Table of contents

Using a direct variational approach, accurate approximate solutions are obtained for the famous Thomas–Fermi nonlinear differential equation describing the screening of the Coulomb potential inside heavy neutral atoms. In addition, the analysis demonstrates the flexibility and power, but also the limitations of the variational approach.

A normal light bulb provides excellent opportunities for learning modern calorimetric techniques. The tungsten filament in a light bulb allows calorimetric measurements to be made over a wide range of high temperatures. The filament serves simultaneously as a sample, a heater and a thermometer. A student experiment employing a pulse calorimetric technique is described. A brief review of existing calorimetric techniques is given, and the temperature dependence of specific heat of solids is considered.

We study the stabilization of a driven inverted pendulum from the point of view of Hamiltonian mechanics. We average the fast time dependence of the Hamiltonian, H(θ, p, t), to obtain an effective Hamiltonian which contains a potential energy term proportional to the fluctuation of the pivot-point velocity, . The pendulum, of length l, is stabilized when . In addition, the frequency of oscillations about the stable point (non-inverted pendulum) is found to increase as a result of the pivot-point motion.

The classical lattice dynamics of honeycomb lattices is studied in the harmonic approximation. Interactions between nearest neighbours are represented by springs connecting them. A short and necessary introduction of the lattice structure is presented. The dynamical matrix of the vibrational modes is then derived and its eigenvalue problem is solved analytically. The solution may provide deeper insight into the nature of the vibrational modes. Numerical results for the vibrational frequencies are presented. To show how effective our method for the honeycomb lattice is, we also apply it to triangular and square lattice structures. A few suggested problems are listed in the concluding section.

Analytic expressions are found for the electric field and potential around a pair of hyperbolic conductors with a potential difference between them. The results also apply to the field and potential between a hyperbolic conductor and a conducting plane, and to the two-dimensional flow of an ideal fluid between hyperbolic barriers or between a flat surface and a hyperbolic barrier. The field strength at a conductor is found to be proportional to the cube root of the local curvature. (The planar case must be obtained as a limit.) The methods and results are simple enough to be used in teaching electrostatics and hydrodynamics, in particular to supply an explicit counter-example to the popular misconception that the field strength at a conductor is proportional to the local value of the curvature.

We report on a teaching approach oriented to the understanding of some relevant concepts of wave propagation in solids. It is based on simple experiments involving the propagation of shock mechanical waves in solid slabs of various materials. Methods similar to the generation and propagation of seismic waves are adopted. Educational seismometers, interfaced with computers, are used to detect and visualize the shock waves and to analyse their propagation properties. A qualitative discussion of the results concerning the propagation and the attenuation of the waves allows us to draw basic conclusions about the response of the matter to solicitation impacts and their propagation.

A formulation of Dirac's equation using complex-quaternionic coordinates appears to yield an enormous gain in formal elegance, as there is no longer any need to invoke Dirac matrices. This formulation, however, entails several peculiarities, which we investigate and attempt to interpret.

In the spirit and style of John S Bell's well-known paper on How to teach special relativity [1] it is argued that a 'Bohmian pedagogy' provides a very useful tool to illustrate the relation between classical and quantum physics and illuminates the peculiar features of the latter.

Understanding the deflection of light by a massive deflector, as well as the associated gravitational lens phenomena, requires the use of the theory of general relativity. I consider here a classical analogy, based on Newton's equation of motion for massive particles. These particles are emitted by a distant source and deflected by the gravitational field of a (opaque) star or of a (transparent) galaxy. The dependence of the deviation angle D on the impact parameter b and the—Euclidean—geometry of the (source, deflector, earth) triplet, imply that different particle trajectories may reach an earth based observer. Since D(b) does not depend on the mass of the particles, a (Newtonian) flavour of gravitational lens phenomena is naively obtained by setting the particles' velocity equal to the speed of light. Orders of magnitude are obtained through this classical approach, and are compared to the general relativity results.

A technical description of the 'Wilson cloud chamber' developed by C T R Wilson in 1911–12 will be given here. This instrument soon became a fundamental tool of research in nuclear, cosmic ray and elementary particle physics. The close examination of the expansion apparatus, the illumination method and the photographic method shows that the cloud chamber is a fine example of experimental ingenuity.

We provide a simple theoretical study of beams non-uniformly polarized across their transverse sections which can be introduced in undergraduate optics courses. In order to generate such beams we propose to use a slightly convergent (or divergent) linearly and uniformly polarized beam impinging on an anisotropic uniaxial material with the beam propagation direction along the optic axis. Analytical expressions for the Jones vector, Stokes parameters, ellipticity and azimuth at each point of the transverse section, perpendicular to the propagation direction, are obtained at the output of this system. By means of these parameters a detailed description of the state of polarization across the transverse profile is given.

In a Michelson interferometer, the contrast of the interference pattern fades away due to incoherence of light when the mirrors are not in equidistant positions. We propose an experiment where the distance between the interference fringes can be determined, even when the difference in length of the interferometer arms is far beyond the coherence length of the light, i.e. when the interference pattern disappears completely for the naked eye. We used a semiconductor laser with two photodiodes as sensors, which enabled us to follow the fluctuations of the light intensity on the screen. The distance between invisible interference fringes was determined from periodic changes of the summed fluctuating signal, obtained by changing the distance between the two sensors.

For a fluid system, obeying a state equation of the van der Waals type, the gas and the liquid phases can coexist in equilibrium, at a given temperature, only if the volume of the system is kept fixed. Thus, in order to study the two-phase equilibria of a fluid system, it seemed quite natural to choose the molar volume as the independent variable, and, consequently, the Helmholtz free energy as the proper thermodynamic potential for the application of the minimum principle. Specific computations are here carried out for a single van der Waals fluid, namely, pure water at 300 °C. As a result, the present treatment indicates a simple and effective way to identify the whole range of molar volumes where the equilibrium preferred by the system is a two-phase equilibrium. This range results to be wider than the interval of strict instability of the van der Waals isotherm. Finally, it is pointed out that all the results, obtained here for the van der Waals state equation, can be extended to all the state equations of the same type.

The problem of determining the state parameters' sub-domain where the behaviour of the classical ideal gas approximates that of the Bose and Fermi ideal gases is tutorially discussed. The entropy of any quantum system being always positive, the classical approximation can only be satisfactory within the parameters' sub-domain where the classical entropy turns out to be positive. We show that the sub-domain determined by this condition is close to that where de Broglie's thermal wavelength is smaller than the mean interparticle distance. The exact determination of the state parameters' region, where the particle number density, the grand potential and the entropy of quantum ideal gases differ from those of the classical gas less than a specified quantity, is also illustrated.

The physically unrealistic implications of negative relaxation in continuum mechanics is explored in the context of Stokes' first problem for Maxwell fluids. In addition, the electrodynamic analogue of negative relaxation is discussed and several lessons for students are presented. Lastly, two of Maxwell's important contributions to the physical sciences are noted.

A classical problem of mechanics involves a projectile fired from a given point with a given velocity whose direction is varied. This results in a family of trajectories whose envelope defines the border of a 'safe' domain. In the simple cases of a constant force, harmonic potential and Kepler or Coulomb motion, the trajectories are conic curves whose envelope in a plane is another conic section which can be derived either by simple calculus or by geometrical considerations. The case of harmonic forces reveals a subtle property of the maximal sum of distances within an ellipse.

Dipole emission mechanisms for energy transfer operate in many important areas of photophysics. A straightforward analysis based on quantum electrodynamics not only reveals the entanglement of mechanisms usually regarded as 'radiative' and 'radiationless'; it also gives significant physical insights into a host of topics in electromagnetism. These include: the designation of real and virtual photons; propagating and non-propagating character in electromagnetic fields; near-zone and wave-zone effects; transverse and longitudinal character; the effects of retardation; the relation between couplings of static and transition dipoles, and manifestations of quantum uncertainty. A simple extension of the theory to accommodate magnetic dipole as well as electric dipole transitions furthermore reveals key differences between the range dependences of the magnetic and electric fields produced by dipolar emission. With important technological applications, this lesson in advanced physics showpieces the interplay of principles associated with quantum mechanics, electromagnetism and photophysics.

We evaluate the electrostatic potential and the electrostatic field created by a point charge and an arbitrarily oriented electrical dipole placed near a grounded perfectly conducting sphere. Induced surface charge distributions and possible variants of the problem are also discussed.

Computerized experiments are described for teaching free and forced oscillations of a tuning fork, as well as the transient process when establishing steady forced oscillations. With positive feedback, it is easy to generate continuous oscillations. The advantage of employing a tuning fork is that the short transient allows one to significantly decrease the time of the experiments and to use them as classroom demonstrations. The experiments might be used instead of similar experiments with a pendulum.

The PDF file provided contains web links to all articles in this volume.