (2005 Eur. J. Phys.26 401).