Table of contents

The observation of peculiar light patterns produced by reflection from a water surface perturbed by falling droplets is reported. The phenomenon is analysed in some detail, with a simplified model of a surface wave packet. A simple experiment reproducing the phenomenon in the laboratory is presented, also showing evidence of pattern distortions that were not reported before. An account is given in terms of the vector nature of the law of reflection, also referring to the time constant of the eye.

We study a system of two RLC oscillators coupled through a variable mutual inductance. The system is interesting because it exhibits some peculiar features of coupled oscillators: (i) there are two natural frequencies; (ii) in general, the resonant frequencies do not coincide with the natural frequencies; (iii) the resonant frequencies of both oscillators differ; (iv) for certain choices of parameters, there is only one resonant frequency, instead of the two expected.

This paper describes a historical and pedagogical survey of siphon explanations through the literature of the nineteenth century up to the first half of the twentieth century. An amazing feature is emphasized that this old and humble device has shown some misunderstanding aspects.

Corrections were made to this article on 11 April 2008. Changes were made to equations (5), (6), (7), (8) and (9) on pages 426 and 427. The corrected electronic version is identical to the print version.

Two complementary media, when put together, produce a perfect optical cancellation. We explore a physical recipe to obtain complementary media in two quite different forms and interpret these results in geometric terms.

In this work, we review two formalisms of coherent states for the case of a particle in a magnetic field. We focus our revision on both pioneering (Feldman and Kahn 1970 Phys. Rev. B 1 4584) and recent (Kowalski and Rembieliński 2005 J. Phys. A: Math. Gen.38 8247) formulations of coherent states for this problem. We introduce a general application, the Husimi function, which takes into account collective and environmental effects, and which can be derived in a variety of ways. However, we calculate this function as the expected value of the density operator in a basis of coherent states.

We present a pedagogic approach aimed at modelling electric conduction in semiconductors in order to describe and explain some macroscopic properties, such as the characteristic behaviour of resistance as a function of temperature. A simple model of the band structure is adopted for the generation of electron–hole pairs as well as for the carrier transport in moderate electric fields. The semiconductor behaviour is described by substituting the traditional statistical approach (requiring a deep mathematical background) with microscopic models, based on the Monte Carlo method, in which simple rules applied to microscopic particles and quasi-particles determine the macroscopic properties. We compare measurements of electric properties of matter with 'virtual experiments' built by using some models where the physical concepts can be presented at different formalization levels.

Nanosciences and nanotechnology (NST) currently constitute a major research field all over the world. NST deal with the study of phenomena and manipulation of materials at atomic, molecular and macromolecular scales, where properties differ significantly from those at the larger scale. The properties of materials can be different at the nanoscale for two main reasons: size and quantum effects. In this work, some effects of geometry at the nanoscale are considered: enhanced specific area, adsorption, chemical reactivity. The cases of nanocrystals, fullerenes and carbon nanotubes are studied.

The Stoner–Wohlfarth (SW) model is the simplest model that describes adequately the physics of fine magnetic grains, the magnetization of which can be used in digital magnetic storage (floppies, hard disks and tapes). Magnetic storage density is presently increasing steadily in almost the same way as electronic device size and circuitry are shrinking, and magnetism in general appears as a new contender for many novel computing applications that were considered traditionally beyond its range. Denser storage leads to finer magnetic grains and smaller size leads to magnetic grains so fine that they contain a single magnetic domain, i.e. a region in the material with a well-defined uniform magnetization best described with the mathematics of the SW model.

The continuous growth of computer and sensor technology allows many researchers to develop simple modifications and/or refinements to standard educational experiments, making them more attractive and comprehensible to students and thus increasing their educational impact. In the framework of this approach, the present study proposes an alternative experimental setup, which allows the confirmation of Hagen–Poiseuille's law, governing the flow of real fluids through tubes, a law with numerous important applications in both technology and medicine. In the proposed educational procedure, experimental measurements of fluid outflow are performed with the use of a motion sensor and a suitable computer program, allowing the determination of both the hydrostatic pressure and the flow rate. The dependence of the flow rate on parameters such as viscosity of the fluid, length and radius of the tube and the pressure difference between the ends of the tube are also studied, providing a laboratory activity which is useful and attractive for first year students, especially those of technologically oriented departments.

In this paper we derive an expression for the dynamic electric polarizability of a particle bound by a double delta potential for frequencies below and above the absolute value of the particle's ground state energy. The derived expression will be used to study some of the fundamental features of the system and its representation of real systems. In addition we derive a general expression of the dynamic electric polarizability for a potential of multi-delta functions.

The Kronig–Penney problem is a textbook example for discussing band dispersions and band gap formation in periodic layered media. For example, in photonic crystals, the behaviour of bands next to the band edges is important for further discussions of such effects as inhibited light emission, slow light and negative index of refraction. However, the standard Kronig–Penney equation does not explicitly state the band edge conditions. This paper derives a new solution for the Kronig–Penney problem that explicitly displays the band edge conditions as well as contains all other essential physics of band formation. Therefore, the present exposition should show the student that the band edge conditions are not simply special cases of the familiar Kronig–Penney equation but, instead, are an integral part of the band theory. For the computationally minded student, the new equation is particularly convenient for calculating the positions of closely spaced band edges. The present results can be taught alongside the Kronig–Penney equation in advanced undergraduate or beginning graduate quantum mechanics, solid state theory and photonics courses dealing with wave propagation through periodic layered media.

The energy levels of the helium atom and isoelectronic ions in two dimensions are considered. The difficulties encountered in the analytical evaluation of the perturbative and variational expressions for the ground state, promote an interesting factorization of the inter-electronic interaction, leading to simple expressions for the energy. This expression provides an insight into the general structure and screening effect in the ground state. Some of the considerations are extended to excited states. The dimensionality properties of the energy spectrum of a helium atom, the screening effect and their implications are of significant pedagogical interest in the study of perurbative and variational approaches to analyse the properties of helium atoms, in quantum mechanics and atomic physics, for general graduate students and specialists in atomic physics.

We present a study of the ascending vertical motion of a self-propelled body under a uniform gravitational field suffering the action of two different types of air friction forces: linear on the velocity, which is valid for slowly moving bodies, and quadratic on the velocity. We study the special case where the thrust force is a decreasing function of mass, which corresponds to the exponential mass exhaustion rate. We present in detail the analytical solutions for the equations of motion for the two types of air friction, and briefly present some techniques for solving the related ordinary differential equations. This paper is intended for undergraduate physics teachers and for graduate students.

A historical discussion of the theories which deal with the formation of real images in mirrors and lenses is presented in this paper. Speculations on mirrors appeared as early as Plato. Euclid's, Hero's and Ptolemy's approaches to visual rays are described. The theory on burning mirrors starts with Diocles and later was continued by the Arabs. Al Haytham extensively studied the reflection of light rays on concave mirrors. Huygens tried to find a shorter way to do the calculations. With lenses Kepler gave a new way of finding the position of images by using approximations. Huygens also gave a solution for the shape of a 'perfect' lens. Huygens' principle on waves can be combined with Fermat's principle to explain the formation of images. These theories can be used in education to help students better understand the formation of images, the propagation of waves and the properties of lenses.

This paper focuses on the determination of the final equilibrium state when two ideal gases, isolated from the exterior and starting from preset initial conditions, interact with each other through a piston. Depending on the piston properties, different processes take place and also different sets of equilibrium conditions must be satisfied. Three cases are analysed, namely, when (case 1) the piston is a heat conductor and free to move, (case 2) the piston allows heat conduction but its position is fixed, and (case 3) the piston is free to move but it is adiabatic (so no heat can be exchanged). Cases 1 and 2 have straightforward solutions, but it is shown that case 3 leads to an undeterminable final state. Even though this last situation seems to be strange and difficult, mechanical and thermodynamical analyses are performed. It is shown that the determinability of the final state depends on whether friction is considered or not. Carried out numerically, both analyses provide consistent results and not only do they enable an interesting and useful discussion regarding the concepts of energy, heat, work and entropy, but they also reinforce some ideas which were recently published.

The quantum-mechanical probability densities are compared with the probability densities treated by the theory of random variables. The relevance of their difference for the interpretation of quantum mechanics is commented.

We consider a fundamental quantum mechanical bound-state problem in the form of the quartic-well potential . The analytical transfer matrix method is applied. This yields a quantization condition from which we can calculate the phase contributions and ground-state energy eigenvalues numerically. We also compare the results with those obtained from other typical means popular among physics students, namely the numerical shooting method, perturbation theory and the standard WKB method.

We describe a simple and very inexpensive undergraduate laboratory experiment for fast determination of Young's modulus at moderate temperatures with the aid of a force sensor. A strip-shaped specimen rigidly bolted to the force sensor forms a clamped–free cantilever beam. Placed in a furnace, it is subjected to free-bending vibrations followed by a fast Fourier transform for identifying the resonant frequency, whereby Young's modulus is calculated from the Euler–Bernoulli beam model. Room temperature moduli obtained for a series of diverse industrial materials (stainless steel, copper, aluminium, Perspex, wood and getinax) are in excellent agreement with the available literature data. The temperature dependence of Young's modulus for stainless steel measured over the 300–600 K interval is analysed.

Representations of centrifugal forces are derived in a variety of rotating frames. Although the rotating angle of a point mass relative to an inertial frame is often confused with the rotating angle of a rotating frame relative to the inertial frame, they should be differentiated.

We describe the construction and operation of three low-cost schlieren imaging systems that can be fabricated using surplus optics and 80/20, an aluminium extrusion based construction system. Each system has a different optical configuration. The low cost and ease of construction makes these systems highly suitable for high-school and undergraduate laboratories. Undergraduate students responded enthusiastically to the experience of assembling and operating these systems. This experience also served as an introduction to issues in optical design, helping the students gain an intuition for geometrical optics.

We present an experiment, well adapted for students of introductory optics courses, for the visualization of the impact of spherical aberration in the point spread function of imaging systems. The demonstrations are based on the analogy between the point-spread function of spherically aberrated systems, and the defocused patterns of 1D slit-like screens.

In the traditional statistical mechanics textbooks, the entropy concept is first introduced for the microcanonical ensemble and then extended to the canonical and grand-canonical cases. However, in the authors' experience, this procedure makes it difficult for the student to see the bigger picture and, although quite ingenuous, the subtleness of the demonstrations to pass from the microcanonical to the canonical and grand-canonical ensembles is hard to grasp. In the present work, we adapt the approach used by Schrödinger to introduce the entropy definition for quantum mechanical systems to derive a classical mechanical entropy definition, which is valid for all ensembles and is in complete agreement with the Gibbs entropy. Afterwards, we show how the specific probability densities for the microcanonical and canonical ensembles can be obtained from the system macrostate, the entropy definition and the second law of thermodynamics. After teaching the approach introduced in this paper for several years, we have found that it allows a better understanding of the statistical mechanics foundations. On the other hand, since it demands previous knowledge of thermodynamics and mathematical analysis, in our experience this approach is more adequate for final-year undergraduate and graduate physics students.

We study different ways of determining the mean distance ⟨r_n⟩ between a reference point and its nth neighbour among random points distributed with uniform density in a D-dimensional Euclidean space. First, we present a heuristic method; though this method provides only a crude mathematical result, it shows a simple way of estimating ⟨r_n⟩. Next, we describe two alternative means of deriving the exact expression of ⟨r_n⟩: we review the method using absolute probability and develop an alternative method using conditional probability. Finally, we obtain an approximation to ⟨r_n⟩ from the mean volume between the reference point and its nth neighbour and compare it with the heuristic and exact results.

An evaluation of a course aimed at developing university students' understanding of the nature of scientific measurement and uncertainty is described. The course materials follow the framework for metrology as recommended in the Guide to the Expression of Uncertainty in Measurement (GUM). The evaluation of the course is based on responses to written questionnaires administered to a cohort of 76 first year physics students both pre- and post-instruction, which were interpreted in terms of 'point' or 'set' reasoning. These findings are compared with responses from a control group of 70 students who completed a similar laboratory course apart from the use of traditional approaches to measurement and data analysis. The results suggest that the GUM framework, together with the specific teaching strategies described, provides opportunities for more effective learning of measurement and uncertainty in the introductory laboratory.

We clarify the contribution of Ragusa and Lazo (2004 Eur. J. Phys.25 127–31) to the problem of covariance of electromagnetic momentum as given by Poynting's vector and the associated problem of the 4/3 in the electromagnetic mass. In particular, we note that the authors are not conscious of von Laue's theorem which makes their discussion mostly unnecessary.

We generalize the approach developed by us for the evaluation of the total time derivative of circular integral (Kholmetskii A L et al 2008 Eur. J. Phys.29 L1–4) and derive an explicit expression for the total time derivative of the area integral for a differentiable vector field. On the basis of this expression, we prove for the first time that Faraday's law can be directly obtained from the Maxwell equations , ∇ ⋅ B = 0 and the Lorentz force law.

The thread-between-spaceships problem is analysed both in its 'mild' variant (after some time the ships' acceleration ceases and they coast at the same constant speed, with respect to the lab frame), and in a special case of its 'tough' variant (the ships' acceleration never ceases). It is pointed out that in the special case of the tough variant the thread connecting spaceships may never break, regardless of how close the ships' speed approaches c.

We determine exact expressions for the amplitudes and vibrational eigenmodes of a taut linear string of identical masses that are subjected to a viscous damping that is proportional to their momentum. We first examine the damped displacements of the string in the more familiar limit of a continuous distribution of mass. We then derive a damped version of the linear, one-dimensional wave equation and describe the dispersion curve. Next, we find exact expressions for the damped normal eigenmodes for a finite system. We show that a fraction of these modes have imaginary frequencies that correspond to overdamped oscillations. Next, we show that the remaining modes exhibit underdamped oscillatory behaviour and represent that portion of the dispersion curve graphically. We use methods that are at an intermediate level of presentation.

Simple diffraction arguments can explain blue suns and moons without needing the full complexity of Lorentz–Mie theory. These rare phenomena give striking evidence of the wave nature of light when the size of the scattering particles is close to the wavelength of light, leading to strong diffraction maxima that allow shorter wavelengths to pass in the forward direction and scatter longer wavelengths into the surrounding sky. The result reverses the colours we normally see in the sky caused by scattering from particles small compared to the wavelength.

There is a correction to pp 378 of this paper. See PDF file for details.