Table of contents

We present a heuristic argument to deduce the non-intuitive Thomas factor for an atomic electron revolving in its orbit about the nucleus. The argument, which is simple, pedagogical and accessible to physics beginners, uses elementary notions of classical electromagnetism rather than the advanced relativity concepts which are needed in any standard textbook derivation.

While the energy bands of solids are often thought of as continuous functions of the wavevector, k, they are in fact discrete functions, due to the periodic boundary conditions applied over a finite number of primitive cells. The traditional approach enforces periodicity over a volume containing N_i primitive unit cells along the direction of the primitive lattice vector a_i. While this method yields a simple formula for the allowed k, it can be problematic computer programs for lattices such as face-centred cubic (FCC) where the boundary faces of the primitive cell are not orthogonal. The fact that k is discrete is of critical importance for supercell calculations since they include only a finite number of unit cells, which determines the number of wavevectors, and have a given geometry, which determines their spacing. Rectangular supercells, with the faces orthogonal to the Cartesian axes, are computationally simplest but are not commensurate with the FCC unit cell, so that the traditional approach for determining the allowed k-values is no longer useful. Here, we present a simple method for finding the allowed k-values when periodic boundary conditions are applied over a rectangular supercell, answering the question in both its practical and pedagogical aspects.

A didactic description of the thermodynamic properties of classical spin systems is given in terms of their quantum counterpart in the Hamiltonian limit. Emphasis is on the construction of the relevant Hamiltonian and the calculation of thermal averages is explicitly done in the case of small systems described, in Hamiltonian field theory, by small matrices. The targeted students are those of a graduate statistical physics course.

It is hard to understand spin-one-half fields without reading Weinberg. This paper is a pedagogical footnote to his formalism with an emphasis on the boost matrix, spinors and Majorana fields.

In the standard classical mechanics textbooks used at undergraduate and graduate levels, no attention is paid to the dimensional aspects of the Kepler–Coulomb problem. We have shown that the orbits of the n-dimensional classical Kepler–Coulomb problem are the usual conic sections in a fixed two-dimensional subspace and the Kepler laws with their well-known forms are valid independent of dimension. The basic characteristics of motion in a central force field are also established in an arbitrary dimension. The approach followed is easily accessible to late undergraduate and recent graduate students.

Introductory courses covering modern physics sometimes introduce some elementary ideas from general relativity, though the idea of a geodesic is generally limited to shortest Euclidean length on a curved surface of two spatial dimensions rather than extremal aging in spacetime. It is shown that Epstein charts provide a simple geometric picture of geodesics in one space and one time dimension and that for a hypothetical uniform gravitational field, geodesics are straight lines on a planar diagram. This means that the properties of geodesics in a uniform field can be calculated with only a knowledge of elementary geometry and trigonometry, thus making the calculation of some basic results of general relativity accessible to students even in an algebra-based survey course on physics.

It is well established that the presence of wind is crucial for the practice of numerous outdoor sports. Among them, the case of cycling competition is especially important because the speeds that are developed are perfectly comparable with moderate or strong winds, giving rise to great variability in race times. Conversely, the sidewind seems to behave as neutral in the race, and nevertheless all the cyclists know the difficulties that it causes. In this brief paper we show that the sidewind also produces an appreciable braking as a consequence of the quadratic dependence of the aerodynamic drag force on the air speed. Our work presents an interesting problem of mechanics and physics of fluids, which may be suitable as a task for intermediate level students in university courses.

We analyse a model of a swing where the swing is regarded as a pendulum with a variable length under the influence of friction. Related to the optimal control of the swing, we discuss the various pumping manners and the effects of their properties such as the pumping frequency, pumping phase and pumping amplitude. The optimal pumping manner can be the step pumping, where sitting (standing) occurs when the swing goes from the highest (lowest) position to the lowest (highest) position with as large a pumping amplitude as possible.

The potentials and fields of hyperbolic and hyperboloidal conductors are available analytically. A detailed comparison of the two-dimensional and three-dimensional problems shows strong similarities, but also interesting differences. The electric field near a hyperboloidal needle is stronger (ceteris paribus) than near a hyperbolic blade, and dies off faster. The field at the hyperbolic conductor varies as the 1/3 power of the local curvature. At the hyperboloid conductor the field varies as the 1/4 power of the local Gaussian curvature (which is the product of the two principal curvatures).

The use of a novel thermoacoustic transducer facilitates the study of ultrasonic fields; with this aim, a description of its simple construction is given. In particular, an experiment with a spherically curved source is presented, where the theoretical acoustic pressure obtained by a numerical approximation shows, by comparing with the experimental results, the importance of taking into consideration the phase of the radiated sound on the field produced by focusing sources. The type of thermoacoustic transducer presented in this paper radiates sound in the audible and ultrasonic regions, without resonances, up to at least 100 kHz. These features, along with their ease of construction, allow the use of large, planar or curved, acoustic radiators in laboratory experiments.

A straightforward derivation of relativistic expressions for the mechanical momentum, kinetic and total energies, and mass–energy equivalence (including potential energy) which does not require any knowledge of the energy–momentum relation for electromagnetic waves or consideration of elastic collisions, but is directly based on Newton's second law and Lorentz's transformations, is presented in this paper. The existence of an invariant force is shown to be important for the validity of the relativistic mechanics.

The pattern embossed on the back of an oriental magic mirror appears in the patch of light projected onto a screen from its apparently featureless reflecting surface. In reality, the embossed pattern is reproduced in low relief on the front, and analysis shows that the projected image results from pre-focal ray deviation. In this interesting regime of geometrical optics, the image intensity is given simply by the Laplacian of the height function of the relief. For patterns consisting of steps, this predicts a characteristic effect, confirmed by observation: the image of each step exhibits a bright line on the low side and a dark line on the high side. Laplacian-image analysis of a magic-mirror image indicates that steps on the reflecting surface are about 400 nm high and laterally smoothed by about 0.5 mm.

How long does it take to boil an egg? Theoretical prediction, based on a simple adaptation of the solution to the exact thermal diffusion equation for a sphere, is consistent with experiments. The experimental data are also used to estimate an average value for the thermal diffusivity of an egg.

A great deal of insight can be gained from the analysis of coupled masses connected to springs in order to understand better the origin of band gaps in physical systems. The approach is based on the application of the superposition principle for finding the general solution in simple mechanical systems involving functions, which vary periodically with time. Graphs show that sums of periodic functions oscillating at different frequencies lead to an exchange of energy from one oscillator to another in a simple mechanical system of three objects connected by identical springs. A system of a large number of masses connected to springs having the same spring constant K is then considered and compared with a system in which the spring constants alternate from K to another value G when connecting one mass to another. Using the results found from the mechanical systems, an analogy of charge oscillations excited on both uniform and corrugated surfaces is presented. The results obtained attempt to expand understanding of the origin of the band gap occurring in some systems involving periodic motions.

We point out a logical possibility, based on special relativity, that the validity of Einstein's 1905 generalization, Δm = ΔE_rest/c², might depend on how the energy of a body is defined in special relativity. The analysis is another illustration of the importance of Ockham's razor in special relativity.

In this work, natural convection heat transfer in vertical channels is experimentally investigated by applying different optical techniques, namely holographic interferometry and schlieren. Both these techniques are based on the temperature dependence of the air refractive index but they detect different optical quantities and their use involves different instrumentation and optical components. Optical methods, non-intrusive in nature, are particularly suitable for the visualization of flow and thermal fields as witnessed by their increasing use in a range of scientific and engineering disciplines; for this reason, the introduction of these experimental tools into a laboratory course can be of high value. Physics and engineering students can get familiarized with optical techniques, grasp the basics of thermal phenomena, usually elusive, which can be more easily understood if they are made visible, and begin to master digital image analysis, a key skill in laboratory activities. A didactic description of holographic interferometry and schlieren is provided and experimental results obtained for vertical, smooth and rib-roughened channels with asymmetrical heating are presented. A comparison between distributions of the local heat transfer coefficient (or its dimensionless counterpart, the Nusselt number) revealed good agreement between the results separately obtained by the two techniques, thus proving their suitability for investigating free convection heat transfer in channels.

A relativistic projectile motion in a vacuum is examined by means of elementary consequences of special relativity. Exact analytical expressions were found for the kinematics variables using basic mathematical tools. The trajectory equation was established and the area under the trajectory traversed by the relativistic projectile was determined. It was found that, unlike non-relativistic projectile motion, the launching angles that maximize both the horizontal range as well as the area under the trajectory are functions of the initial speed. It is anticipated that this paper will be consistent with the intuition of students and serve as a resource for further problems usually encountered in the special theory of relativity.

Algebraic, numerical and conceptual problems in a recently published paper by Dabaghian and Jensen (2005 Eur. J. Phys.26 423–439) are discussed. A convergent scheme for summing periodic orbits in the tunnelling regime is suggested.

We provide a short response to Blümel's comment on our paper (Dabaghian and Jensen 2005 Eur. J. Phys.26 423), concerning the periodic orbit summation techniques in the tunnelling regime.