Volume 78, Issue 3, March 2010
 PAPERS


Cylindrical magnets and ideal solenoids
View Description Hide DescriptionBoth wirewound solenoids and cylindrical magnets can be approximated as ideal azimuthally symmetric solenoids. We present an exact solution for the magnetic field of an ideal solenoid in an easy to use form. The field is expressed in terms of a single function that can be rapidly computed by means of a compact efficient algorithm, which can be coded as an addin function to a spreadsheet, making field calculations accessible to introductory students. These expressions are not only accurate but are also as fast as most approximate expressions. We demonstrate their utility by simulating the dropping of a cylindrical magnet through a nonmagnetic conducting tube and comparing the calculation with data obtained from experiments suitable for an undergraduate laboratory.

A more thorough analysis of water rockets: Moist adiabats, transient flows, and inertial forces in a soda bottle
View Description Hide DescriptionAlthough waterrockets are widely used to illustrate first year physics principles, accurate measurements show that they outperform the usual textbook analysis at the beginning of the thrust phase. This paper gives a more thorough analysis of this problem. It is shown that the air expansion in the rocket is accompanied by water vaporcondensation, which provides an extra thrust; the downward acceleration of water within the rocket also contributes to the thrust, an effect that is negligible in other types of rockets; the apparent gravity resulting from the acceleration of the rocket contributes as much to water ejection as does the pressure difference between the inside and outside of the rocket; and the water flow is transient, which precludes the use of Bernoulli’sequation. Although none of these effects is negligible, they mostly cancel each other, and the overall accuracy of the analysis is only marginally improved. There remains a difference between theory and experiment with waterrockets.

Local pinning of trial wave functions: An optimization method without integrals for finding approximate solutions of field equations
View Description Hide DescriptionWe present an optimization method that requires no integrations for obtaining approximate solutions of field equations of the form . The expression would be identically zero everywhere if were an exact solution. The method consists of selecting an appropriate trial function , which depends on several parameters, and requiring that the values of these parameters be such that equal zero in certain regions of space, especially around singularities. This requirement yields a locally pinned trial wave function that approximates the exact solution. We illustrate the method by applying it to a simple harmonic oscillator, a vortex in a superfluid, to the ground state of a Bose–Einstein condensate and to the ground state of the helium atom.

Nonlinear dynamics of a ball rolling on a surface
View Description Hide DescriptionAn underlying potential energy function can provide visual and intuitive insight into a system’s stability and overall behavior. In particular, the motion of a ball moving along a curve or surface in a gravitational field provides a macroscale demonstration of interesting dynamics. We investigate the motion of a small ball rolling along a smooth twodimensional potential surface. A direct experimental realization of this situation is suitable for demonstrating some classic features of nonlinear dynamics. The results of numerical simulations are directly compared with experimental data. To better characterize the dynamical behavior of the ball, especially when it is undergoing chaotic motion, several descriptive measures are discussed, including timelag embedding, initial condition maps, power spectra, Lyapunov exponents, and fractal dimensions.

Interferometry using a noisy moving source
View Description Hide DescriptionInterferometry using a broadband noise source and a narrow band receiver is studied. It is shown that interferometry kinematic information relating to trains may be obtained using only a radio receiver and a watch.

Application of exterior calculus to waveguides
View Description Hide DescriptionExterior calculus is a powerful tool for finding solutions to the electromagnetic field equations. Its strength can be better appreciated when applied to nontrivial configurations. We show how to exploit this tool to obtain the TM and TE modes in hollow cylindrical waveguides. The use of exterior calculus and Lorentz boosts leads straightforwardly to the solutions and their respective power transmitted along the waveguide.

Optical Faraday rotation
View Description Hide DescriptionThree calculations of optical Faraday rotation are presented in which a linearly polarized field is incident on a medium of harmonic oscillators in the presence of a longitudinal magnetic field. The rotation of the plane of polarization of the field is evaluated using classical oscillators and the Lorentz force equation, quantum oscillators and the Heisenberg equations of motion, and quantum oscillators and a Schrödinger equation approach. It is shown that a simple argument, based on the assumption that a circularly polarized field drives either or transitions on absorption ( is the magnetic quantum number), leads to an incorrect result for the Verdet constant.

The profile of a capillary liquid bridge between solid surfaces
View Description Hide DescriptionScanning force microscopy, such as atomic force microscopy(AFM) is complicated by the capillary force of a water meniscus formed in air between the probe tip and the sample. This small liquid bridge between the hydrophilic sample and the sharp AFM tip can be formed by capillary condensation from the vapor phase. We present an analytical model that describes the shape of the meniscus in which the pressure difference across the curved liquid air interface is taken into account. The analysis is based on a minimization of the liquid surface energy, together with the boundary condition of a given pressuredrop across the surface as determined by the relative humidity of the vapor. The capillary forces that the wetting liquid exerts on the AFM tip are derived from the model. The resulting expressions can be used to describe arbitrary axial symmetric liquid air interfaces with nonzero total curvature, such as fluid bridges between two surfaces and droplets under a uniform force. The model illustrates some of the basic concepts of capillarity, such as surface tension forces and interfacial pressuredrops.

Solution of Laplace’s equation for the confining end potentials of a coaxial Malmberg–Penning trap
View Description Hide DescriptionLaplace’s equation is solved for the confining potentials at the ends of a coaxial form of the Malmberg–Penning charged particle trap. The solution gives insight into the confinement and dynamics of the trapped particles. The solution employs several mathematical methods that are often studied in isolation. The connections between these methods are illustrated by solving the problem in different ways. This process also produces several new integral and series identities.

Lorentzinvariant threevectors and alternative formulation of relativistic dynamics
View Description Hide DescriptionBesides the wellknown scalar invariants, there also exist vectorial invariants in special relativity. It is shown that the threevector is invariant under the Lorentz transformation. The subscripts and denote the respective components with respect to the direction of the velocity of the body , and is the relativistic momentum. We show that this vector is equal to a force , which satisfies the classical Newtonian law in the instantaneous inertial rest frame of an accelerating body. Therefore, the relation , based on the Lorentzinvariant vectors, may be used as an invariant (not merely a covariant) relativistic equation of motion in any inertial system of reference. An alternative approach to classical electrodynamics based on the invariant threevectors is proposed.

Electronic structure, Raman tensors, and resonance phenomena in a simple molecular model
View Description Hide DescriptionSome concepts in modern spectroscopy are very specialized, and explanations based on simple examples are not readily available. An example is the changes in the intrinsic symmetry of Raman tensors in molecules produced by resonance or nearresonance conditions. Many of these effects can be obtained from commercial and open source programs that solve the electronic structure of the molecules with density functional theory and compute the Raman tensors of the vibrations. The origin of these changes is hidden by the complexity of these calculations and by the many intermediate computational steps that are not presented to the user. We discuss a simplified model for the electronic structure of a molecule and correlate what is observed with a calculation using density functional theory for a specific molecule. The model yields insight into resonance phenomena, symmetryrelated aspects of Raman tensors, and the microscopic origin of the Raman effect itself.

The force, power, and energy of the 100 meter sprint
View Description Hide DescriptionAt the 2008 Summer Olympics in Beijing, Usain Bolt broke the world record for the 100 m sprint. Just one year later, at the 2009 World Championships in Athletics in Berlin he broke it again. A few months after Beijing, Eriksen et al. [Am. J. Phys.77, 224–228 (2009)] studied Bolt’s performance and predicted that Bolt could have run about onetenth of a second faster, which was confirmed in Berlin. In this paper we extend the analysis of Eriksen et al. to model Bolt’s velocity time dependence for the Beijing 2008 and Berlin 2009 records. We deduce the maximum force, the maximum power, and the total mechanical energy produced by Bolt in both races. Surprisingly, we conclude that all of these values were smaller in 2009 than in 2008.

Thermal fluctuations near a phase transition probed through the electrical resistivity of hightemperature superconductors
View Description Hide DescriptionWe present a simple and inexpensive experimental setup that can determine the thermal fluctuations near a superconducting transition. By using equipment common in undergraduate laboratories, the inplane dc electrical resistivity of a cuprate superconductor as a function of temperature is measured with resolution sufficient to analyze the fluctuation conductivity above the superconducting critical temperature, including the values of the critical exponents. We also present a simple calculation of the fluctuation conductivity, including its dependence on the layered structure of the material, within the Gaussian–Ginzburg–Landau formalism.
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 BOOK REVIEWS


How Physics Confronts Reality: Einstein was Correct but Bohr Won the Game
View Description Hide DescriptionThis article reviews How Physics Confronts Reality: Einstein was Correct but Bohr Won the Game by Roger G. Newton 
The Black Hole War: My Battle with Stephen Hawking to Make the World Safe for Quantum Mechanics
View Description Hide DescriptionThis article reviews The Black Hole War: My Battle with Stephen Hawking to Make the World Safe for Quantum Mechanics by Leonard Susskind 
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