Volume 56, Issue 8, August 1988
 Letters To The Editor


‘‘Comment on ‘A study of barrier penetration in quantum mechanics’ by J. D. Chalk’’
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On the so‐called minimum time problem revisited
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Comment on Cohen’s article on the Chernobyl disaster
View Description Hide DescriptionThe amount of uranium enrichment needed to make a reactor critical is discussed. (AIP)
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 Editorial


Editorial: Wanted— Tutorial articles for the A m e r i c a n J o u r n a l o f P h y s i c s
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 Papers


Resource Letter RMSL‐1: Recent measurements of the speed of light and the redefinition of the meter
View Description Hide DescriptionThis Resource Letter provides a guide to the literature on recent measurements of the speed of light and the redefinition of the meter. The letter E after an item indicates elementary level or material of general interest to persons becoming informed in the field. The letter I, for intermediate level, indicates material of somewhat more specialized nature; and the letter A indicates rather specialized or advanced material. An asterisk (*) indicates those articles to be included in an accompanying Reprint Book.

The force on a magnetic dipole
View Description Hide DescriptionThe classical magnetic force on a magnetic dipole depends upon the model for the dipole. The usual electric current loop model for a magnetic dipole leads to the force F=∇(m⋅B) on a magnetic dipole m in a magnetic fieldB. The separated magnetic charge model for a magnetic dipole leads to the force F=(m⋅∇)B on a magnetic dipole. The latter expression is analogous to the force experienced by an electric dipole in an electric field. Here, some elementary examples are given where the force expressions yield entirely different forces on a magnetic dipole. Electromagnetism textbooks usually do not emphasize the difference between these force expressions; however, occasionally the difference is important for understanding experimental results. In the 1930s and 1940s the difference in force expressions was involved in a determination of the nature of the neutrondipole moment. At present, in the 1980s, the difference in the force expressions is central to a controversy over an experiment to test the proposed Aharonov–Casher effect.

Perfect disturbing measurements
View Description Hide DescriptionThe Wigner–Araki–Yanase theorem is often interpreted as meaning that there must be an error in measurement under certain conditions (when the commutator C between an additive conserved quantity and the discrete‐spectrumed measured quantity does not vanish), and that the error may only be reduced by increasing the size of the apparatus. By explicit example, it is shown that it i s possible to have a perfectly accurate measurement if the system being measured is disturbed, regardless of the size of the apparatus. Also illustrated is how the error in an imperfectly accurate measurement may be reduced by decreasing the magnitude of C, without affecting the size of the apparatus.

What really happens in the Franck–Hertz experiment with mercury?
View Description Hide DescriptionRecent studies using spin‐polarized electrons have revealed the role that various interactions play in inelastic collisions of slow electrons with complex atoms. It has been shown that electron exchange and resonances (temporarily negative compound ion states) strongly influence the excitation cross sections of the first excited states of mercury very close to excitation thresholds. The particular result obtained in the Franck–Hertz experiment is determined by the magnitude of the excitation cross sections involved and depends on the geometrical design of the tube.

Dielectric slab in a parallel‐plate condenser
View Description Hide DescriptionThe problem of a dielectric slab inside a parallel‐plate capacitor is considered from the point of view of a simple force calculation. The usual method of presenting this problem, found in most textbooks, is via energy considerations. The method presented here allows corrections to the well‐known result to be obtained.

Lowest modes of a bottle
View Description Hide DescriptionHelmholtz’s model predicts correctly the frequency of the lowest mode of a bottle. A simple generalization of Helmholtz’s model correctly predicts this mode and also the next few modes—‘‘flute modes’’—of flasks with long uniform necks but arbitrarily shaped bodies. Wine bottles have additional low‐frequency ‘‘cavity modes’’ that require a further easy generalization. For a bottle with slowly varying cross section an additional generalization can be made that retains the one‐dimensional (1‐D) character of the previous models and gives results that are in good agreement with experiment for the lowest mode—the ‘‘diametral mode’’—of a hollow sphere. For higher modes of a sphere, the 1‐D model is inadequate and must be discarded in favor of exact solutions of the 3‐D wave equation.

Electrostatic fields in inhomogeneous dielectrics
View Description Hide DescriptionConditions under which simple, closed‐form solutions to the electrostatic field equations in linear inhomogeneous dielectrics can be obtained are investigated. It is shown that such solutions can be found in general orthogonal coordinates (q _{1},q _{2},q _{3}), in which equipotential surfaces are given by constant values of the axial q _{1} coordinate, whenever the permittivity can be expressed as ε(q _{1},q _{2},q _{3})=ε_{0}(h _{1}/h _{2} h _{3})f(q _{1}) g(q _{2},q _{3}). In this formula, h _{1},h _{2}, and h _{3} are arc length factors, while f(q _{1}) and g(q _{2},q _{3}) are essentially arbitrary functions of the coordinates. These solutions have the property that the E and D fields have components only in the axial q _{1} direction. Explicit formulas for field, displacement, polarization, potential, capacitance, etc. are obtained in terms of the generalized coordinates and the functions f and g. These expressions encompass virtually all the ‘‘workable’’ capacitor problems posed in standard texts. In the case of Cartesian coordinates where x is the axial coordinate, the conditions for closed‐form solutions of this type reduce to ε=ε_{0} f(x)g(y,z), where f(x) and g(y,z) are arbitrary functions, and it is found that E _{ x } depends only on x, while D _{ x } depends only upon y and z. Effects arising from the presence of volume distributions of free charge are investigated, and it is found that their presence severely restricts the conditions under which simple solutions may exist. The methods used, and the results obtained, are applicable to structures such as capacitors,superlattices,optical fibers, inhomogeneous thin films, semiconductor heterojunctions, etc.

Experiment in time‐average holographic interferometry for the undergraduate laboratory
View Description Hide DescriptionAn experiment suitable for an undergraduate laboratory on time‐average holography is described. The object analyzed is a rectangular bar driven at one end and supported at its geometric center by means of a vertical rod. The nodal point, as indicated by the brightest fringe, is found to be away from the geometric center. A simple analysis explains the observed results.

Dimensions of relativistic quantum mechanical paths
View Description Hide DescriptionThe notion of the dimension of a relativistic quantum mechanical path is illustrated. An elementary introduction to the concept of a fractal curve allows the discussion of the evolution of the quantum mechanical wave packet for free particles in terms of Hausdorff dimensions of quantum ‘‘trajectories.’’ The results of a previous work by Abbott and Wise can be generalized to relativistic quantum mechanics of spin‐1/2 particles in the Foldy–Wouthuysen representation. A further extension to spin‐zero particles is briefly outlined. By simple considerations, an understanding of the notion of a Hausdorff dimension and its values can be introduced.

Thomson scattering and plasma oscillations
View Description Hide DescriptionA direct calculation shows that Thomson scattering—on the microscopic level—manifests itself as plasma oscillations—on the macroscopic level.

Acoustically induced square law forces and some speculations about gravitation
View Description Hide DescriptionThe goal of this article is twofold: (1) To review and clarify a derivation of the force between two objects in an acoustic field, demonstrating the circumstances under which this force takes a special form—an inverse‐square‐law behavior like that of gravitational attraction; and (2) to carry this analogy to gravitational forces, speculating (in perhaps an outlandish manner) that so‐called mock gravity (from radiationpressure) is the source of real gravitational forces—that is, that this force arises from the response of bodies to a reverberating field of gravity waves perhaps produced in the big bang.

The apparent shape of a rapidly moving sphere
View Description Hide DescriptionSeveral articles have considered the apparent shape of a relativistic sphere. One interesting finding showed that, under certain conditions, the surface of the sphere may appear concave. This article concentrates on the case where the observer is at the origin and the sphere moves in the positive x direction with its center on the x axis. The sphere actually hits the observer, and one of the findings presented here is that for any speed v, 0<v≤c, part of the surface appears concave for some part of the motion along the x axis.

A simple example related to the time‐reversal paradox
View Description Hide DescriptionAn example is given of a reversible, deterministic system in which unexpected and seemingly irreversible behavior occurs. The system is simple enough so that states can be counted explicitly and the behavior can be clearly understood as nonparadoxical.

The ‘‘faster than gravity’’ demonstration revisited
View Description Hide DescriptionThe initial acceleration of a rigid rod rotating around one end, and the time required for the rod to fall onto a horizontal table, is compared with the acceleration and falling time of a free particle originally placed on the rod. The conditions under which the rod falls faster than the particle are established. Three cases are identified. The third case, in which the particle catches up with the rod, has not, to the author’s knowledge, been published previously.

Air track with a distributed infrared detector system
View Description Hide DescriptionA novel air track system is described which contains many permanently mounted emitter–detector pairs. This enables a computer to make quickly a large number of accurate position and time measurements for each run. Because the emitter–detectors employ infrared light, the system is essentially insensitive to normal ambient radiation. (In addition, the system design is simple enough for a student project.) Details of the apparatus will be sketched and a variety of experiments will be discussed.
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 Notes and Discussions


A ripple tank demonstration of the conditions for interference of waves
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