bluuebunny/crossref_metadata_2025 huggingface dataset:
Approach of gas and radiation to equilibrium
Authors: Munley Frank
Published: September 2005

Einstein described equilibrium between two-level gas molecules and radiation in terms of a Brownian motion model involving frictional and agitational effects arising from photon absorptions and emissions. This paper extends his treatment to the case of disequilibrium when molecules and radiation are in cavities with either black or perfectly reflecting walls. The precise role of the black walls in promoting equilibrium is explicated in terms of Einstein’s formula for the frictional effect and the nature of changes in radiation patterns caused by Doppler shifts of photon frequencies. It is then shown that equilibrium between the two-level gas molecules and radiation cannot be achieved for a container with perfectly reflecting walls. The blockage of equilibrium occurs because a change in the slope of the Planck spectrum at the resonant frequency causes a change in the frictional force.
Estimate of the critical mass of a fissionable isotope
Authors: Derringh Edward
Published: September 2005

A relatively simple method for estimating the critical mass of a fissionable isotope in a spherical configuration is presented. Good agreement with an experimentally determined value is obtained.
Resource Letter MD-1: Maxwell’s demon
Authors: Leff Harvey S. , Rex Andrew F.
Published: September 2005

This Resource Letter provides a comprehensive guide to the voluminous literature that has developed around Maxwell’s demon, and offers a perspective on issues for which the hypothetical character Maxwell introduced over 120 years ago has inspired continuing research and debate. The code (E) indicates elementary level or general interest material useful to persons just learning the field; (I) indicates intermediate level or somewhat specialized material; and (A) indicates advanced or highly specialized material. No accompanying AAPT reprint book will be available, because an extensive reprint collection (Ref. 29) edited by the authors will be published separately.
Nitrogen temperature superconducting ring experiment
Authors: Liu Fuhan , Tucker Rochelle R. , Heller Peter
Published: September 2005

A student experiment is described for studying persistent currents in a commercially obtained ring of the ‘‘123’’ superconducting material at liquid-nitrogen temperature. The currents are easily detected with a standard analog Hall probe. From observations extended over a 3-week period, an upper limit on the possible resistance of one such ring was set at about 2×10−16 Ω. For the rings studied, the induced current saturated at about 2 A as the applied flux change was increased. An ac technique for checking the continuity of the superconducting path around the ring is also described. These experiments provide an interesting supplement for topics in first-year electricity and magnetism. The effects are striking and easily discussed at an introductory level. For example, the current induced by turning the ring over in the Earth’s field is readily seen.
The fighter pilot’s egg
Authors: Fry Dale W.
Published: September 2005

Of common interest to fighter pilots is what maneuver should be flown to accomplish a 360° heading change in the least time. Among the infinite possibilities, analytic expressions for a very commonly used maneuver, a roughly circular planar turn that accomplishes the required heading change, are derived and examined under four realistic assumptions. Contrary to common belief, the time to complete the stated heading change turns out to be independent of the inclination of the plane of the turn.
Comparison of the gyromagnetic ratios of the 1H and 19F nuclei—A student experiment
Authors: Davies G. R.
Published: September 2005

A technique to determine gyromagnetic ratios of nuclei is demonstrated that does not require a magnetic field measurement or a probe head change. (AIP)
Noether’s theorem for local gauge transformations
Authors: Karatas Debra L. , Kowalski Kenneth L.
Published: September 2005

The variational methods of classical field theory may be applied to any theory with an action that is invariant under local gauge transformations. What is the significance of the resulting Noether current? This article examines such currents for both Abelian and non-Abelian gauge theories and provides an explanation for their form and limited range of physical significance on a level accessible to those with a basic knowledge of classical field theory. Several of the more subtle aspects encountered in the application of the residual local gauge symmetry found by Becchi, Rouet, Stora, and Tyutin are also considered in detail in a self-contained manner.
Maxwell’s demon, power, and time
Authors: Leff Harvey S.
Published: September 2005

What rate (Pd) of energy transfer is attainable by a Maxwell’s demon who sorts gas molecules serially; and how much time (td) does it take to achieve a designated temperature difference ΔT across a partition? Two estimates are made, using (i) the energy–time form of Heisenberg’s uncertainty principle and (ii) classical kinetic theory. For a dilute gas, at ∼300 K, the uncertainty principle implies Pd <1.5×10−6 W. If the gas volume is the size of a large room, and ΔT=2 K, then td >103 years. These bounds are loose, but are of interest because this is one of few elementary applications of the energy–time uncertainty principle. With similar assumptions, classical kinetic theory implies much tighter bounds: Pd <10−9 W and td >4×106 years. The kinetic theory approach allows an extension of Brillouin’s demonstration that the second law of thermodynamics cannot be violated by a Maxwell’s demon using light signals.
Tunneling through a truncated harmonic oscillator potential barrier
Authors: Chalk Jeff D.
Published: September 2005

The sum of a one-dimensional, truncated harmonic oscillator potential and square well of the same range, defined in the positive half-space, serves as a convenient and instructive example for which the Schrödinger equation may have both bound-state and continuum solutions. A superposition of these solutions is used in a study of barrier penetration by a wave packet representing a particle with an initial position in the region of the potential well. The presence or absence of a bound state in the superposition is shown to be the key factor determining the evolution of the wave packet. If no bound state exists, the probability of the particle having a position within the potential well is a monotonically decreasing function of time. If the superposition includes a bound state, however, this probability oscillates slightly because of an interference between the bound-state and continuum components of the wavefunction.
A short proof of the generalized Helmholtz theorem
Authors: Heras José A.
Published: September 2005

By making use of the Stokes operator and its corresponding dual operator, which are introduced here, a short proof of the Helmholtz theorem for antisymmetric second-rank tensor fields in Minkowski space-time is presented.
Entropy and homogeneity
Authors: Rodewald Bernd
Published: September 2005

The intuitive concept of homogeneity is developed into a physical quantity to provide a qualitative level of description for thermodynamic systems. The treatment gives some insight into the relation of entropy, disorder, and homogeneity.
Reconciliation of esu and mksa units in nonlinear optics
Authors: Skinner I. M. , Garth S. J.
Published: September 2005

An explanation is provided of the differences between the mksa and esu systems used in the study of nonlinearities in the optical properties of materials used in optical waveguides. Means of converting both units and quantities between the two are presented, and the relationship between the electric susceptibility tensors defined in the two systems is demonstrated.
Bell’s theorem without inequalities
Authors: Greenberger Daniel M. , Horne Michael A. , Shimony Abner , Zeilinger Anton
Published: September 2005

It is demonstrated that the premisses of the Einstein–Podolsky–Rosen paper are inconsistent when applied to quantum systems consisting of at least three particles. The demonstration reveals that the EPR program contradicts quantum mechanics even for the cases of perfect correlations. By perfect correlations is meant arrangements by which the result of the measurement on one particle can be predicted with certainty given the outcomes of measurements on the other particles of the system. This incompatibility with quantum mechanics is stronger than the one previously revealed for two-particle systems by Bell’s inequality, where no contradiction arises at the level of perfect correlations. Both spin-correlation and multiparticle interferometry examples are given of suitable three- and four-particle arrangements, both at the gedanken and at the real experiment level.
Lindhard’s paradox—Diffusion of magnetic field into a perfect conductor
Authors: Pippard A. B.
Published: September 2005

This paper develops the argument, presented briefly by Lindhard in 1953, that a perfectly conducting metallic sample is unable to keep out a steady magnetic field, since even in the absence of collisions the field energy can still be reduced by transfer to the electron assembly. For a plane slab the analysis, which is given in detail, is an extension of the theory of the anomalous skin effect, and the ineffectiveness concept plays the same role in elucidating the physical mechanism. In samples of other shapes, interaction with the field may not be so narrowly confined to a small fraction of effective electrons, but it is suggested that the diffusion of magnetic field nevertheless proceeds at a rate that is usually rather indifferent to shape.
Measurements of the superfluid transition in helium by means of a vibrating reed
Authors: Martinez E. N. , Esquinazi P. , Luzuriaga J.
Published: September 2005

A relatively simple method is presented that has been used in an undergraduate experimental course for observing the superfluid transition in 4He. The experimental technique consists of submerging a vibrating reed in liquid helium and observing the changes in frequency and damping (energy dissipation) as the temperature is regulated above and below the superfluid transition. The results of the measurements are interpreted by means of a simplification of the acting hydrodynamic forces, and it is seen that most of the relevant physics is well described by the approximations.
Torque and force on a magnetic dipole
Authors: Vaidman Lev
Published: September 2005

Recent controversies about torque and force on a magnetic dipole are discussed. Three essentially different current loop models are analyzed. Although all models yield the same expression for the torque, N=m×B, the detailed mechanisms that give rise to the torque in each case are very different. The expression for the force on a magnetic dipole is derived and analyzed for all models. The force expression is the same for all current loop models but it differs from the force on a magnetic charge dipole. The expression, obtained for the force on a current loop magnetic dipole, FCL=■(m⋅B)−(d/dt)(m×E/c), differs from what usually appears in the educational literature. The standard ‘‘naive’’ calculation of the force yields the correct expression for the rate of change of the total momentum, dP/dt=∇(m⋅B). However, the current loop in an externa l electric field has an internal ‘‘hidden momentum’’ m×E/c, which is not related to the motion of the center of mass of the dipole. Thus, for the force, defined as mass times acceleration, it is necessary to subtract the time derivative of this ‘‘hidden momentum.’’
On the skin effect approximation
Authors: Smith Glenn S.
Published: September 2005

A simple derivation is presented for the classical skin effect approximation in a good conductor of general shape. The requirements for the validity of the approximation are clearly presented. While two of these requirements are often stated, the third, which is a restriction on the variation of field along the surface of the conductor, is seldom mentioned in intermediate level textbooks on electromagnetism. The requirements are illustrated with a few simple examples.
Hot water, fresh beer, and salt
Authors: Crawford Frank S.
Published: September 2005

In the ‘‘hot chocolate effect’’ the best musical scales (those with the finest tone quality, largest range, and best tempo) are obtained by adding salt to a glass of hot water supersaturated with air. Good scales can also be obtained by adding salt to a glass of freshly opened beer (supersaturated with CO2) provided you first (a) get rid of much of the excess CO2 so as to produce smaller, hence slower, rising bubbles, and (b) get rid of the head of foam, which damps the standing wave and ruins the tone quality. Finally the old question, ‘‘Do ionizing particles produce bubbles in fresh beer?’’ is answered experimentally.
Pioneer women in nuclear science
Authors: Rayner-Canham M. F. , Rayner-Canham G. W.
Published: September 2005

It is a commonly accepted myth that Marie Curie and Lise Meitner were the only women working in the field of nuclear science during the early part of this century. In fact, there were at least 14 others who published work in this field between 1900 and 1915. This paper provides biographical notes on these women and explores the role of the supervisors. Part of the reason for the significant number of women researchers could have been the supportive attitude of Ernest Rutherford toward female physics graduates. In addition, we argue that several of these women provide better role models for potential women physicists than Marie Curie.
An inexpensive apparatus for the measurement of the group velocity of light in transparent media using a modified Helium–Neon laser
Authors: Barr R. , Armstrong T. R.
Published: September 2005

An apparatus is described for the measurement of the group velocity of light in transparent media. The apparatus uses a modified He–Ne laser as a stable, amplitude-modulated light source. With the aid of a single beam oscilloscope, results for the velocity of light in air accurate to ∼0.02% have been obtained. This accuracy is sufficient to enable the relationship between the group and phase velocities in transparent dispersive media to be experimentally verified.
On the electromotive force in moving conductors
Authors: Delellis R. , Kelly H. , Márquez A.
Published: September 2005

This article attempts to clarify the role of the magnetic force and the ‘‘rotational’’ component of the electric field in calculating the induced electromotive force in a circuit having a moving part. For this purpose, an example where both terms occur is analyzed.
Computer simulation of ergodicity and mixing in dynamical systems
Authors: Sperandeo-Mineo R. M. , Falsone A.
Published: September 2005

This article discusses the microcomputer simulation of dynamical systems that have a few degrees of freedom but which illustrate the properties associated with ergodicity and mixing shared by more realistic many-particle systems. The models discussed are a stadium-shaped billiard table model, and a particle moving in a regular array of hard disks or two-dimensional Lennard–Jones particles. The examples show the need for a statistical description and are very useful for pedagogical purposes.
Complex coordinates in central force analyses
Authors: Finkel Robert Warren
Published: September 2005

The standard vector treatment of central force motion is facilitated by representing vectors with complex numbers. When a radius vector r is represented by the polar form r exp (iθ), no vector theorems are required to generate the equations of motion. The formalism is applied to establish conservation of energy and angular momentum. A third conserved quantity, a complex number, is introduced for the inverse-square law. The orbit expression r=r(θ) follows from this quantity without formulating or solving a differential equation.
A square-well potential model to describe lambda-hypernuclei
Authors: Siegel P. B. , Farrow-Reid Mark
Published: September 2005

The binding energy of a number of nuclei is calculated using as an approxi− mation the three-dimensional square-well potential. (AIP)
Simple derivation of Lagrange’s three-body equilibrium
Authors: Styer Daniel F.
Published: September 2005

This article presents a simple derivation of the fact that three noncollinear gravitating particles can move as a rigid body only if they are located at the vertices of an equilateral triangle. The derivation does not assume that the motion is two-dimensional but it does assume that the rotation is uniform.
Measuring small vibrations with interferometry
Authors: Diamond Joshua B. , Donnelly Denis P. , Breault James D. , McCarthy Mary E.
Published: September 2005

A mirror in a Michelson interferometer is attached to a vibrating object, producing a moving fringe pattern and a time-varying light intensity incident on a small-aperture photodetector. Analysis of the photodetector output yields the amplitude and frequency of the object’s vibrations. The technique has previously been applied to measure amplitudes as small as a few light wavelengths. As part of an undergraduate project, an apparatus of this kind has been set up and used to study the resonant vibration of a speaker, which is found to approximate a forced, damped harmonic oscillator.
The oscilloscope as a measuring instrument
Authors: Cooke B. A. , Harris T. J. , Derbyshire H. S.
Published: September 2005

As part of a wide-ranging revision and modernization of the first-year laboratory program a work package was put into effect based solely on introducing our students to the oscilloscope as a measuring instrument. The features of the instrumentation developed to allow the students to investigate the oscilloscope bandwidth and rise time are described and the correct use and value of the variable trigger level control, of external triggering, and the necessity of an oscilloscope probe are learned. The work package is designed to take 6 h to complete, spread over two afternoons.
Asymmetry and convergence in the central limit theorem: An approach for physicists
Authors: Pury Pedro A.
Published: September 2005

A version of the central limit theorem is presented that allows the study of the rate of convergence to the normal probability density of the average of independent identically distributed random variables. Particular emphasis is put on the effect due to the asymmetry of the probability density of the variables. An example is worked out that gives a convincing visual display of the theorem and its convergence.
A relativistic extension of the concepts of the central and conservative forces
Authors: Nishikawa S.
Published: September 2005

Usually in classical dynamics the fundamental quantities of electromagnetism are considered to be the electric and magnetic fields. However, in quantum dynamics, the fundamental quantities of electromagnetism are the potentials. To solve this confusion there is a discussion of how the concepts of the central force and the conservative force in Newtonian dynamics are extended if the Galilean transformation is replaced by the Lorentz transformation. As a result, if only the conservativeness, linearity, and wave nature of field are assumed, a relativistic potential can be axiomatically defined and the dynamical equations that are similar to the equation of motion, the Lorentz condition, and the field equation for the potentials of electromagnetism can be composed. Then, the concept of charge can be introduced leading to electromagnetism without the electric and magnetic fields and the inverse-square law. Thus even in classical dynamics the fundamental quantities of electromagnetism are the potentials. The discussion above is important in the understanding of the mutual relations among Newtonian dynamics, the special theory of relativity, and electromagnetism.
The role of kinematics in Hamiltonian dynamics: Momentum as a Lagrange multiplier
Authors: Whatley M. C.
Published: September 2005

The domain space of the Lagrangian for n degrees of freedom is formally a 2n-dimensional space—e.g., for n=1, L=L(x,v)—and the virtual paths associated with Hamilton’s principle are formal paths in that space of states. The requirement that those paths be consistent with physical kinematics can be regarded as a constraint: A particle’s parametric velocity function v(t) and its position function x(t) must satisfy dx/dt−v(t)=0. When the role of this constraint is explicated by maintaining the logical distinction between v and ẋ, the momentum p emerges as a Lagrange multiplier. When the Lagrangian and Hamiltonian are regarded as state functions, it is seen that the integrand in the customary ‘‘modified Hamilton’s principle’’—which involves nonkinematical paths—is not the Lagrangian function L=pv−H but is rather pẋ−H=L+p(ẋ−v). The relationship of that variational condition to canonical transformations is reviewed, and a parsimonious proof is given for the canonical invariance of the Poisson brackets.
Generating functions and the statistics of radioactive chains
Authors: Vitela E. Javier , Coronado Manuel
Published: September 2005

The statistics of a radioactive chain composed of father nuclei decaying into two daughter nuclei is analyzed. To illustrate the usefulness of generating functions, a master equation is used for the probability density and then an equation is derived and solved for the corresponding generating function. From this function the time evolution of the probability density is obtained. Two cases are considered: (1) when an external source gives rise to the father nuclei and (2) with no external source present. The effect of different initial conditions is also analyzed. The results clarify some misleading statements appearing in the literature, and provide teachers with interesting examples of the use of generating functions.
A plea for a correct translation of Newton’s law of inertia
Authors: Marquit Erwin
Published: September 2005

The formulations of Newton’s law of inertia in university-level textbooks usually follow the Motte–Cajori translation of Newton’s Principia. The rich content of Newton’s law of inertia as a causality principle for mechanics is partly hidden by Motte’s mistranslation of the Latin phrase nisi quatenus as unless instead of except insofar as. One effect of this error is a tendency to isolate from the law some of the ‘‘Definitions’’ that precede Newton’s statement of the three laws of motion. These definitions, when taken together with the laws, provide the basis for Newton’s quantitative and qualitative elaboration of the coupled concepts of mass and force.
Measurement of thermal expansion coefficients using a strain gauge
Authors: Fox John N.
Published: September 2005

A strain gauge is employed to determine the coefficient of thermal expansion of a solid. The apparatus is interfaced to a computer to obtain a linear plot of displacement against temperature.
An alternative treatment of the problem of image formation of an object through plane or spherical interfaces
Authors: Eftaxias K. , Londos C. A. , Vallianatos F.
Published: September 2005

The problem of the virtual image of an object being itself in a different medium than that of the observer is discussed. In dealing with the above problem, plane or spherical refraction surfaces are replaced by lenses. By using this ‘‘trick,’’ one can first study the lens and then proceed to study images formed by an interface. This is the reverse of what most textbooks do. This analysis is simple and suitable for students of introductory physics courses.
Simple forms for equations of rays in gradient-index lenses
Authors: Evans James
Published: September 2005

The calculation of the shape of an optical ray in a gradient-index lens is often greatly simplified by the use of formalism in which the equation governing the ray assumes the form of Newton’s law of motion. A new justification of this method is presented, and the method is applied to a gradient-index lens with cylindrical symmetry.
On heat conduction in one-dimensional solids
Authors: Díaz-Guilera Albert
Published: September 2005

A simple and intuitive procedure to derive the equation of heat conduction in a one-dimensional solid is presented. A boundary condition at the surface for a three-dimensional solid becomes a new term of heat exchange with an external bath in the equation for the temperature in one dimension.
Oscillation of a gas balloon due to a temperature gradient
Authors: Mellen Walter Roy
Published: September 2005

A gas balloon floats at some level if the temperature increases with height. If the balloon is disturbed, it oscillates about this level with a damped harmonic motion. The derived period of the motion due to a temperature gradient dT/dy is 2π[(T/g)/(dT/dy)]1/2, where T is the absolute temperature and g is the acceleration of gravity. With no temperature gradient, the derived period of oscillation due to the pressure gradient is 2π[TR/Mg2]1/2, where R is the universal gas constant and M is the molecular mass of air. The primary experimental difficulties are temperature measurements, air currents, and damping.
Everett’s theory and the ‘‘many-worlds’’ interpretation
Authors: Ben-Dov Yoav
Published: September 2005

A distinction is made between the ‘‘popular many-worlds interpretation’’ and Everett’s ‘‘relative state’’ formulation of quantum mechanics. This distinction is claimed to be relevant to arguments raised against both schemes.
Wilberforce’s pendulum revisited
Authors: Köpf Ulrich
Published: September 2005

If its radius of gyration is properly tuned, a mass attached to a simple helical spring is able to perform longitudinal and torsional oscillations that are coupled with one another and exhibit the beats characteristic to coupled oscillations. The mechanism of coupling as well as the corresponding differential equations are derived. For the case of resonance, the solutions are compared with experimental results.
The Hall effect in copper: An undergraduate experiment
Authors: Armentrout Charles E.
Published: September 2005

This experiment on the Hall effect uses only apparatus ordinarily found in an elementary laboratory, and the Hall effect device itself can be constructed easily from ordinary materials without special equipment. It illustrates the vector nature of the Lorentz force equation, shows that the charge carriers in copper are negatively charged, and permits determination of the charge-carrier density and other related parameters in ordinary commercial copper.
The fields of a bounded electromagnetic beam propagating through an air gap between two dielectrics for frustrated total reflection
Authors: Pomer Francisco , Navasquillo Joaquin
Published: September 2005

Matrix relations for the Cartesian components of all the fields associated with a plane electromagnetic wave impinging on a dielectric slab between two media are derived in terms of the incident fields, and then applied to the case of an air gap between two identical dielectric media and angles of incidence greater than the critical angle. The fields and the Poynting vector for a two-dimensional beam represented by means of the angular spectrum method are obtained, showing that when the gap width is less than a wavelength, some energy crosses the gap in the form of a transmitted beam.
The motion of wave packets through their expectation values and uncertainties
Authors: Styer Daniel F.
Published: September 2005

The Ehrenfest equations govern the time evolution of quantal expectation values, and they have a widely known generalization that governs the time evolution of quantal uncertainties. Unfortunately, these results are usually applied only to free particles. Here, these methods are reviewed and applied to the simple harmonic oscillator, revealing a simple and beautiful picture of wave-packet motion with an obvious classical limit.
About the statistical meaning of group velocity
Authors: Wang Wen-Cheng , Chen Cheng-Ming
Published: September 2005

The propagation velocity of light in a medium is the transport velocity of the energy. According to the theory of wave motion, this velocity is also the group velocity of light. But, by using quantum theory, it is proved that the group velocity is not the instantaneous velocity of a photon, but rather the statistical average velocity of photons in a medium.
Equation-solving software packages: Uses in the undergraduate curriculum
Authors: Donnelly Denis
Published: September 2005

Equation-solving software packages provide undergraduate students considerable power in performing the numerical tasks that they encounter. The range of possible uses is great; these include plotting functions, solving simultaneous equations, performing numerical differentiation and integration, and solving differential equations by iterative methods. MathCAD and eureka are two equation-solving packages of possible interest. These packages have some overlapping of capabilities, but MathCAD has many more features, e.g., seeded iterations, and is more powerful than eureka. eureka has a slightly more friendly interface and may seem less imposing to the beginner.
Direct calculation of electric and magnetic forces from potentials
Authors: Jefimenko Oleg D.
Published: September 2005

A new method for calculating electric and magnetic forces is presented. The method makes it possible to calculate forces directly from scalar and vector potentials rather than from field vectors. Several new force equations are derived. They are grouped in four categories: equations for calculating electric forces from electric scalar potentials; equations for calculating electric forces from electric vector potentials; equations for calculating magnetic forces from magnetic vector potentials; and equations for calculating magnetic forces from magnetic scalar potentials. Since the potentials are usually easier to compute than the corresponding fields, the new equations provide an effective alternative to the previously available techniques for force calculations. From the theoretical point of view, these equations reveal a physical significance of electric and magnetic potentials not heretofore apparent and provide a new insight into the nature of electric and magnetic forces. Illustrative examples on the use of the equations are given.
On dynamical equations and conservation laws in quasistatic electromagnetic systems
Authors: Aguirregabiria J. M. , Hernández A. , Rivas M.
Published: September 2005

The necessary attribution of linear and angular momenta to the electromagnetic field, even in quasistatic situations, is illustrated by discussing the dynamical conservation laws of an interacting system composed of two point charges and a magnetic dipole. The evaluation of the trajectories gives an interesting example for numerical computation.
Interference patterns from circularly polarized light using a Michelson interferometer
Authors: Mellen Walter Roy
Published: September 2005

Coherent light in the two arms of a Michelson interferometer are made circularly polarized but with opposite rotations. When the two beams recombine, the light is linearly polarized but the direction of polarization changes depending on the phase between the two beams. When a linear polarizer is used on the output and rotated, the observed interference fringe pattern shifts. If the field of view contains circular fringes, the continuous rotation of the polarizer in one direction makes the circles continuously expand or contract.
Right and wrong use of the Lenz vector for non-Newtonian potentials
Authors: McDonald K. T. , Farina Carlos , Tort Alexandre
Published: September 2005

An article that recently appeared in this Journal1 propagates an error first published here more than 20 years ago.2 This concerns a proposed use of the Lenz vector3 in obtaining the rate of precession of the perihelion for a central-force problem in which the potential has a small departure from the Newtonian form, V(r)=k/r. Three correct solutions to this problem are presented, based on the effective-potential technique, on the method of averages, and on a careful use of the Lenz vector.4
A 0.7-mW magnetic heat engine
Authors: Mattocks P. G.
Published: September 2005

A small magnetic heat engine is described that clearly demonstrates the principles involved and possible usefulness on a larger scale. It attracts considerable attention at open house demonstrations. Operating between 0 and ∼4 °C, it produces a measurable output of 0.7 mW corresponding to 0.5 W kg−1 of working substance (gadolinium).
Spectrum analysis of optical resonators
Authors: Whitlatch Norman
Published: September 2005

The oscilloscope display of frequency spectra for both passive and active He–Ne laser cavities is described. The experimental methods exploit the ability of a common He–Ne laser tube to serve as an optical spectrum analyzer. The only additional apparatus required is typically found in the introductory laboratory. Accuracies within 2% can be obtained, and frequencies can be resolved to better than one part in 108.
An algebraic approach for solving mechanical problems
Authors: Farina de Souza Carlos , Mendes Gandelman Miriam
Published: September 2005

An algebraic approach is used to obtain the time evolution of physical systems based on the Poisson bracket formulation of classical mechanics. This method is applied in many examples, including three-dimensional problems.
On seeing instantaneous centers of velocity
Authors: Tea Peter L.
Published: September 2005

The instantaneous center of velocity is seen in still photographs for a variety of rigid bodies in plane motion.
Canonical transformation in quantum mechanics
Authors: Kim Y. S. , Wigner E. P.
Published: September 2005

The phase-space picture of quantum mechanics and some examples illustrating it are presented. Since the position and momentum are c numbers in this picture, it is possible to introduce the concept of phase space in quantum mechanics. The uncertainty relation is stated in terms of an area element in phase space, whose minimum size is Planck’s constant. Area-preserving canonical transformations in phase space are therefore uncertainty-preserving transformations. The wave-packet spread, coherent-state representation, and squeezed states of light are discussed as illustrative examples.
Einstein before 1905: The early papers on statistical mechanics
Authors: Gearhart Clayton A.
Published: September 2005

Albert Einstein’s work on the quantum and Brownian motion, which he began to publish in 1905, was preceded by three papers on kinetic theory and statistical mechanics published between 1902 and 1904. In these early papers, which give us considerable insight into Einstein’s early education and development, Einstein independently derived many of Boltzmann’s and Gibbs’ results, including the canonical ensemble, an analysis of fluctuations, and the relation between entropy and probability. This article discusses those papers and their background in 19th-century physics.
Doppler effect for sound via classical and relativistic space-time diagrams
Authors: Reynolds Robert E.
Published: September 2005

The longitudinal classical and relativistic Doppler effects for sound are derived through use of space-time diagrams. Diagrams appropriate to Galilean and Lorentz transformations are reviewed in order to stress their simplicity and instructional value. The relativistic limiting case in which the signal is light is also considered.
Random multiplicative processes: An elementary tutorial
Authors: Redner S.
Published: September 2005

An elementary discussion of the statistical properties of the product of N independent random variables is given. The motivation is to emphasize the essential differences between the asymptotic N→∞ behavior of a random product and the asymptotic behavior of a sum of random variables—a random additive process. For this latter process, it is widely appreciated that the asymptotic behavior of the sum and its distribution is provided by the central limit theorem. However, no such universal principle exists for a random multiplicative process. In this case, the ratio between the average value of the product 〈P〉 and the most probable value Pmp diverges exponentially in N as N→∞. Within a continuum approximation, the classical log-normal form is often invoked to describe the distribution of the product. It is shown, however, that the log-normal provides a poor approximation for the asymptotic behavior of the average value and, also, for the higher moments of the product. A procedure for computing the correct leading asymptotic behavior of the moments is outlined. The implications of these results for simulations of random multiplicative processes are also discussed. For such a simulation, the numerically observed ‘‘average’’ value of the product is of the order of Pmp, and it is only when the simulation is large enough to sample a finite fraction of all the states in the system that a monotonic crossover to the true average value 〈P〉 occurs. An idealized, but quantitative account for this crossover is provided.
Coriolis and magnetic forces: The gyrocompass and magnetic compass as analogs
Authors: Opat Geoffrey I.
Published: September 2005

The Coriolis force on a particle (acting within a rotating frame) and the magnetic force on a charge, are both proportional to and act perpendicular to the velocity. As a consequence, the action of the Coriolis force on rotating mass, and the action of the magnetic force on rotating charge are formally identical. Just as the action of a magnetic field is to align the axis of the rotating charge distribution (magnetic dipole) with itself, so the Coriolis force aligns the axis of a rotating mass distribution (angular momentum) with the angular velocity of the rotating frame. This enables us to understand the gyrocompass by analogy with the magnetic compass. A simple way to demonstrate the gyrocompass to a class is also presented.
Symmetry in electrodynamics: A classical approach to magnetic monopoles
Authors: Zeleny W. B.
Published: September 2005

A simple means of deriving Maxwell’s equations and the Lorentz force law from their symmetries is presented. The same approach is then used to derive the classical theory of magnetic monopoles and dual-charged particles.
Analogy between general relativity and electromagnetism for slowly moving particles in weak gravitational fields
Authors: Harris Edward G.
Published: September 2005

Starting from the equations of general relativity, equations similar to those of electromagnetic theory are derived. It is assumed that the particles are slowly moving (v≪c), and the gravitational field is sufficiently weak that nonlinear terms in Einstein’s field equations can be neglected. For static fields, the analogy to electrostatics and magnetostatics is very close. Results are compared with those of a previous derivation by Braginsky, Caves, and Thorne [Phys. Rev. D 15, 2047–2068 (1977)]. These results lead to very simple derivations of the Lense–Thirring precession [Phys. Z. 19, 156–163 (1918)] and the spin-curvature force of Papepetrou [Proc. R. Soc. London Ser. A 209, 248–258 (1951)] and Pirani [Acta Phys. Pol. 15, 389–405 (1956)].
Transition between particle behavior and wave behavior
Authors: Bardou François
Published: September 2005

Qualitatively the same experiment can reveal either the particle behavior, or the wave behavior, or even intermediate behaviors, if the experiment is performed in each case with adequate parameters. A simple parametrized double-slit apparatus can be built for this purpose. This helps to make more explicit the idea of particle–wave duality. An ‘‘interpretation function’’ is also defined to estimate the possibilities of interpretation.
Eddy current distributions: Their calculation with a spreadsheet and their measurement with a dual dipole antenna probe
Authors: Hart Francis X. , Wood Kenneth W.
Published: September 2005

A spreadsheet is used to calculate numerical values for the eddy current/induced electric field distribution produced by a time-varying magnetic field in a saline solution. The value of each cell corresponds to the current density of a loop in the object. It is related to the current densities in its neighboring cells and the resistivities of their borders. The resulting set of equations is solved by iteration. Comparisons of the two components of the induced field are made with analytical solutions and with experimental values measured by a dual dipole antenna system. Such analyses can be used to provide students with a better understanding of the induced field distribution in irregularly shaped objects.
A simple derivation of the Doppler effect for sound
Authors: Cook G. , Lesoing T.
Published: September 2005

The Doppler formulas for sound waves are derived in a particularly simple way from a space-time diagram. The method used emphasizes use of space-time diagrams in a quantitative way.
The Runge–Lenz vector for the two-dimensional hydrogen atom
Authors: Yang X. L. , Lieber M. , Chan F. T.
Published: September 2005

The recent discovery of novel properties possessed by two-dimensional systems has led to the investigation of properties of the hydrogen atom in two dimensions. With proper definition of the system, the so-called Runge–Lenz vector may be defined for this system, and shown to be related to the underlying O(3) symmetry, just as for the three-dimensional system, although some interesting aspects are revealed.
Electromagnetic wave propagation in media with complex refractive indices and transmission through absorbing films
Authors: Lodenquai J. F.
Published: September 2005

The usual textbook treatment of electromagnetic (em) wave propagation across the boundary separating two dielectrics is generalized to materials characterized by complex refractive indices. Such materials are being investigated intensively and possess interesting and potentially useful properties in the infrared (IR) and optical bands. The sum of the transmission and reflection coefficients at the boundary in these materials does not add to unity, as in the case of perfect dielectrics, and the reason for this unexpected behavior is examined. Since these materials are usually fabricated in film form, appropriate transmission and reflection coefficients of thin and thick films with arbitrary refractive indices are derived (taking account of multiple reflections) to point out the differences introduced by the complex nature of the refractive index which have not been included in previous results or which have generally not been emphasized. Results are also presented for films deposited on thick substrates of arbitrary refractive indices since this is the common mode of fabrication. Graphs are presented to illustrate the transmission behavior of a thin absorbing film with and without a substrate.
New quantum mechanical perturbation technique using an ‘‘electronic scratchpad’’ on an inexpensive computer
Authors: Lynch R. , Mavromatis H. A.
Published: September 2005

The authors have developed a new method for doing numerical quantum mechanical perturbation theory. It has the flavor of Rayleigh–Schrödinger perturbation theory (division of the Hamiltonian into an unperturbed Hamiltonian and a perturbing term, use of the basis formed by the eigenfunctions of the unperturbed Hamiltonian) while turning out to be a variational technique. Furthermore, it is easily implemented by means of the widely used ‘‘electronic scratchpad,’’ MathCAD 2.0, using an inexpensive computer. As an example of the method, the problem of a harmonic oscillator with a quartic perturbing term is examined.
Cumulant expansions in statistical and quantum physics
Authors: Gersch H. A.
Published: September 2005

Two examples are given of the use of cumulant expansions for systems of particles with hard-core two-body interactions. One of these, in classical statistical mechanics, is shown to lead to virial coefficients. An analogous procedure in quantum mechanics produces energy level shifts due to hard-core interactions that are expressed in terms of the t matrix.
Use of a saline oscillator as a simple nonlinear dynamical system: Rhythms, bifurcation, and entrainment
Authors: Yoshikawa Kenichi , Oyama Noboru , Shoji Masaru , Nakata Satoshi
Published: September 2005

An oscillatory flow of water is generated when a cup, with a small orifice (ca. 1 mm diam) in its base, and filled with saline water, is placed within an outer vessel containing pure water. This simple system exhibits various nonlinear characteristics, such as a limit cycle, bifurcation of the oscillatory flow, and entrainment among the oscillators. These characteristics can be reproduced by a numerical simulation based on simple nonlinear differential equations.
A general physics laboratory investigation of the thermodynamics of a rubber band
Authors: Savarino G. , Fisch M. R.
Published: September 2005

The appropriateness of the standard equation of state of a rubber band is studied in an undergraduate laboratory experiment. There are two parts to the experiment: In the first part, a band is stretched at constant temperature; in the second, the band is kept at constant length while heated. The two coefficients in the equation of state of the rubber band are determined in four different ways: numerical differentiation of the data, numerical integration of the data, regression analysis, and linear regression on a ‘‘straight-line’’ form of the equation of state. The advantages and disadvantages of each technique are discussed. The characteristic constants describing the rubber band appear to vary by up to 15% depending on the technique of analysis employed. The origins of these variations are discussed.
Alchemy in 1+1 dimension: From bosons to fermions
Authors: Galić H.
Published: September 2005

Canonical massless fermion field is constructed from canonical boson field in 1+1 dimensional space. Single-fermion states are expressed in terms of eigenstates of boson operators.
Classical limit of quantum mechanics (electron in a magnetic field)
Authors: Rowe E. G. Peter
Published: September 2005

Different classical limits of quantum mechanics are discussed in the setting of a simple example, the two-dimensional motion of a spinless electron in a uniform magnetic field. Both the time-independent energy eigenstates and the time-dependent states developing from initial localized states are exactly solvable. In the first example, an ℏ→0 limit with fixed energy leads to a simple ensemble of classical orbits. In the second, ℏ→0 with fixed initial mean position and momentum leads to a unique classical orbit.
Magnetic drag in the quasi-static limit: A computational method
Authors: Marcuso M. , Gass R. , Jones D. , Rowlett C.
Published: September 2005

The method of successive approximations is applied to Maxwell’s equations to calculate the magnetic drag on a conducting disk rotating under the influence of a localized nonuniform magnetic field. An expression for the damping torque produced by the magnetic field is obtained in the low-velocity (quasi-static) limit of the disk’s motion: The damping force, in the case of rectilinear motion, is also calculated. When the theoretical expression for the damping torque is specialized to the case of a uniform magnetic field, the result is found to be identical with that of an existing textbook treatment. In the Appendix, a simplified treatment of the magnetic drag problem suitable for an introductory-level laboratory class is given. This treatment yields a final expression for the damping torque which is identical in form to the rigorous result except for a scaling factor.
A general method for calculating reflection and transmission coefficients in multiple slabs
Authors: Morales Juan J. , Nuevo María J.
Published: September 2005

For normal incidence onto a stack of multiple slabs, a method is described that relates the reflection and transmission coefficients in an inner slab and the reflection coefficients in two consecutive slabs, and that gives an expression for the reflection and transmission coefficients in the last slab. This method is more general than that traditionally used, and becomes easier and quicker as the number of slabs increases.
Nuclear potential in the Yukawa model
Authors: Gauthier N. , Sherrit S.
Published: September 2005

A realistic expression for the net nuclear potential is obtained as the sum of the contributions due to the inner core and the outer shell. (AIP)