Volume 67, Issue 11, November 1999
 GUEST COMMENT
 PAPERS


Some paradoxes, errors, and resolutions concerning the spectral optimization of human vision
View Description Hide DescriptionThe peak brightness of the solar spectrum is in the green when plotted in wavelength units. It peaks in the nearinfrared when plotted in frequency units. Therefore the oftquoted notion that evolution led to an optimized eye whose sensitivity peaks where there is most available sunlight is misleading and erroneous. The confusion arises when density distribution functions like the spectral radiance are compared with ordinary functions like the sensitivity of the eye. Spectral radiance functions, excepting very narrow ones, can change peak positions greatly when transformed from wavelength to frequency units, but sensitivity functions do not. Expressing the spectral radiance in terms of photons per second, rather than power, also causes a change in the shape and peak of the distribution, even keeping the choice of bandwidth units fixed. The confusion arising from comparing simple functions to distribution functions occurs in many parts of the scientific and engineering literature aside from vision, and some examples are given. The eye does not appear to be optimized for detection of the available sunlight, including the surprisingly large amount of infrared radiation in the environment. The color sensitivity of the eye is discussed in terms of the spectral properties and the photo and chemical stability of available biological materials. It is likely that we are viewing the world with a souvenir of the human evolutionary voyage.

Understanding the penetration of electromagnetic velocity fields into conductors
View Description Hide DescriptionThe electromagnetic fields due to a point charge can be broken into velocity fields and acceleration fields. The acceleration fields give rise to electromagnetic waves whose penetration into ohmic conductors is described by exponential damping with a characteristic skin depth. The velocity fields allow a steadycurrent limit where the magnetic field penetrates even a good conductor. Here we note the contrasts between velocity fields and wave fields in their interactions with conductors. We derive a new timeintegral invariant of the magnetic field in an ohmic conductor. Finally we note the disparate analyses in the literature and suggest the following summary regarding the penetration of the electromagnetic fields of a point charge moving parallel to a conducting surface. (1) The falloff of the electromagnetic fields is algebraic, not exponential, and cannot be characterized by a skin depth. (2) In the limit of low velocity, the magnetic field penetration is independent of the conductivity of the material. (3) In general the penetration of the electromagnetic fields depends upon the velocity of the particle and becomes vanishingly small for a perfect conductor. (4) The time integral of the magnetic field at a fixed spatial point inside (or outside) an ohmic conductor is independent of the conductivity of the material; thus as the conductivity of the material becomes larger and the magnetic field inside becomes smaller, the time of penetration becomes longer. The penetration of timedependent velocity fields into conductors has become of interest largely in connection with the Aharonov–Bohm effect. It is curious that the classical explanation for the Aharonov–Bohm effect depends upon the time integral of the magnetic field, which is independent of the conductivity of any ohmic shielding material.

From Maupertuis to Schrödinger. Quantization of classical variational principles
View Description Hide DescriptionIn his first paper on wave mechanics, Schrödinger presented a heuristic argument which led from the Hamilton–Jacobi equation through the quantum variational principle to his famous wave equation. In his second paper, Schrödinger withdrew this heuristic argument as “incomprehensible.” We show by using a recently generalized form of Maupertuis’ principle that Schrödinger’s original heuristic argument can be made more logical. Aside from pedagogical interest, this path is useful as a method of quantization of general mechanical systems.

An accurate measurement of gusing falling balls
View Description Hide DescriptionWe describe an experiment to determine the acceleration due to gravity, g, with an accuracy of about 1 part in The experiment was designed to expose students to critical thinking in collecting, selecting, and analyzing data, and interpreting the results.

Motion of a wave packet in the Klein paradox
View Description Hide DescriptionComputer simulations for the scattering of a relativistic wave packet by a step potential are performed. Under the condition of the Klein paradox, where the height of the potential barrier becomes supercritical, the scattering induces the creation of electron–positron pairs. The “pair creation” is demonstrated in the present simulation by making use of the hole theory.

A spin network primer
View Description Hide DescriptionSpin networks, essentially labeled graphs, are “good quantum numbers” for the quantum theory of geometry. These structures encompass a diverse range of techniques which may be used in the quantum mechanics of finite dimensional systems, gauge theory, and knot theory. Though accessible to undergraduates, spin network techniques are buried in more complicated formulations. In this paper a diagrammatic method, simple but rich, is introduced through an association of matrices with diagrams. This spin network diagrammatic method offers new perspectives on the quantum mechanics of angular momentum,group theory,knot theory, and even quantum geometry. Examples in each of these areas are discussed.

Experiments with electrical resistive networks
View Description Hide DescriptionTheoretical problems involving equivalent resistances of large or infinite networks of resistors have received substantial attention. We consider two actual networks. In the first, the resistance is measured across one end of a ladder whose number of loops is incremented until the precision of the multimeter is exceeded. In the second, resistances are measured across nodes near the center of a 12 by 12 square grid of resistors. These experiments are useful in the introductory physics laboratory as interesting examples of equivalent resistance, and can be added to a standard Ohm’s law experiment. The square grid apparatus also can be employed for lecture demonstrations. In addition, this apparatus offers approximate experimental confirmation of complicated theoretical calculations for the equivalent resistance between two nonadjacent nodes of an infinite square grid. These experimental results are verified numerically.

Relationship of the commutation rules to classicallike representations of quantum angular momenta addition
View Description Hide DescriptionWe perform an analysis of a graphical representation for the addition of two angular momenta, focusing our attention on the angle δ between the xy components of two angular momenta. Then we propose a new complete set of commuting observables, which differ from the usual sets that are connected by the Clebsch–Gordan coefficients. This set shows that the angle δ can be a welldefined variable in quantum mechanics. An empirical analysis of the graphical representations of the angular momenta relations, which may include the angle δ, followed by quantum mechanical considerations, leads to the vanishing of certain quantum angular momentum commutators for specific states. Therefore, although the commutators are not null in general, the quantum addition of angular momenta may be represented using classicallike diagrams.

A simple calculation of the rate of emission of energy and of linear and angular momentum by a point charge in arbitrary motion
View Description Hide DescriptionWe calculate directly the rates of emission of energy and linear momentum by a point charge in arbitrary motion using mathematical results that render the calculation of solidangle integrals very simple. We show that the results of these explicit calculations agree with those based on covariance, illustrating how deeply special relativity is rooted in classical electrodynamics. Then, we use these covariance arguments to calculate the emission of angular momentum, which is a new result. We also indicate how this calculation can be done in the direct, but much longer, way.

Measuring the interaction force between a high temperature superconductor and a permanent magnet
View Description Hide DescriptionRepulsive and attractive forces are both possible between a superconducting sample and a permanent magnet, and they can give rise to magnetic levitation or freesuspension phenomena, respectively. We show experiments to quantify this magnetic interaction, which represents a promising field with regard to shortterm technological applications of high temperature superconductors. The measuring technique employs an electronic balance and a rareearth magnet that induces a magnetic moment in a melttexturedsuperconductor immersed in liquid nitrogen. The simple design of the experiments allows a fast and easy implementation in the advanced physics laboratory with a minimum cost. Actual levitation and suspension demonstrations can be done simultaneously as a help to interpret magnetic force measurements.

Relativistic contraction of an accelerated rod
View Description Hide DescriptionThe relativistic motion of an arbitrary point of an accelerated rigid rod is discussed for the case when velocity and acceleration are directed along the rod’s length. The discussion includes the case of a timedependent force applied to a single point, as well as a timeindependent force arbitrarily distributed along the rod. The time dependence of the rod’s relativistic length depends on the application point of the force, but after the termination of acceleration the final velocity and length do not depend on it. An observer on a uniformly accelerated rod feels an inertial force which decreases in the direction of acceleration. The influence of nonrigidity of realistic rods on our results is qualitatively discussed.

Diffraction by disordered gratings and the Debye–Waller effect
View Description Hide DescriptionWe analyze the effect of disorder in a slit array upon the resulting laser diffraction pattern. This problem is analogous to the Debye–Waller effect, which applies to xray scattering by crystals, where disorder is due to uncorrelated thermal vibrations. We have made quantitative measurements using equipment available at our undergraduate laboratory. The measurements were analyzed with the aid of phasor diagrams.

Free fall and angular momentum
View Description Hide DescriptionThe problem of free fall (rising or falling) of an object near the surface of the rotating earth is revisited. Expressions for the eastward and westward deflections of the object, which are normally explained by Coriolis force, are derived directly from conservation of angular momentum of the object with respect to the center of the earth. Launching conditions for zero lateral (east–west) deflections are also discussed. The treatment is simple enough, both conceptually and mathematically, to be presented in an introductory physics course.
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 APPARATUS AND DEMONSTRATION NOTES


A study of an acrylic Cerenkov radiation detector
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A demonstration apparatus for an acoustic analog to the Casimir effect
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Microrockets for the classroom
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 BOOK REVIEWS


Minireview: Science Study Series
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