Table of contents

It is common to modify valuable, sophisticated equipment, originally acquired for other purposes, to adapt it for the needs of educational experiments, with great didactic effectiveness. The present project concerns a setup developed from components of a portable system for energy dispersive x-ray fluorescence spectroscopy (EDXRF). Two educational modules have been developed on the basis of this setup. Module 1 comprises a series of x-ray laboratory exercises investigating basic principles, such as the verification of Moseley's law, Compton's law and the Lambert–Beer law. Module 2 concerns the calibration of the XRF with reference materials, aiming to get quantitative measurements of the elemental composition of objects of cultural interest. The application of the calibrated experimental setup is demonstrated with indicative measurements of metal objects and pigments of wall paintings, in order to discuss their spectra, and their qualitative and quantitative analyses. The setup and the applied experiments are designed as an educational package of laboratory exercises on the one hand for students in natural sciences, and on the other for the education of students who will work in the field of cultural heritage, such as conservation science or archaeological science.

A system of two magnets hung from two vertical springs and oscillating in the hollows of a pair of coils connected in series is a new, interesting and useful example of coupled oscillators. The electromagnetically coupled oscillations of these oscillators are experimentally and theoretically studied. Its coupling is electromagnetic instead of mechanical, and easily adjustable by the experimenter. The coupling of this new coupled oscillator system is determined by the currents that the magnets induce in two coils connected in series, one to each magnet. It is an interesting case of mechanical oscillators with field-driven coupling, instead of mechanical coupling. Moreover, it is both a coupled and a damped oscillating system that lends itself to a detailed study and presentation of many properties and phenomena of such a system of oscillators. A set of experiments that validates the theoretical model of the oscillators is presented and discussed.

This paper describes theory and experiments, taken from biophysics and physiological measurements, to illustrate the technique of signal averaging. In the process, students are introduced to the basic concepts of signal processing, such as digital filtering, Fourier transformation, baseline correction, pink and Gaussian noise, and the cross- and autocorrelation functions. From these computations, the students estimate physically interesting parameters such as the pulse rate and blood flow velocity. They also learn about some of the pitfalls encountered in quantifying the signal and noise components for a meaningful computation of the signal-to-noise ratio.

The Dirac belt trick is often employed in physics classrooms to show that a 2π rotation is not topologically equivalent to the absence of rotation whereas a 4π rotation is, mirroring a key property of quaternions and their isomorphic cousins, spinors. The belt trick can leave the student wondering if a real understanding of quaternions and spinors has been achieved, or if the trick is just an amusing analogy. The goal of this paper is to demystify the belt trick and to show that it suggests an underlying four-dimensional parameter space for rotations that is simply connected. An investigation into the geometry of this four-dimensional space leads directly to the system of quaternions, and to an interpretation of three-dimensional vectors as the generators of rotations in this larger four-dimensional world. The paper also shows why quaternions are the natural extension of complex numbers to four dimensions. The level of the paper is suitable for undergraduate students of physics.

A standard topic in an advanced undergraduate classical mechanics course is the determination of the orbits in a gravitational field. In the present paper we report on the calculation of bound orbits in the gravitational field of a spiral galaxy. Calculations such as these could serve to focus attention on an area of cutting edge astrophysics and could serve as an instructive exercise for advanced undergraduates. In the computations given in this paper, use is made of real data on the flat rotation curve of NGC 3198 obtained by Begeman et al (van Albada et al 1985 Astrophys. J.295 305–13; Begeman 1989 Astron. Astrophys.223 47–60; Begeman 1987 PhD Thesis University of Groningen http://irs.ub.rug.nl/ppn/291578543), and a fitting of that data to a theoretical model outlined in a previous paper (Bacon and Sharrar 2010 Am. J. Phys. at press). The galaxy is modelled as a thin exponential disc of baryonic matter combined with a spherically symmetric dark matter halo. The bound orbits in the plane of the galaxy are investigated. The computations are carried out using an icon-driven systems-modelling program that avoids the need for extensive programming expertise. The range of orbits investigated includes bound circular orbits and bound closed and open orbits that precess. The bound closed and open orbits are bounded by circles generated by the loci of the apsides of the orbit.

Archimedes' principle is well known to state that a body submerged in a fluid is buoyed up by a force equal to the weight of the fluid displaced by the body. Herein, Archimedes' principle is derived from first principles by using conservation of the stress–energy–momentum tensor in general coordinates. The resulting expression for the force is applied in Schwarzschild coordinates and in rotating coordinates. Using Schwarzschild coordinates for the case of a spherical mass suspended within a perfect fluid leads to the familiar expression of Archimedes' principle. Using rotating coordinates produces an expression for a centrifugal buoyancy force that agrees with accepted theory. It is then argued that Archimedes' principle ought to be applicable to non-gravitational phenomena, as well. Conservation of the energy–momentum tensor is then applied to electromagnetic phenomena. It is shown that a charged body submerged in a charged medium experiences a buoyancy force in accordance with an electromagnetic analogue of Archimedes' principle.

We present a physically interesting toy, which is easily constructed and operated—the so-called buzzer. In spite of its simplicity, its physical analysis turns out to be rather complex. Thus, it comes as no surprise that most of its users are not familiar with the underlying physical mechanism. In this paper we propose a physical model which allows for the qualitative and quantitative description of the fundamental physical properties of the buzzer and report on the good agreement between theoretical and experimental data. The model is designed to give a basis for further simplification.

An analytic approximation of the solution to the differential equation describing the oscillations of a simple pendulum at large angles and with initial velocity is discussed. In the derivation, a sinusoidal approximation has been applied, and an analytic formula for the large-angle period of the simple pendulum is obtained, which also includes the initial velocity of the pendulum. This formula is more accurate as compared to most of what has previously been published, and gives the period with an accuracy better than 0.04% for angles up to , and within 0.2% for angles up to 2.8 radians. The major advantage of the present derivation of the expression for the pendulum period is probably the simplicity, which makes the formula useful for analysing pendulum experiments with initial velocities in introductory physics labs. For a given set of initial conditions, the formula also predicts a critical velocity, at which the period of the pendulum becomes infinite as the pendulum will exactly come to rest at the upper, unstable equilibrium. In the small-angle regime, the formula becomes equivalent to the result for the period of the linear pendulum. For initial angles up to , the sinusoidal approximation of the solution is rather good, but deviation is increasingly observed at larger angles, as the motion of the pendulum becomes anharmonic.

The rotational speeds of stars in the disc of a spiral galaxy are virtually independent of the distances of the stars from the centre of the galaxy. In common parlance, the stellar speed versus distance plot known as a galactic rotation curve is by observation typically nearly flat. This observation provides strong evidence that most galactic matter is dark, i.e. undetectable through luminosity measurements. This article is intended to serve the pedagogical purpose of using the dark matter problem as the basis of exercises accessible to university students. Gauss's law is first used to explain why luminosity measurements lead to the prediction of incorrect speed distributions for disc stars in spiral galaxies. The question of how mass can be distributed within a galaxy to enable flat rotation curves is then considered. Gauss's law concisely answers this question for an idealized galaxy with spherical symmetry. For an idealized planar galaxy with circular symmetry, constructing and evaluating an integral for the gravitational field as a function of distance from the centre of the galaxy provides the answer to the question.

Wood is transparent for microwaves and due to its anisotropic structure has anisotropic dielectric properties. A laboratory experiment that allows for the qualitative demonstration and quantitative measurements of linear dichroism and birefringence in the microwave region is presented. As the proposed experiments are based on the anisotropy (of wood), which is evident from the observable anisotropic structure of wood, they may serve as a demonstration for explaining the anisotropic properties in crystals in the optical region.

This paper was first motivated by the following question: 'A pair of twins, R and S, each gives the same hard push on a block. R's block is on a rougher floor than S's. Who does more work?' It is shown that S will do more work on his block if there is no constraint on the distance over which the force is applied. On the other hand, if the assumption is relaxed to 'the same hard push within some distance', say due to the length of their arms, then it is possible that R does more work than S. To resolve this issue, general formulae for the velocity, displacement and work done are obtained for a Gaussian impulsive force acting on a body moving on a horizontal surface.

In this paper, we analyse a simple experiment to study the effects of polarized light. A simple optical system composed of a polarizer, a retarder (cellotape) and an analyser is used to study the effect on the polarization state of the light which impinges on the setup. The optical system is characterized by means of a Jones matrix, and a simple procedure based on Jones vectors is used to obtain an expression for the intensity after the light passes through the optical system. The light intensity is measured by a photodetector and the expression obtained theoretically is experimentally validated. By fitting the experimental intensity data, the value of the retardation introduced by the retarder can also be obtained.

The speed of light is an essential topic in the teaching of physics at school and at university, either with respect to the type of experiment or of course with respect to its genuine inherent importance. In reality, the various available experiments are hardly ever performed in class for many reasons. Therefore, we offer this experiment as a remotely controlled laboratory (RCL). An RCL is a real experiment setup at location A which can be controlled via the Internet by a user at a distant location B. It allows several actions like in the hands-on experiment and delivers convincing results. Finally, we present experiences of the use of the RCL, describe the added value of this experiment as an RCL and give hints for implementing the RCL in teaching.

Over time the account of how the Kramers–Kronig (dispersion) relations between the real and imaginary parts of response functions were derived in 1926 and 1927 has been transmogrified into anecdotes about what might have been done but was not. Although Kramers obtained both members of a pair of relations, Kronig obtained only one. Both authors appealed to specific models of an atomic gas rather than to the general arguments about linearity, causality and analyticity in modern model-independent derivations. Kramers merely speculated on whether the specific results he obtained might have a more general validity. Neither author showed that a signal cannot travel faster than cin any medium for which the dispersion relations are satisfied. Indeed, they did not mention, even obliquely, signal speeds and causality. Despite their magical aura, Kramers–Kronig relations are translations into somewhat cryptic frequency language of statements clearer in time language.

A common moving-coil (dynamic) loudspeaker is a device very suitable for teaching the general features of oscillating systems. As an addition to a previous paper (Kraftmakher 2009), this paper includes the following topics: (i) a new design of the optical sensor for measuring the cone oscillations; (ii) positive feedback and self-excited oscillations; (iii) the determination of the loudspeaker parameters without direct measurements of the cone oscillations; and (iv) the loudspeaker in a vacuum chamber. The measurements are performed without a lock-in amplifier. The tight connection between the mechanical and electrical parameters of a loudspeaker is confirmed. The experiments are well related to the university courses of electricity and magnetism and can be used in undergraduate laboratories.

We take the Bose–Hubbard model to illustrate exact diagonalization techniques in a pedagogical way. We follow the route of first generating all the basis vectors, then setting up the Hamiltonian matrix with respect to this basis and finally using the Lanczos algorithm to solve low lying eigenstates and eigenvalues. Emphasis is placed on how to enumerate all the basis vectors and how to use the hashing trick to set up the Hamiltonian matrix or matrices corresponding to other quantities. Although our route is not necessarily the most efficient one in practice, the techniques and ideas introduced are quite general and may find use in many other problems.

The static equilibrium deformation of a heavy spring due to its own weight is calculated for two cases: first for a spring hanging in a constant gravitational field, and then for a spring which is at rest in a rotating system where it is stretched by the centrifugal force. Two different models are considered: first a discrete model assuming a finite number of point masses connected by springs of negligible weight, and then the continuum limit of this model. In the second case, the differential equation for the deformation is obtained by demanding that the potential energy is minimized. In this way a simple application of the variational calculus is obtained.

We solve the Schrödinger equation for a hydrogen atom within a spherical box approximately by means of the variational method. We propose two simple trial functions for the case in which both the nucleus and the electron move within the box. Present results are sufficiently accurate for all values of the box radius and therefore an improvement on an earlier calculation based on perturbation theory. We compare the energies of three alternative approaches for the moving-nucleus model with that of the nucleus clamped at origin. We also outline some physical applications of the model.

We stress the usefulness of the work reservoir in the formalism of thermodynamics, in particular in the context of the first law. To elucidate its usefulness, the formalism is then applied to the Joule expansion and other peculiar and instructive experimental situations, clarifying the concepts of configuration and dissipative work. The ideas and discussions presented in this study are primarily intended for undergraduate students, but they might also be useful to graduate students, researchers and teachers.

In the conventional interpretation of quantum mechanics the interference of particles in a two-beam interferometer is closely related to the problem of which-way information. One of the mysteries of quantum mechanics relies on the assumption that the wavefunction of each photon propagates simultaneously along both classically allowed paths, and that interference arises as a consequence of the indistinguishability of those paths. Any attempt to obtain which-way information by putting individual labels on the photons in each pathway inevitably destroys interference. However, even in cases in which the photons carry which-way labels, it is possible to erase those labels after the particle has left the interferometer. The erasing process (partly or completely) destroys the which-way information, and thereby restores interference. This phenomenon is known as quantum erasing. Here we present a lecture demonstration experiment of quantum erasing based on a Mach–Zehnder interferometer operated with single photons.

Electrons moving in a tilted periodic potential perform a periodic motion, known as Bloch oscillation. Within a semiclassical description, the crystal momentum increases linearly with time until it reaches the boundary of the first Brillouin zone in reciprocal space. Then, it reenters the first Brillouin zone by the opposite edge. This periodic motion in reciprocal space is accompanied by an oscillation in real space. The angular frequency of the oscillations and their amplitude can be calculated within the semiclassical framework. Nevertheless, the semiclassical approach cannot explain the rich phenomenology of the Bloch oscillations, such as the breathing of the electronic wave packet. We present a simple description of the Bloch oscillations of tightly bound electrons in biased lattices at a basic level and calculate exactly the wavefunction as a function of time.

Time series of atmospheric pressure data, collected over a period of several years, were analysed to provide undergraduate students with educational examples of application of simple statistical methods of analysis. In addition to basic methods for the analysis of periodicities, a comparison of two forecast models, one based on autoregression algorithms, and the other making use of an artificial neural network, was made. Results show that the application of artificial neural networks may give slightly better results compared to traditional methods.

We consider the simple and classical theorem of the motion of the centre of mass, pointing out that many textbooks append a wrong corollary to it: that the motion of the centre of mass is always independent from the internal forces. We give an explicit example showing that this corollary is wrong. We discuss using a historical approach the genesis of such a misunderstanding. The contents of the paper may be used at different levels of complexity. The explicit example may be used to discuss the theorem at an undergraduate level in a clearer way than usual, but the paper also contains much for an advanced course on classical mechanics. Moreover, the historical approach may also be of interest in the study of the philosophy and sociology of science.

The van der Waals equation of state does not sufficiently represent a gas unless a thermodynamic potential with two proper and independent variables is simultaneously determined. The limiting procedures under which the behaviour of the van der Waals gas approaches that of an ideal gas are letting two van der Waals coefficients be zero rather than letting the molar volume become infinitely large; otherwise, the partial derivative of internal energy with respect to pressure at a fixed temperature does not vanish.

A simple, easy to build electronic circuit consisting of an amplifying chain followed by a state-variable bandpass filter has been used to measure the thermal noise produced by a resistor. Using Johnson's theory for thermal noise, Boltzmann's constant could be derived with an accuracy of 1.4% of its accepted value. This simple instrument can be very useful for modern physics laboratories due to its low cost and use of standard instrumentation.

Due to the great impact that quantum information theory has and its applicability for condensed matter systems, we study the concurrence measure of entanglement for the finite Ising chain with the Dzyaloshinsky–Moriya interaction. The explicit calculation of the concurrence and the reduced density matrix necessary to calculate it is presented, in order to use them as a pedagogical tool to make such important concepts clearer. A growth of the concurrence was observed as the parameter of the interaction increased, reaching a maximum at D = 1, which indicates the phase transition of the system from an antiferromagnetic phase to a chiral one.

A numerical Gram–Schmidt orthonormalization procedure is presented for constructing an orthonormal basis function set from a non-orthonormal set, when the number of basis functions is large. This method will provide a pedagogical illustration of the Gram–Schmidt procedure and can be presented in classes on numerical methods or computational physics.

A closed expression for the chemical potential of Bose gases in an external one-dimensional harmonic trap, reported recently in this journal (Mungan 2009 Chemical potential of one-dimensional simple harmonic oscillators Eur. J. Phys.30 1131–6), is approximate and not applicable for temperatures lower than a characteristic value below which the ground state becomes occupied by a macroscopic number of particles. In this letter, the correct behaviour of the chemical potential at low temperature is addressed.

We compute the action of a static magnetic field on a current loop. From this we recover the definition of as given in textbooks. The presentation is accessible to undergraduate students with a knowledge of the basic notions of classical electromagnetism.

It is shown by direct computation that angular momentum cannot be reduced to two dimensions by assuming lack of dependence on the z coordinate. Formally, this is because the independence of z breaks rotational invariance. As a physical example, it is shown that a gauge- and rotation-invariant description of planar electrons in a perpendicular magnetic field must have a three-dimensional form, even though all the terms refer to planar coordinates alone. The physical concepts underlying common formal manipulations with the quantum angular momentum operator are treated at a level directly useful to students encountering them thoroughly for the first time. The letter may also serve as a contextual introduction to advanced quantization issues, usually studied at graduate level.

A closed-form approximate expression for the solution of a simple pendulum in terms of elementary functions is obtained. To do this, the exact expression for the maximum tension of the string of the pendulum is first considered and a trial approximate solution depending on some parameters is used, which is substituted in the tension equation. We obtain the parameters for the approximate by means of a term-by-term comparison of the power series expansion for the approximate maximum tension with the corresponding series for the exact one. We believe that this letter may be a suitable and fruitful exercise for teaching and better understanding nonlinear oscillations of a simple pendulum in undergraduate courses on classical mechanics.