Volume 64, Issue 10, October 1996
 Papers


Experimental projects in graduate theoretical physics courses
View Description Hide DescriptionSome beginning graduate courses in physics at the University of Chicago have been taught with final projects in addition to or in place of written final examinations. Although these courses and many of the projects are theoretical, experimental projects have been encouraged, with some success. A few examples are discussed.

When charge meets flux: Classical view of a quantum paradigm
View Description Hide DescriptionQuantum mechanics is known to forbid charged, spinless particles from entering an infinitely thin tube of magnetic flux, though particles which do possess spin may even be bound inside. For spin 1/2 the particles may be ‘‘quasibound,’’ so that some penetration from outside to inside is allowed. We use freshman physics and plane geometry to explore such charge‐flux systems in classical electrodynamics, finding remarkable similarities to the quantum case, along with characteristic differences. Both aspects illuminate the relationship between classical and quantum physics.

High‐altitude free fall
View Description Hide DescriptionThe problem of an object falling from high altitudes where the variation of atmospheric pressure cannot be neglected is investigated. The equation of motion for the variation of the velocity of the object as a function of altitude is solved exactly. The results show that, unlike an object falling in a uniform atmosphere whose speed monotonically increases and approaches the terminal speed, the speed of a high‐altitude falling object first increases, goes through a maximum, and then decreases and approaches the terminal speed from above. The results also show that if the initial altitude of the object is greater than a critical value, the object always strikes the ground with a speed that is higher than its terminal speed by a finite value, in contrast to the case of a freely falling object in a uniform atmosphere.

The multivariate Langevin and Fokker–Planck equations
View Description Hide DescriptionA novel derivation of the Langevin equation that was recently presented in this journal for a univariate continuous Markov process is generalized here to the more widely applicable multivariate case. The companion multivariate forward and backward Fokker–Planck equations are also derived. The derivations require just a few modest assumptions, and are driven by a self‐consistency condition and some established theorems of random variable theory and ordinary calculus. The constructive nature of the derivations shows why a multivariate continuous Markov process must evolve according to equations of the canonical Langevin and Fokker–Planck forms, and also sheds new light on some uniqueness issues. The need for self‐consistency in the time‐evolution equations of both Markovian and non‐Markovian stochastic processes is emphasized, and it is pointed out that for a great many non‐Markovian processes self‐consistency can be ensured most easily through the multivariate Markov theory.

Simple demonstrations for introducing spatial coherence
View Description Hide DescriptionWhen two quasiparallel identical beams, originating from the same laser, are superposed, the crossing volume becomes the seat of a system of interference fringes. The spatialcoherence of the radiation filling this region may be evaluated by observing the interference pattern generated on a far screen. When the beams originate from two identical but different lasers, the fringe system in the crossing region will be washed out by mutual phase fluctuations: the radiation field, however, retains its ability to produce an interference pattern when it goes through a double slit. This remark may be the starting point for a simpler and intuitive approach to the van Cittert–Zernike theorem. Simple laboratory experiments are described which illustrate these results.

Thermodynamic entropy: The spreading and sharing of energy
View Description Hide DescriptionA new approach to thermodynamic entropy is proposed to supplement traditional coverage at the junior–senior level. It entails a model for which: (i) energy spreads throughout macroscopic matter and is shared among microscopic storage modes; (ii) the amount and/or nature of energy spreading and sharing changes in a thermodynamic process; and (iii) the degree of energy spreading and sharing is maximal at thermodynamic equilibrium. A function S that represents the degree of energy spreading and sharing is defined through a set of reasonable properties. These imply that S is identical with Clausius’ thermodynamic entropy, and the principle of entropy increase is interpreted as nature’s tendency toward maximal spreading and sharing of energy. Microscopic considerations help clarify these ideas and, reciprocally, these ideas shed light on statistical entropy.

The impact of video motion analysis on kinematics graph interpretation skills
View Description Hide DescriptionVideo motion analysissoftware was used by introductory physics students in a variety of instructional settings. 368 high school and college students took part in a study where the effect of graduated variations in the use of a video analysis tool was examined. Post‐instruction assessment of student ability to interpret kinematics graphs indicates that groups using the tool generally performed better than students taught via traditional instruction. The data further establishes that the greater the integration of video analysis into the kinematics curriculum, the larger the educational impact. An additional comparison showed that graph interpretation skills were significantly better when a few traditional labs were simply replaced with video analysis experiments. Hands‐on involvement appeared to play a critical role. Limiting student experience with the video analysis technique to a single teacher‐led demonstration resulted in no improvement in performance relative to traditional instruction. Offering more extensive demonstrations and carrying them out over an extended period of time proved somewhat effective. The greatest impact came from a combination of demonstrations with hands‐on labs. The curricular modifications employed in the different classrooms and the methods used to evaluate them are discussed.

Measurement of the efficiency of a BF_{3} counter
View Description Hide DescriptionA final year undergraduate nuclear experiment is described in which the efficiency of a boron trifluoride (BF_{3}) counter is determined. This involves attaching neutron sensitive activation foils to the surface of the counter, exposing the foils to the neutron flux from an Am—Be neutron source for a given time, and measuring the total number of neutrons entering the sensitive volume of the counter. Several basic principles of nuclear physics are incorporated into one single experiment which has proved instructive and challenging to final year undergraduate students. The student is expected to understand and apply topics such as radioactive half‐life, absorption of nuclear radiation, nuclear detection, neutron activation and nuclear decay schemes in a single experiment. The experiment can also be converted into an open ended project where, in addition, the student can study several other aspects of the subject including the kinetics of (α,n) reactions.

Updating Monte Carlo algorithms
View Description Hide DescriptionUsing the long‐range Ising model, we present modern Monte Carlo techniques—single and multiple histogram and entropic sampling—which permit increasing the amount of information obtained from a simulation. Numerical results for the density of states, mean energy and specific heat are compared with exact calculations, easily handled in this case. As a consequence of the simplicity of the model, the ability of those methods to generate continuous plots of thermodynamical quantities can be appreciated even by students taking basic courses of statistical physics.

Effective Lagrangians and nonlinear electromagnetism
View Description Hide DescriptionLorentz invariance together with gauge invariance provide strong constraints on possible generalizations of classical electrodynamics. We show here how to construct the lowest‐order corrections to the MaxwellLagrangian, and examine the implications in a few interesting cases. While the above invariance principles are not sufficient to completely determine the model, specific predictions may be obtained if we allow the remaining arbitrary parameters to take on the values obtained within a given fundamental theory—QED in the context of the present paper. This straightforward approach provides the student with a new perspective on classical electromagnetism, and on the role played by the vacuum.

Mandated energy dissipation—e pluribus unum
View Description Hide DescriptionA single mathematical model describes disparate phenomena involving colliding masses, interacting capacitors, or ‘‘dueling’’ springs. In each case the dissipation of a definite fraction of system energy is mandated when a conservation principle and a final state constraint are imposed. The mathematical ‘‘oneness’’ points not only to the physical similarity but also to a notable physical difference among the various phenomena. An important by‐product is an enrichment of the usual treatment of two particle collisions.

A novel geometry for Rutherford scattering
View Description Hide DescriptionWe describe a new geometry for a Rutherford scattering experiment which utilizes an annular detector and movable target. The apparatus provides both a good counting rate and simple data reduction. Both the angular dependence and the absolute value of the cross section can be obtained for a reasonable range of scattering angles.

Rayleigh–Gans scattering from polydisperse colloidal suspensions
View Description Hide DescriptionThe scattering of polarized light from a collection of dielectric spheres with model disorder in the particle dimensions is derived in the first Born approximation. The paper is intended as a brief pedagogical review of some of the essential physics behind static light scattering from complex fluids, and attempts to elucidate the natural emergence of the statistical‐structure and form factors while emphasizing the role of polydispersity in the particle size. The criterion for the convergence of the Born series is examined. The problem is cast in the language of Fourier transforms and correlation functions at a level that should be accessible to advanced undergraduates.

Using music to teach physics
View Description Hide DescriptionThe analysis and synthesis of tones produced by acoustic musical instruments is used as a tool for teaching physics to students majoring in disciplines other than the sciences. A series of laboratory exercises, based on soundanalysis and synthesis software running on microcomputer‐based workstations, is described and placed in context with more traditional physics laboratory experiences and written assignments in a one‐semester course in the physics of musical sound taught without advanced mathematics. Student reaction to the exercises is discussed.

An exactly soluble Fresnel diffraction model of two‐slit interference
View Description Hide DescriptionUsing the formalism of Fresnel diffraction, we examine the diffraction and interference of two initially widely separated parallel Gaussian light beams. This two‐beam configuration is a version of Young’s two‐slit interference problem that is free of the mathematical complexity that occurs in most Fresnel diffraction calculations. As a result, it clearly and simply illustrates the transition from ray optics to wave interference.

Rigorous Newtonian cosmology
View Description Hide DescriptionIt is generally believed that it is not possible to rigorously analyze a homogeneous and isotropic cosmological model in Newtonian mechanics. I show on the contrary that if Newtonian gravity theory is rewritten in geometrical language in the manner outlined in 1923–1924 by Élie Cartan [Ann. Ecole Norm. Sup. 40, 325–412 (1923); 41, 1–25 (1924)], then Newtonian cosmology is as rigorous as Friedmann cosmology. In particular, I show that the equation of geodesic deviation in Newtonian cosmology is exactly the same as equation of geodesic deviation in the Friedmann universe, and that this equation can be integrated to yield a constraint equation formally identical to the Friedmann equation. However, Newtonian cosmology is more general than Friedmann cosmology: Ever‐expanding and recollapsing universes are allowed in any noncompact homogeneous and isotropic spatial topology. I shall give a brief history of attempts to do cosmology in the framework of Newtonian mechanics.

More than misconceptions: Multiple perspectives on student knowledge and reasoning, and an appropriate role for education research
View Description Hide DescriptionThis article analyzes an excerpt of a discussion from a high schoolphysics class from several different perspectives on students’ knowledge and reasoning, illustrating a range in what an instructor might perceive in students’ work and take as tasks for instruction. It suggests a view of current education research as providing perspectives to expand, refine, and support instructors’ perceptions and judgment, rather than as providing definitive principles or proven methods.

Ripples in the energy of a damped oscillator: The experimental point of view
View Description Hide DescriptionAn irreversible transformation of mechanical into thermal energy takes place during the motion of a damped harmonic oscillator, with the result that the level of the total mechanical energy of the system, as a first approximation, decays exponentially with time. A detailed description of this decrease, however, is not usually supplied in textbooks of classical mechanics or general physics. As Karlow has recently pointed out, the negative‐exponential decay is modulated by a sequence of energy ripples, owing to the dissipation rate being not constant during the motion. Up to now, the analysis of this behavior has been based only on theoretical considerations; in this work we present the corresponding experimental evidence based on a couple of laboratory demonstrations that use an electrical RLC circuit and a hydrodynamically damped mechanical pendulum.
