Volume 54, Issue 5, May 1986
 Letters To The Editors



Charge momentum?
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Reduced mass and the Bohr radius
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Tenure, salary, grants et al.
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 Editorial


Editorial: ‘‘...to see it as it is...to know it as it isn’t...’’
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 Papers


Sustained availability: A management program for nonrenewable resources
View Description Hide DescriptionThe continued extraction from the earth of nonrenewable mineral and fuel resources is a cause for concern, particularly where the rates of extraction are growing. If the rate of extraction declines a fixed fraction per unit time, the rate of extraction will approach zero, but the integrated total of the extracted resource between t=0 and t=∞ will remain finite. If we choose a rate of decline of the rate of extraction of the resource such that the integrated total of all future extraction equals the present size of the remaining resource then we have a program which will allow the resource to be available in declining amounts for use forever. This program is called Sustained Availability (SA) and it is somewhat analogous to the program of ‘‘sustained yield’’ in the management of renewable resources such as agriculture. The mathematics of this program, the opportunities it presents, and its consequences are examined in detail.

Simple electrostatic model of the structural phase transition
View Description Hide DescriptionA purely classical system of four equal point charges q which are free to move over the surface of a sphere is considered. In the equilibrium configuration all charges are located at the vertices of the regular tetrahedron inscribed into the sphere. If a weak uniform electric field E is directed along the z axis, one charge q will move to the position (0,0,R), while three others will form an equilateral inscribed triangle with the plane perpendicular to the field. An increase of the field will distort the tetrahedron [three charges uniformly shift towards the charge at (0,0,R)], keeping the field (z axis) as a C _{3}‐symmetry axis. However, when E reaches the ‘‘critical’’ value E _{0}=0.732q/R ^{2}, the system abruptly changes its symmetry: For E>E _{0} the equilibrium (least energy) configuration corresponds to a l l f o u r charges located symmetrically at the corners of the inscribed square whose plane is perpendicular to E. Consequently, the field will now form a C _{4}‐symmetry axis. This jumpwise C _{3}→C _{4} symmetry change provides a simple classical model for the structural phase transitions when the spontaneous discontinuity arises during a c o n t i n u o u s change of one of the intensive variables (in this model the electric field E). Several related open problems which may appear attractive for the modeling on personal computers are also outlined.

Velocities, refraction, and particle aspects of water waves
View Description Hide DescriptionExamples are presented which show that water waves behave in some ways like ‘‘fictitious particles.’’ The velocities and refraction equations are developed for a wave packet composed of nonparallel wave components. The relationships between Fermat’s principle, the Euler–Lagrange equation for rays, the ray curvature expression, Snell’s law, and Hamilton’s equations are discussed. The energy transmission rate of a hydron (water wave packet) is used to derive the shoaling and refraction coefficients. The total reflection of water waves is examined. The results for a hydron are compared with those of monochromatic waves.

Microcomputers as digital electronics
View Description Hide DescriptionAn instructional unit in which students build a microcomputer out of individual integrated circuits has been developed. It introduces the basic operations and design principles of computers, demonstrates the sequential cycles through which computers interface with external circuits, and provides, through machine code, insight into software at the most fundamental level. The computer consists of a Z‐80 microprocessor plus memory, I/O, load/test, and applications circuits. A special breadboard system is used to facilitate construction and testing.

Extending special relativity via the perplex numbers
View Description Hide DescriptionIn analogy to the complex numbers z=x+i y, where the ‘‘imaginary’’ i is such that i ^{2}=−1, a system of perplex numbers z=x+h y is introduced, where the ‘‘hallucinatory’’ h is such that ‖h‖=−1. This system, invented by four freshmen at St. Olaf College, appears to have relevance in physics. In particular, it provides a natural way to extend the usual formalism of special relativity to the case ‖v‖>c. This is done by means of a velocity parameter φ, such that v=c tanh φ, where tanh is an extension of the ordinary hyperbolic tangent function. The fact that this extension has two different angles φ for each velocity v accounts for the different approaches in the literature to superluminal phenomena.

Faraday’s law—Quantitative experiments
View Description Hide DescriptionA cylindrical bar magnet is dropped through a coil which encircles a glass tube. A Commodore 64 computer equipped with an analog to digital converter digitizes and stores the resulting voltage pulse. The magnet velocity and the number of turns on the coil are varied and simple measurements on the voltage pulse give quantitative tests of Faraday’s law. Integration produces the total flux through the coil. Flux integral calculations which treat the magnet either as a dipole or as a uniformly magnetized cylinder lead to equations for the voltage pulses which give a good fit to the experimental data obtained with a set of variable area coils. Either model is good for radial distances >40% the magnet length. Other results include determination of dipole separation and pole strength and surface current density. A versatile program for data acquisition and inspection is described. Suggestions are made for demonstrations and laboratory work for students in introductory noncalculus and calculus based courses and for advanced experimental work for majors.

An experiment with two air tracks
View Description Hide DescriptionThe analysis of a mechanical system with two air tracks is presented. The theoretical results are compared with the experimental data.

Derivation of the mirror equation
View Description Hide DescriptionThe mirror equation, giving the relationship between the object distance, the tangential image distance, the radius of curvature of a concave mirror, and the angle of incidence, is derived in a way which is different from that appearing in the literature. The derivation given here is for an ideally shaped mirror and involves no approximations. This new approach shows the equation to be more generally applicable than might otherwise be expected.

Production and use of a light beam with an intensity proportional to the distance from the source
View Description Hide DescriptionA laser beam arriving at normal incidence on a sharp edge produces a diffraction edge wave which appears on an observation screen as a bright line perpendicular to the edge. Provided some conditions are satisfied, it is shown that the intensity of this edge wave is proportional, within a finite range, to the distance from the diffracting edge to the screen. A brief history of the edge wave theory, which was first proposed by Thomas Young in 1802 to explain diffraction phenomena, is presented. Experimental results using this interesting proportionality between the diffraction edge wave intensity and distance are shown. These results concern the monitoring of a glider’s position on an air track under various conditions.

On an apparent paradox in the motion of a smoothly constrained rod
View Description Hide DescriptionThe motion in a vertical plane of a straight rod constrained to move between two frictionless concentric circles is investigated. An apparent paradox arises because the reaction forces at the contact points are purely radial, and together with the rod’s weight, which acts at the center of mass, are uncapable of supplying the torque about the center of mass necessary for the rod’s rotation. A tentative explanation has been given by Carroll [Am. J. Phys. 5 2, 1010 (1984)], where a mysterious tangential contact force arising from an infinite radial contact force is invoked. We disagree with these conclusions. In postulating that the motion takes place according to certain prescribed geometrical constraints, one must always make certain that the laws of physics are respected, and the model must be chosen accordingly. A simple model consisting of an H‐shaped frame with four contact points is used to determine the nature of the contact forces and the origin of the torque. As the link between the two half‐rods of the H frame is reduced, a single rod is obtained which can be thought of as the limit of a sector whose angular width is made vanishingly small. The dilemma is resolved by showing that a singular angular density of contact force distribution prevails, but that the contact force itself is finite.

A simple solution of the two‐dimensional relativistic Kepler problem
View Description Hide DescriptionA two‐dimensional electron is described by an appropriate spin 1/2 bi‐dimensional Dirac equation. The special case of a hydrogenic atom in two dimensions is solved in order to obtain its energy spectrum and the total spread of its fine structure levels.

The dielectrophoresis force
View Description Hide DescriptionWe present a simple and physically transparent calculation of the dielectrophoresis force, i.e., a force acting on neutral bodies in nonuniform electric fields in the cases of dc and ac fields.

Optical display of the ambiguity function and the diffraction pattern of a screw—An analogy
View Description Hide DescriptionThe Fraunhofer diffraction pattern of a screw is interpreted as an optical display of essential features of the ambiguity function of a chirp signal. Since no special preparation and optical processing are needed for diffraction, the optical display is very simple to demonstrate and impressive to visualize.

Integral approach to scattering by a surface impurity
View Description Hide DescriptionThe integral approach to quantum mechanical scattering is exhibited heuristically in a one‐dimensional context and applied to the multiple scattering problem of conduction electronscattering by a surface impurity, where the impurity is modeled as a complex delta‐function potential. It is shown to provide a much more physically insightful solution than direct solution to the Schrödinger equation. The total reflection amplitude is revealed, and its behavior physically interpreted, as a multiple scattering series of impurity and barrier amplitudes coupled by appropriate phase shifts as the electron wave propagates back and forth between the impurity and surface barrier.

Mechanical analog of optical retarders
View Description Hide DescriptionAn anisotropic system, which is composed of only springs and metal balls, is shown to be similar to an optical retarder. A mechanical analog of a half‐wave plate is thus constructed and its performance is found in qualitative agreement with the theory. Such a system may serve as a useful demonstration in teaching wave propagation in anisotropic media.
