Volume 61, Issue 4, April 1993
 Guest Comment
 Awards


Robert G. Fuller: Recipient of the Robert A. Millikan Medal
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Millikan Lecture 1992: Hypermedia and the knowing of physics: Standing upon the shoulders of giants
View Description Hide DescriptionHypermedia is defined and its roots briefly discussed. Models of knowing physics and of intrinsically motivating instruction are presented. Uses of hypermedia to increase the knowing of physics and the motivation of learners are proposed.
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 Papers


A magnetic suspension system for atoms and bar magnets
View Description Hide DescriptionA three dimensional magnetic confinement system is presented which will trap both macroscopic and atomic magnetic dipoles. The dipole is confined by dc and oscillating magnetic fields, and its motion is described by the Mathieu equation. Most aspects of the dynamics of the trapped objects depend only on the ratio of the magnetic moment to the mass of the dipole. Similar motion was observed for masses varying over 21 orders of magnitude (from 1 atom to 0.2 g). The trap is constructed from inexpensive permanent magnets and small coils which are driven by 60 Hz line current. The design of the trap as well as the behavior of the trapped particle are discussed herein.

The Smith chart and quantum mechanics
View Description Hide DescriptionThe Schrödinger equation and the equation describing the behavior of voltage on a transmission line are both linear second‐order equations, which may be solved by convenient matrix methods. By drawing analogies between these two problems, it is shown that a method used for antenna impedance matching based on the Smith chart corresponds in quantum mechanics to a simple conformal transformation of the logarithmic derivative of the wave function. One thereby can arrive at an elementary derivation of the Wentzel–Kramers–Brillouin quantization condition.

Relativistic equipartition via a massive damped sliding partition
View Description Hide DescriptionA cylinder partitioned by a massive sliding slab undergoing nonrelativistic damped one‐dimensional (1D) motion under bombardment from the left (i=1) and right (i=2) by particles having rest mass m _{ i }, speed v _{ i }, relativistic momentum (magnitude) p _{ i }, and (let c≡1) total energy E _{ i }=(p _{ i } ^{2}+m _{ i } ^{2})^{1/2} is considered herein. The damped slab of mass M transforms the system from its initial p _{ i } distributions (i=1,2) to a state, first, of pressure (P) equilibrium with P _{1}=P _{2}, but temperature T _{1}≠T _{2}, then, to P‐T equilibrium with P _{1}=P _{2} and T _{1}=T _{2}, given by the (1D) ‘‘first moment’’ equipartition relation (κ is Boltzmann’s constant), <q _{1}≳=<q _{2}≳≡κT [Eq. (A1)], where q _{ i }≡p _{ i } v _{ i }=E _{ iv } _{ i } ^{2}=p _{ i } ^{2}/E _{ i }. In achieving first‐moment equilibrium at a given κT the slab M can be taken sufficiently large, hence slab oscillation period τ sufficiently long (τ≫t _{max} where t _{max}=2L _{ i }/v _{min} is the round trip period of the slowest particle) to give ‘‘mechanical adiabatic invariance’’ (MAI), hence conservation of mechanical ‘‘action’’ p _{ iL } _{ i } of each particle. This first‐moment equilibrium is not yet ‘‘thermal’’ equilibrium, since the MAI process leaves the higher moments <q _{ i } ^{2}≳, <q _{ i } ^{3}≳, etc., with their original values, relative to <q _{ i }≳.
To achieve thermal equilibrium the slab damping is turned off and slab mass M is reduced, hence τ decreases, until τ≪t _{max}, whereupon MAI becomes ‘‘broken’’ and we achieve complete thermal equilibrium, given by Eq. (A1) plus the appropriate higher moments. Using straightforward extension of the relativistic technique used by Menon and Agrawal to find the first‐moment relation (A1) we find that all of the moments of q _{1} satisfy the recursion relation <q _{ i } ^{ n }≳= nκT<q _{ i } ^{ n−1}≳ +(n−1)m _{ i } ^{2}κT<q _{ i } ^{ n−1}/E _{ i } ^{2}≳, i=1 or 2, n=1, 2, 3, 4,... [Eq. (A2)].

The Science Bag^{ T } ^{ M } at the University of Wisconsin–Milwaukee: A successful forum for science outreach
View Description Hide DescriptionA series of science programs for the public has been running for 19 years with an accumulated attendance of 95 000 people.

Self‐organized criticality: An experiment with sandpiles
View Description Hide DescriptionIn 1987, Bak, Tang, and Wiesenfeld introduced the notion of self‐organized criticality (SOC) in the guise of a computer simulation: a ‘‘sandpile cellular automaton machine.’’ They supposed that a real, many‐bodied, physical system in an external field assembles itself into a critical state. The system then relaxes about the critical state creating spatial and temporal self similarities which give rise to fractal objects and 1/fnoise. Their computer modeling was of a system like a sandpile at its critical angle of repose. This provided a new paradigm for many‐body dynamics. Understanding SOC may well allow substantial strides to occur in the theory of flow and transport. The simplest model system, one for which computer simulations and corresponding real experiments are feasible, is a ‘‘sandpile’’ near its critical angle of repose. The size and duration of avalanches occurring as subsequent ‘‘sand’’ grains are added can provide detailed information about the ‘‘sandpile’’ as a model of SOC, and for SOC as a basis for many‐body dynamics. This article describes a fairly simple, junior‐level experiment in this new field involving the measurement of the distribution of avalanche sizes which occur as grains of sand are added to a ‘‘sandpile.’’ The universality of the phenomena can be observed and a power law relationship can be deduced.

Confusion by representation: On student’s comprehension of the electric field concept
View Description Hide DescriptionIt is argued that university students’ diffuse ideas about the two related concepts of force and force field could be due to lack of mastery of the graphical representation of these and related concepts.

The cosmological spectral shift
View Description Hide DescriptionThe formula for the cosmological spectral shift in a Robertson–Walker universe is derived by applying the method of separation of variables to the solution of Maxwell’s field equations. A proof is also given of the formula for the variation of the apparent brightness of a source with distance.

Phenomenological quantum models for under and critically damped oscillators
View Description Hide DescriptionAccording to Rayleigh, if one sets a linear oscillator in motion in a viscous fluid, the fluid not only opposes the motion with a velocity‐dependent dissipative force, but the fluid accretes to the oscillator changing its effective mass. A perturbation linking a mass accretion operator with the usual complex spontaneous decay operator is applied to a nonrelativistic quantum undamped oscillator. It results in a quantum model of Rayleigh’s oscillator for which the equation of motion for the expectation value of position and the Heisenberg uncertainty relation between the position and linear momentum are the anticipated ones for a quantum mechanical underdamped oscillator. Another model based on a variation of these ideas which allows for all possibilities, no change or diminution or accretion, for the perturbed mass is presented for the critically damped oscillator. In both models, the situation is fixed by the experimental dissipative ‘‘forces.’’

The ray form of Newton’s law of motion
View Description Hide DescriptionThrough the use of the optical‐mechanical analogy, Newton’s law of motion may be cast into the same form as the equation for the ray in the geometrical optics of gradient‐index media. The resulting equation is called the ray form of Newton’s law of motion. The same equation may be derived by taking the geometrical optics limit of quantum mechanics. The ray form of Newton’s law of motion is derived in three different ways and is applied in the solution of several problems.

Electrostatic field bounds for model dielectric configurations
View Description Hide DescriptionThe normal component of an electric field near a convex shaped object of zero net charge and dielectric constant ε embedded in a vacuum with an asymptotically uniform electric field has an upper bound (ε/ε_{0})‖E _{0}‖, where ε_{0} is the vacuum permittivity, and E _{0} is the asymptotic field. This limit may be closely approached in the vicinity of regions of high surface curvature.

Longitudinal standing waves on a vertically suspended slinky
View Description Hide DescriptionThe vertically suspended slinky is a system where variable tension, and variable mass density, combine to produce a simple solution for the longitudinal normal modes. The time taken for a longitudinal wave to traverse a single turn of the slinky is found to be constant for a variety of slinky configurations. For the freely suspended slinky this constant traverse time yields standing wave frequencies that depend only on the length of the hanging slinky and not on the material, radius, or stiffness of the slinky. Data, obtained by students in a laboratory setting, are presented to illustrate the application of these results.

The charge densities in a current‐carrying wire
View Description Hide DescriptionIn the lab frame the total linear charge density of a current‐carrying wire must be zero, while in the rest frame of the electrons making up the current the total volume charge density must be zero. These two pieces of information enable the determination of the volume, surface, and linear charge densities of such a wire in both of these frames using only straightforward relativistic length contractions and simple mathematics.

The computerized student laboratory: Motion in a potential well
View Description Hide DescriptionThe motion of a body in a potential well is studied in the computerized undergraduate student laboratory. Both large and small oscillations are studied. Small oscillations are found to be harmonic, and the frequency of the oscillations are found to tend to a constant value as the amplitude tends to zero. The trajectory in phase space is studied for both small and large oscillations. The potential energy is found as a function of position by two methods.

Modelling tidal effects
View Description Hide DescriptionTwo models for demonstrating tides and experimenting with various tidal effects are presented. The first takes advantage of the approximately inverse‐square nature of the force law for magnetic poles and exhibits symmetric tidal bulges on opposite sides of the planet, analogous to the tides of the earth. The second demonstration apparatus is a realization of the ‘‘rubber sheet’’ geometry analogy that is often used to model potential wells of massive bodies in space. By rolling various objects on the surface of a stretched elastic cloth one can effectively demonstrate tidal bulges, the Roche limit, and other well‐known astrophysical phenomena.

Classical electromagnetism and relativity: A moving magnetic dipole
View Description Hide DescriptionThe force on a stationary electric charge due to a moving current carrying coil and the torque on a moving current carrying coil due to the electrostatic field of a stationary electric charge can both be interpreted quantitatively by including the effects due to the second order charge distribution on the moving coil predicted by the charge transformations of special relativity.
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 Notes and Discussions



Measure the earth’s radius while boating on one of its lakes
View Description Hide DescriptionA very simple and enjoyable method of measuring the earth’s radius while boating on a lake is described. Apart from equipment that is usually carried while boating, no additional apparatus is required and the cost of the experiments is zero.
