Table of contents

Theories of wetting of liquids on solid surfaces under the condition that van der Waals force is dominant are briefly reviewed. We show theoretically that Zisman's empirical equation for wetting of liquids on solid surfaces is a linear approximation of the Young–van der Waals equation in the wetting region, and we express the two parameters in Zisman's empirical equation in terms of the dielectric polarizabilities of the solid and liquids. The materials contained in this paper are suitable for physics teaching of wetting phenomena for undergraduate, graduate, general physicist, etc.

This paper presents an experiment in which students determine the mass sensitivity of three crystal quartz resonators, designed to be carried out in 'Physics Laboratory II' at the Institute of Experimental Physics, University of Wroclaw. The authors discuss the process of setting up the experiment and the results of the measurements. They clearly show that the set-up is simple, easy to assemble and allows obtaining satisfactory results. The advantages and the disadvantages of the apparatus are discussed along with the implementation into the teaching process. The results support the gathering of knowledge and skills necessary for experimental physicists.

Entropy generation in a chemical reaction is analysed without using the general formalism of non-equilibrium thermodynamics at a level adequate for advanced undergraduates. In a first approach to the problem, the phenomenological kinetic equation of an elementary first-order reaction is used to show that entropy production is always positive. A second approach assumes that the reaction is near equilibrium to prove that the entropy generation is always greater than zero, without any reference to the kinetics of the reaction. Finally, it is shown that entropy generation is related to fluctuations in the number of particles at equilibrium, i.e. it is associated with a microscopic process.

We analyse two-mirror resonators in terms of their fractional Fourier transform (FRFT) properties. We use the basic ABCD ray transfer matrix method to show how the resonator can be regarded as the cascade of two propagation–lens–propagation FRFT systems. Then, we present a connection between the geometric properties of the resonator (the g parameters) and those of the equivalent FRFT systems (the FRFT order and scaling parameters). Expressions connecting Gaussian beam q-transformation with FRFT parameters are derived. In particular, we show that the beam waist of the resonator's mode is located at the plane leading to two FRFT subsystems with equal scaling parameter which, moreover, coincides with the mode Rayleigh range. Finally we analyse the resonator's stability diagram in terms of the fractional orders of each FRFT subsystem, and the round trip propagation. The presented analysis represents an interesting link between two topics (optical resonators and Fourier optics) usually covered in optics and photonics courses at university level, which can be useful to teach and connect the principles of these subjects.

We study the hydrogen atom confined to a spherical box with impenetrable walls but, unlike earlier pedagogical articles on the subject, we assume that the nucleus also moves. We obtain the ground-state energy approximately by means of first-order perturbation theory and show that it is greater than that for the case in which the nucleus is clamped at the centre of the box. The present approach is valid for strong confinement and resembles the well-known treatment of the helium atom with clamped nucleus.

The meaning of Lorentz contraction in special relativity and its connection with Bell's spaceships parable is discussed. The motion of Bell's spaceships is then compared with the accelerated motion of a rigid body. We have tried to write this in a simple form that could be used to correct students' misconceptions due to conflicting earlier treatments.

A basic setup for data acquisition and analysis from an oscillatory circuit is described, with focus on its application as either low-pass, high-pass, band-pass or band-reject frequency filter. A homemade board containing the RLC elements allows for the interchange of some of them, in particular, for the easy change of the R value, and this makes apparent for the student its influence on the damping factor. The function generator operates in the swap frequency mode over a suitable frequency range and all the circuit parameters are chosen to provide a reasonable set of data for all the electronic filters studied. The output data are acquired through a commercially available DAQ board and data analysis is performed using a graphing and fitting workspace. The main objective is to develop a methodology of teaching the laboratory material through a computer-based environment devised to help students to appreciate how the governing equations work and to visualize their practical applications.

Two results support the idea that the scalar and vector potentials in the Lorenz gauge can be considered to be physical quantities: (i) they separately satisfy the properties of causality and propagation at the speed of light and do not imply spurious terms and (ii) they can naturally be written in a manifestly covariant form. In this paper we introduce expressions for the Lorenz-gauge potentials: at the present time in terms of electric and magnetic fields at the retarded time. These expressions provide a third result in favour of a physical interpretation of the Lorenz-gauge potentials: (iii) they can be regarded as causal effects of the observed electric and magnetic fields.

The behaviour of a quantum rod, pivoted at its lower end on an impenetrable floor and restricted to moving in the vertical plane under the gravitational potential, is studied analytically under the approximation that the rod is initially localized to a 'small-enough' neighbourhood around the point of classical unstable equilibrium. It is shown that the rod evolves out of this neighbourhood. The time required for this to happen, i.e. the tipping time, is calculated using the semi-classical path integral. It is shown that equilibrium is recovered in the classical limit, and that our calculations are consistent with the uncertainty principle.

It is shown that a state that is factorizable in the Hilbert space corresponding to some choice of degrees of freedom becomes entangled for a different choice of degrees of freedom. Therefore, entanglement is not a special case but is ubiquitous in quantum systems. Simple examples are calculated and a general proof is provided. The physical relevance of the change of tensor product structure is mentioned.

This paper enlarges the reservoir of solved tutor problems in non-holonomic mechanics at the undergraduate level of physics education. Unlike other, rather artificial, solved problems typically used, the streetboard-rider locomotion problem presented here represents an appealing contemporary real-world problem with interesting applications in a popular sport and in robotics. In this paper, the streetboard motion is discussed from the physical point of view. We show that the interesting snake-like motion performed by streetboard riders stems from its non-holonomic nature. The related non-holonomic constraints are derived and the problem of the mechanical system subjected to these non-holonomic constraints is solved using methods appropriate to the undergraduate university level. The analytical solution for the circular motion and the numerical solution for the general motion are obtained, the physical meaning of the derived constrained forces is investigated and the results are discussed with respect to observed streetboard-rider locomotion. The brief outline of the paper can be used as a demonstration example in non-holonomic mechanics lessons, while the paper itself establishes an original undergraduate computational student project in theoretical mechanics.

An experiment analogous to that devised by Grimaldi and subsequently repeated by Young to study the nature of light has been realized with electrons. Following the Grimaldi and Young line of thought, an original approach is presented to introduce undergraduate physics students to the wave behaviour of electrons. An electron microscope equipped with a low coherent source of electrons and a thin platinum wire, acting as an opaque obstacle, is used to reproduce the experimental conditions adopted by Grimaldi and Young with light. Electron interference fringes produced in the geometrical shadow of the obstacle are interpreted by assuming that electrons behave like a sound or a light wave. This hypothesis is confirmed by the modelling of the experimental electron interference patterns.

The solution to the Euler–Lagrange equation is an extremal functional. To understand that the functional is stationary at local extrema (maxima or minima), we propose a physics experiment that involves using a soap film to form a catenoid. A catenoid is a surface that is formed between two coaxial circular rings and is classified mathematically as a minimal surface. Using the soap film, we create catenoids between two rings and characterize the catenoid in situ while varying the distance between the rings. The shape of the soap film is very interesting and can be explained using dynamic mechanics. By observing the catenoid, physics students can observe local extrema phenomena. We stress that in situ observation of soap-film catenoids is an appropriate physics experiment that combines theory and experimentation.

Two important questions concerning cosmic rays are: Why are electrons in the cosmic rays less efficiently accelerated than nuclei? How are particles accelerated to great energies in ultra-high energy cosmic rays? In order to answer these questions we construct a simple model of the acceleration of a charged particle in the cosmic ray. It is not meant as a detailed model, which is expected to be rather complicated, but rather as a 'pedagogic model' pointing out some important elements of a more complete model. Furthermore, the present model is sufficiently simple that it may be suitable as an 'astrophysical example' in the teaching of the special theory of relativity. In this model a particle is accelerated by ultrarelativistic shocks in a source of gamma ray bursts. No assumption as to the details of the accelerating mechanism is made except that the force acting on a charged particle depends only upon the charge of the particle and not upon its mass, and the product of the force and the thickness of the shock waves must be sufficiently great. It is important for the success of the model that the energy radiated by the particles is taken mainly from the Schott energy and not from the kinetic energy of the particles. It is shown how this model of the accelerating process can explain why electrons are accelerated to less energy than protons and heavier nuclei. The mechanism also explains how particles may be accelerated to energies greater than 10²⁰ eV.

We present a teaching module dealing with the thermal effects of interaction between radiation and matter, the infrared emission of bodies and the greenhouse effect devoted to university level and teacher education. The module stresses the dependence of the optical properties of materials (transparency, absorptivity and emissivity) on radiation frequency, as a result of interaction between matter and radiation. Multiple experiences are suggested to favour a progressive construction of knowledge on the physical aspects necessary to understand the greenhouse effect and global warming. Some results obtained with university students are briefly reported.

The Landau–Zener formula provides the probability of non-adiabatic transitions occurring when two energy levels are swept through an avoided crossing. The formula is derived here in a simple calculation that emphasizes the physics responsible for non-adiabatic population transfer.

We study the equilibrium thermodynamics of the electromagnetic radiation in a cavity of a given volume and temperature. We found three levels of description, the thermodynamics of one mode, the thermodynamics of the distribution of frequencies in a band by summing over the frequencies in it and the global thermodynamics by summing over all the frequencies. One equation relating frequency and volume is used to define the thermodynamics of one mode, and to explain the mystery of the frequency-dependent quantities having a similar behaviour to the non-frequency-dependent quantities for some thermodynamic equations and different behaviour for others. Besides, this frequency–volume relation is used to count the number of modes in a band of frequency.

Condensation and evaporation are ubiquitous phenomena in nature. When we have a liquid partially filling a closed container, evaporation starts to happen, causing vapour formation in the remaining space. As the gaseous molecules bounce around, some of them hit the liquid surface and condense back into their original phase. This process evolves until there are as many molecules condensing as there are evaporating, that is, when the system reaches equilibrium. In this paper we introduce a very simple model addressed to undergraduate students and teachers that provides the temporal behaviour of this process. Using the results we propose an experimental technique to measure condensation coefficients.

Complementarity and the commutation relation of position (x) and momentum (p) imply much more than the fundamental x–p uncertainty inequality. Here, we display some further consequences of the former that could have certain pedagogical interest and, so, contribute to the teaching of quantum mechanics. Inspired by an elementary derivation of the x–p uncertainty inequality, based upon a positive quadratic polynomial, we explore one possible extension, via quartic polynomials and simple algebra and integrations. Our analysis, aimed at providing some further pedagogic expression of genuine quantum behaviours, yields other quantum inequalities for expectation values, expressed through suitable discriminants associated with quartic algebraic equations, which differ from (and are not a strict consequence of) the x–p uncertainty inequality. Those quantum inequalities are confirmed, and genuine non-classical behaviours are exhibited, for simple cases: a harmonic oscillator, a hydrogenic atom and free Gaussian wave packets. The physical interest of the expectation values involved in the quantum inequalities and of the latter is discussed, in the framework of quantum optics and squeezing phenomena.

We comment on the recent paper by Qing-Xin and Yin-Xiao (2008 Eur. J. Phys.29 N43–N45).

We comment on a recent paper by Slüsarenko et al (2008 Eur. J. Phys.29 1177–82).

Importance sampling (IS) is a well-known random simulation technique to estimate rare event probability. It is designed to reduce the variance of the Monte Carlo estimators for a given sample size. IS consists in generating random weighted samples from an auxiliary distribution rather than the distribution of interest. The crucial part of this algorithm is the choice of an efficient auxiliary PDF that should be able to simulate more rare random events. In this letter, we propose to analyse how to simply approach the IS optimal auxiliary density with non-parametric importance sampling (NIS). This letter is intended for graduate students and general physicists.

A simple example of renormalization in electrostatics accessible to undergraduate students is presented as a complement to a letter recently published in this journal.