Volume 71, Issue 9, September 2003
Index of content:
 EDITORIAL


 PAPERS


Can a single bubble sink a ship?
View Description Hide DescriptionWe combine simple experiments and numerical simulations to study the effect of a large, single gas bubble rising underneath a floating body. Our motivation is the possible hazards arising from naturally occurring methane gas hydrates in the North Sea. For floating bodies that possess a hull length of similar scale to the bubble’s radius of curvature, we identify the conditions for the floating body to sink. Our experiments allow us to extend and benchmark numerical simulations using smoothed particle hydrodynamics. There is good agreement between the simulations and experiment.

A dynamic new look at the lambda transition
View Description Hide DescriptionWe discuss aspects of the theory of critical phenomena and explore the superfluid transition in We review some of the recent experimental and theoretical work on helium in nonequilibrium conditions and summarize some future space experiments that might shed light on disagreements between theory and experiment.

Watching a drunkard for 10 nights: A study of distributions of variances
View Description Hide DescriptionFor any physical observable in statistical systems, the most frequently studied quantities are its average and standard deviation. Yet, its full distribution often carries extremely interesting information and can be invoked to put the properties of the individual moments into perspective. As an example, we consider a problem concerning simple random walks. When a drunk is observed over nights, taking steps per night, and the number of steps to the right is recorded for each night, an average and a variance based on these data can be calculated. When the variance is used to estimate the probability for the drunk to step right, complex values for are frequently found. To put such obviously nonsensical results into context, we study the full probability distribution for the variance of the data string. We discuss the connection of our results to the problem of data binning and provide two other examples to demonstrate the importance of full distributions.

Thermodynamics of the harmonic oscillator: Wien’s displacement law and the Planck spectrum
View Description Hide DescriptionA thermodynamic analysis of the harmonic oscillator is presented. The motivation is provided by the blackbody radiation spectrum, because radiation modes take the harmonicoscillator form. We use the behavior of a thermal harmonic oscillator system under a quasistatic change of oscillator frequency ω to show that the thermodynamic functions can all be derived from a single function of analogous to Wien’s displacement theorem. The high and lowfrequency limits yield asymptotic forms involving the temperature alone or frequency ω alone, corresponding to energy equipartition and zeropoint energy. We suggest a natural interpolation between the limiting forms. The Planck spectrum with zeropoint energy corresponds to the function satisfying the Wien displacement result which provides the smoothest possible interpolation between energy equipartition at low frequency and zeropoint energy at high frequency.

Optimal exit: Solar escape as a restricted threebody problem
View Description Hide DescriptionWe analyze solar escape as a special case of the restricted threebody problem. We systematically vary the parameters of our model solar system to show how the optimal launch angle and minimum escape speed depend on the mass and size of the earth. In some cases, it is best to launch near the direction of the earth’s motion, but slightly outward; in other cases, it is best to launch near the perpendicular to the earth’s motion, but inward toward the sun (so as to obtain a solar gravity assist). Between direct escapes for high launch speeds and trapped trajectories for low launch speeds is an irregular band of chaotic orbits that reveals some of the complexity of solar escape and the threebody problem.

Apparatus to measure relativistic mass increase
View Description Hide DescriptionAn apparatus that uses readily available material to measure the relativistic mass increase of beta particles from a radioactive source is described. Although the most accurate analysis uses curve fitting or a Kurie plot, students may just use the raw data and a simple calculation to verify the relativistic mass increase.

Visualization and measurement of quantum rotational dynamics
View Description Hide DescriptionAn introduction to quantum rotational tunneling and libration is presented with an emphasis on obtaining a qualitative understanding of this phenomenon through visualization of the dynamics, simple approximations, and measurements. The tunneling and librational dynamics of small molecular rotors are discussed using a very simple model of the rotational potential. Numerical calculations of the evolution of probability packets are carried out for the lowlying states and the connection is made between the quantum and classical librational dynamics. Finally, we present measurements of these quantum rotations using inelastic neutron scattering and show in particular how neutron scatteringmeasurements of the ground state tunnel splitting and first librational transition can be used to characterize the magnitude and shape of the potential hindering the motion of the rotor. Some conceptual and computational problems are included that are suitable for undergraduate students.

Dispersive properties of probability densities in quantum mechanics
View Description Hide DescriptionIt is shown that the imaginary part of ψ^{*} Pψ plays a crucial role in describing the dispersive properties of probability densities in quantum mechanics. The quantity Im(ψ^{*} Pψ) is proportional to the dispersive velocity of the probability density and has the same form as the diffusion current. The dispersive velocity does not contribute to the expectation value of the momentum operator, but contributes to the expectation value of the kinetic energy operator. The interpretation of the dispersive velocity is given for the onedimensional Gaussian wave packet, the stationary states of a particle in a onedimensional box, and the stationary states of the hydrogen atom.

Coupledchannel integral equations for quasionedimensional systems
View Description Hide DescriptionAn integral equation approach is developed for the propagation of electrons in twodimensional quantum waveguides. The original twodimensional problem is transformed into a set of onedimensional coupled equations by expanding the full wave function in terms of simple transverse basis functions. The equivalence of the Schrödinger equation with suitable boundary conditions in configuration space to an integral equation approach in momentum space can thus be illustrated in a coupled channel situation with a minimum of geometrical complications. The application to scattering from a point defect embedded in a waveguide is considered. In this case the scattering integral equations reduce to a set of algebraic equations, and typical coupled channel phenomena can be discussed through straightforward mathematical techniques. The convergence problems due to a singular perturbation are briefly considered, and the differences between genuine onedimensional problems and the present twodimensional case are discussed.

Derivation of relativistic corrections to bounded and unbounded motion in a weak gravitational field by integrating the equations of motion
View Description Hide DescriptionThe deviation of light by a star and the precession of the perihelion of a planetary orbit are usually derived by independent methods. In the present paper the equations of motion for weak gravitation are integrated approximately and both effects are derived simultaneously.

An extinction theorem for electromagnetic waves in a point dipole model
View Description Hide DescriptionA transfer matrix method is used to show that an electromagnetic wave in a cubic lattice of point dipoles propagates as if its wave speed has been reduced from in vacuo to where is the index of refraction, while in the interstices the wave speed is Because the lattice is not a continuum, there are evanescent waves between lattice planes. These waves ultimately give rise to the Lorentz–Lorenz result for and produce rapid variations of the polarization in the first few lattice planes. These rapid variations indicate that there is an extinction length over which the external wave is converted to the internal wave propagating at the reduced speed. This length is the distance over which the local field assumes its bulk value. The possibility of an extinction length in recent work on the Ewald–Oseen theorem was missed because only continuum models were considered.

Electrically induced magnetic fields; a consistent approach
View Description Hide DescriptionElectromagnetic radiation exists because changing magnetic fields induce changing electric fields and vice versa. This fact often appears inconsistent with the way some physics textbooks solve particular problems using Faraday’s law. These types of problems often ask students to find the induced electric field given a current that does not vary linearly with time. A typical example involves a long solenoid carrying a sinusoidal current. This problem is usually solved as an example or assigned as a homework exercise. The solution offered by many textbooks uses the approximation that the induced, changing electric field produces a negligible magnetic field, which is only valid at low frequencies. If this approximation is not explicitly acknowledged, then the solution appears inconsistent with the description of electromagnetic radiation. In other cases, when the problem is solved without this approximation, the electric and magnetic fields are derived from the vector potential. We present a detailed calculation of the electric and magnetic fields inside and outside the long solenoid without using the vector potential. We then offer a comparison of our solution and a solution given in an introductory textbook.

Shielding by perfect conductors: An alternative approach
View Description Hide DescriptionGiven conductors and charges outside the conductors, is the superposition of the electric potentials due to induced charges on the conductorsurfaces and other charges constant in the conducting region? Although it is a natural question raised by students of electromagnetism, it does not seem to be addressed directly in textbooks. An identity that relates the potential due to the induced charge on a closed conductorsurface to other potentials is useful for answering the question and also for making clearer the phenomenon of electrostatic shielding.

The potential, electric field and surface charges for a resistive long straight strip carrying a steady current
View Description Hide DescriptionWe consider a long resistive straight strip carrying a constant current and calculate the potential and electric field everywhere in space and the density of surfacecharges along the strip. We compare these calculations with experimental results.

Two magnets and a ball bearing: A simple demonstration of the method of images
View Description Hide DescriptionMotivated by an observation of a popular children’s toy, I investigate the behavior of a bar magnet with a steel ball bearing held on one of its poles as it approaches another bar magnet. Mapping the problem onto electrostatics, I give an explanation of my observations based on the behavior of point charges near an isolated, uncharged sphere. This magnetic analog offers a simple demonstration of the familiar “method of images” in electrostatics.

A simple approach to phase holography
View Description Hide DescriptionA simple mathematical description of thick hologram recording and playback is given using a basic wave front representation. Results are obtained for amplitude and phase holography. This approach is very accessible to students and predicts the important results that are obtained in early and some more advanced holography experiments. The approach is also quite general and applies to transmission and reflection holography.

 NOTES AND DISCUSSIONS


The Dirac equation with a confining potential
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Comment on “The falling chain and energy loss,” by David Keiffer [Am. J. Phys. 69 (3), 385–386 (2002)]
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Electrodynamics and elasticity
View Description Hide DescriptionWe consider a Lorenzlike gauge theory with a second speed parameter that differs from the speed of light. This theory extends electrodynamics so that it has two speeds corresponding to the speeds of transverse and longitudinal waves in vacuum. The twoparameter extension of electrodynamics can be formally mapped onto the linear theory of elasticity. Because the case of equal speeds is not realized in an elastic medium, a compressible medium cannot model electromagnetism. In the Coulomb gauge electrodynamics can be formally mapped onto incompressible elasticity, which suggests that a linearelastic incompressible medium can serve as a model of electromagnetism.
