Table of contents

The connection between quantum zero point fluctuations and a density maximum in water and in liquid He⁴ has recently been established. Here we present a description of a simple and rapid method of determining the temperatures at which maximum densities in water and aqueous solutions occur. The technique is such as to allow experiments to be carried out in one session of an undergraduate laboratory thereby introducing students to the concept of quantum zero point energy.

The force between two parallel current-carrying wires is investigated in the rest frames of the ions and the electrons. A straightforward Lorentz transformation shows that what appears as a purely magnetostatic force in the ion frame appears as a combined magnetostatic and electrostatic force in the electron frame. The derivation makes use of a reasonably well-known problem of a charged particle moving parallel to a current-carrying wire, which is often used to illustrate that what appears as a purely electrostatic force in one frame appears as a purely magnetostatic force in another. This paper, aimed at undergraduate electromagnetism and special relativity courses, serves to dispel the notion that this makes either the electrostatic or the magnetostatic force redundant.

An analysis of cylinders rotating at relativistic speeds is considered from the point of view of observers living on the cylinders and from the point of view of observers in an inertial frame at rest with respect to translational motion of the cylinder. All of the observers measure time and distance using the recently introduced floor mirrored Einstein–Langevin light clock (FMEL). Two 'obvious' choices for synchronizing clocks, the traditional Einstein method and the well-known 'global' method, will be compared. It is shown that Selleri's paradox does not actually illustrate a contradiction, and it is shown that the Einstein method seems to allow apparent time ordering violations of causality. The global method leads to a disagreement with those in the inertial frame about velocities, and to a non-isotropic value for the speed of light. Ehrenfest's paradox is explained from the point of view of observers using each choice of synchronization.

X-ray fluorescence is a non-destructive technique that allows elemental composition analysis. In this paper we describe a prescription to obtain the elemental composition of homogeneous coins, like 50 cent Euro coins, and how to get the quantitative proportions of each element with the help of Monte Carlo simulation. Undergraduate students can carry out both the laboratory work and the simulation, thus exploiting an analysis technique widely employed in contemporary physics.

In an inhomogeneous magnetized plasma the transport of energy and particles perpendicular to the magnetic field is in general mainly caused by quasi two-dimensional turbulent fluid mixing. The physics of turbulence and structure formation is of ubiquitous importance to every magnetically confined laboratory plasma for experimental or industrial application. Specifically, high-temperature plasmas for fusion energy research are also dominated by the properties of this turbulent transport. Self-organization of turbulent vortices to mesoscopic structures like zonal flows is related to the formation of transport barriers that can significantly enhance the confinement of a fusion plasma. This subject, of great importance in research, is rarely touched on in introductory plasma physics or continuum dynamics courses. Here a brief tutorial on 2D plasma turbulence is presented as an introduction to the field, appropriate for inclusion in undergraduate and graduate courses.

It is shown how the same physically appealing method can be applied to find analytic solutions for two difficult and apparently unrelated problems in optics and electrostatics. They are: (i) the diffraction of a plane wave at a perfectly conducting thin half-plane and (ii) the electrostatic field associated with a parallel array of stripes held at opposite potentials and lying in a half-plane. In the latter case, the solution of the problem is essential in order to calculate the electron optical phase shift experienced by the electron beam in electron microscopy experiments, where the model mimics an array of reverse biased p–n junctions. This paper is intended, in particular, for the undergraduate student who has completed his programme of calculus and electromagnetism, but it can also be of interest to the general physicist.

An expression is found that relates the initial and final volumes and temperatures for any adiabatic process. It is given in terms of a parameter r that smoothly interpolates between a free adiabatic expansion (r = 0) and a quasi-static one (r = 1). The parameter has to be evaluated numerically, but an approximate expression is given.

This paper describes a computational exercise at undergraduate level that demonstrates the employment of Monte Carlo simulation to study the conformational statistics of flexible polymer chains, and to predict solution properties. Three simple chain models, including excluded volume interactions, have been implemented in a public-domain computer program that is the tool on which this exercise is based. The first stage considers fundamental aspects of polymer chain statistics, such as the distribution of end-to-end distance and the exponents in the power laws that relate properties to chain length, for both ideal, phantom chains and also with excluded volume effects. The numerical results are employed to predict properties of real polymer/solvent systems, such as polystyrene in cyclohexane and toluene, which are then compared to experimental data.

The time of flight (TOF) of a light pulse travelling back and forth and optical path length measurements are used to estimate the velocity of light. The light pulse has a duration of 10 ns, and is obtained from a suitably modified CW commercial laser diode. The TOF is read with a digital oscilloscope connected to two fast sensors, detecting the signals at the switch-on of a laser diode and at the arrival of the light pulse reflected from a plane mirror placed at a known distance.

Light from a rough sample surface illuminated with a laser consists of a speckle pattern. If the surface evolves with time, the pattern becomes dynamic, following the activity of the sample. This phenomenon is used both in research and in industry to monitor processes and systems that change with time. The measuring equipment generally includes high-performance CCD cameras and other expensive devices. Based on the same principle, here we describe a simplified approach where components and devices commonly available in undergraduate physics laboratories are instead used. The technique is described in detail, also reviewing the basic properties of speckle fields. Examples of the application are given, and suggestions for further investigation are provided.

We present a simple device for demonstrating the essential aspects of polymers in flow in the classroom. Rubber bands are used as a macroscopic model of polymers to allow direct visual observation of the flow-induced changes in orientation and conformation. A transparent Perspex Couette cell, constructed from two sections of a tube, is used to create the simple flow developed by the relative motion of the two cylinders. This simple, qualitative visual demonstration of the flow behaviour of rubber bands in a transparent Perspex Couette flow cell shows flow alignment at low concentrations in glycerol in accord with expectation. At higher concentrations, an intriguing flow-induced 'balling' of the rubber bands is observed. The initial random state is realized upon reversal of the flow direction for these systems.

The motion of two bodies, along a straight line, under the inverse square law of gravity is considered in detail, progressing from simpler cases to more complex ones: (a) one body fixed and one free, (b) both bodies free and identical mass, (c) both bodies free and different masses and (d) the inclusion of electrostatic forces for both bodies free and different masses. The equations of motion (EOM) are derived starting from Newton's second law or from conservation of energy. They are then reduced to dimensionless EOM using appropriate scales for time and distance. Solutions of the dimensionless EOM as well as the original EOM are given. The time interval for the bodies to fall is expressed as a function of the distance fallen. Formulae for the inverse were obtained. The coalescence times for the different cases are (a) where L is the initial separation of the two bodies and m₁ is the mass of the fixed body, (b) and (c) where m_T is the total mass of the two bodies and (d) where and is a measure of the ratio of the electrostatic force to gravity. The last formula may also be used when with the interpretation that there is no collision if t is infinity or imaginary. We also discuss this motion along the straight line as a special case of the general elliptic motion of two bodies. I believe that this paper will be useful to university tutors as well as undergraduate and even graduate students who prefer to consider the special case before the general case, and their relationship.

This study aims to analyse university students' reasoning regarding two laws of electromagnetism: Gauss's law and Ampere's law. It has been supposed that the problems seen in understanding and applying both laws do not spring from students' misconceptions. Students habitually use reasoning known in the literature as 'common sense' methodology that leads to incorrect forms of reasoning. To test our hypothesis, questionnaires were designed emphasizing explanations. The results obtained show the low level of students' reasoning in both electricity and magnetism in terms of Gauss's and Ampere's laws.

Colour science is based on the sensation of monochromatic light. In contrast to that, surface colours are caused by reflection of wide sections of the daylight spectrum. Non-spectral colours like magenta and purple appear homologous to colours with spectral hue, if the approach of mixing monochromatic light is abandoned. It is shown that a large region of the colour space can be covered by mixing three primary colours derived from lossless spectral decomposition of daylight. These primaries are specified by hue, saturation and luminosity. Duality of additive and subtractive mixing is formulated quantitatively. Experimental demonstrations of calculated results are suggested. This paper is intended for undergraduate optics courses, and advanced interdisciplinary seminars on arts and physics.

The classical model of an oscillator linearly coupled to a string captures, for a low price in technique, many general features of more realistic models for describing a particle interacting with a field or an atom in an electromagnetic cavity. The scattering matrix and the asymptotic in and out-waves on the string can be computed exactly and the phenomenon of resonant scattering can be introduced in the simplest way. The dissipation induced by the coupling of the oscillator to the string can be studied completely. In the case of a d'Alembert string, the backreaction leads to an Abraham–Lorentz–Dirac-like equation. In the case of a Klein–Gordon string, one can see explicitly how radiation governs the (meta)stability of the (quasi)bounded mode.

A novel method of localizing the direction of a source of sound has evolved in the auditory system of certain small parasitic flies. A mechanical model of this design has been shown to describe the system well. Here, a simplified version of this mechanical model is presented which demonstrates the key feature: direction estimates of high accuracy are obtained via a mechanism that is much smaller than the wavelengths of sound that it detects. The derivation of this unusual result, and the explanation of why it is unusual, is presented at a level suitable for an undergraduate physics programme.

The reciprocal lattice is derived through the Fourier transform of a generic crystal lattice, as done previously in the literature. A few key derivations are this time handled in detail, and the connection with x-ray diffraction is clearly pointed out. The Ewald sphere is subsequently thoroughly explained and a few comments on its representation in a mixed real–reciprocal space are made. In particular, it appears that the majority of textbooks or papers on the subject are limited in their way of picturing it. This paper will be useful to solid state and/or crystallography teachers. It is also suitable for graduate students researching these subjects and for talented undergraduate students.

This paper, as its main didactic objective, shows the conditions needed for the validity of Faraday's law of induction. Inadequate comprehension of these conditions has given rise to several paradoxes about the issue; some are analysed and solved in this paper in the light of the theoretical deduction of the induction law. Furthermore, an experimental set-up, in which such conditions are experimentally tested, is included. The experiment is not complicated and the method we use, and similar methods used elsewhere, is widely considered as suitable laboratory practice for students of first university courses in physics and engineering.

A simple student experiment investigating dependence on air pressure of the attenuation of alpha particles in air is described. An equation giving the pressure needed to absorb all alpha particles of a given energy is derived from the Bethe–Bloch formula. Results are presented for the attenuation of alpha particles from americium 241 and radium 226. The experimental results are in close agreement with the theoretical predictions.

As recently recalled in this journal, Einstein based in 1905 his argument for the existence of quanta on the volume dependence of the black-body entropy in the Wien limit. A similar analysis, starting from the (physically correct) Planck 1900 radiation law, is reputed to be highly complicated. We show that, in contrast, it can be made simple and instructive, if one introduces the notion of volume of coherence. In particular, it gives an additional opportunity to acknowledge the importance of the notion of mode and to discuss the distinguishability of quanta. This paper is intended for physicists interested in the history of quanta and graduate students.

By combining a logarithmic approximate formula for the pendulum period derived recently (valid for amplitudes below π/2 rad) with the Cromer asymptotic approximation (valid for amplitudes near to π rad), a new approximate formula accurate for all amplitudes between 0 and π rad is derived here. It is shown that this formula yields an error that tends to zero in both the small and large amplitude limits, a feature not found in any previous approximate formula. Some ways of refining this formula are also presented. Interestingly, when one of the improved expressions is taken for building a sinusoidal (harmonic) approximation to the solution of the pendulum equation of motion very good agreement is found. The simple log formulae derived here require only a few elementary function calls in a pocket calculator for accurate evaluations, being useful for analyzing pendulum experiments in introductory physics labs. They may also be of interest for those specialists working with nonlinear phenomena governed by pendulum-like differential equations, which arise in many fields of science and technology (e.g., analysis of acoustic vibrations, oscillations in small molecules, optically torqued nanorods, Josephson junctions, electronic filters, gravitational lensing in general relativity, advanced models in field theory, oscillations of buildings during earthquakes, and others).

We demonstrate that by using a combination of a Wollaston prism and two photodiodes the accuracy in the measurements of Faraday rotation with ac magnetic fields can be greatly improved. Our experiments were performed on microscope cover glass plates with thicknesses between 0.13 and 0.16 mm. We show that our setup is capable of distinguishing between the Faraday rotation signals of glass plates having a difference in thickness of a few micrometers, corresponding to Faraday rotations of hundreds of microdegrees per Tesla only.

By combining Fourier optics with classical radiometry, a simple, compact formula is derived for computing the absolute irradiance of diffracted and aberrated images in optical systems. Within appropriate limits, the formula reduces to the familiar equations of classical radiometry and of physical optics. It is argued that the approach presented is pedagogically appealing as it combines the principal results of geometrical optics, physical optics and radiometry into a single equation, thus, providing a convenient and simple means of describing imaging phenomena at the level of an advanced undergraduate or introductory graduate course on radiometry. A practical example concerning the image irradiance of stars is discussed.

We show the impossibility of harvesting power from rotational motions by devices attached to the rotating object. The presentation is suitable for students who have studied Lagrangian mechanics.

Any free-particle wavepacket solution of Schrödinger's equation can be converted by differentiations to wavepackets rotating about the original direction of motion. The angular momentum component along the motion associated with this rotation is an integral multiple of ℏ. It is an intrinsic angular momentum: independent of origin and unchanged by Galilean boosts along the quantization direction. An example is given based on the three-dimensional Gaussian wavepacket, suitable for presentation to an undergraduate class on quantum mechanics.

In a recent article, Ates and Cataloglu (2007 Eur. J. Phys.28 1161–71), in analysing results for a course in introductory mechanics for prospective science teachers, found no statistically significant correlation between students' pre-instruction scores on the Lawson classroom test of scientific reasoning ability (CTSR) and post-instruction scores on the force concept inventory (FCI). As a possible explanation, the authors suggest that the FCI does not probe for skills required to determine reasoning abilities. Our previously published research directly contradicts the authors' finding. We summarize our research and present a likely explanation for their observation of no correlation.

We respond to the comment by Coletta et al (2008 Eur. J. Phys.29 L25) on our work on the effects of students' reasoning abilities on conceptual understanding and problem-solving skills in introductory mechanics.

A new simple derivation of the critical region for the dog-and-rabbit chase problem is presented, which uses the concept of relative motion in introductory mechanics courses.

The resonant configurations and normal frequencies of a heavy elastic cable that is hanging in the field of gravity and is rotating uniformly about the vertical are examined from a theoretical perspective. The cable is assumed pinned at both extremities, with an extra load added to the lower one for stability. The equation of motion for this system is obtained and solved exactly. It is shown that the various resonant configurations of the cable are described by Bessel functions of order zero. The normal configurations and frequencies are obtained and the configuration of the cable in its lowest mode is presented as a photograph of demonstrations effected on three different samples. The results presented in this note would be useful for students in intermediate or advanced courses in classical mechanics, computation physics or waves and vibrations.