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Top 10 Most Read Articles
April 2012
The 10 articles with the most full-text downloads during the month, in descending order.
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In an expanding universe, what doesn’t expand? American Journal of Physics -- May 2012 -- Volume 80, Issue 5, pp. 376
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The expansion of the universe is often viewed as a uniform stretching of space that would affect compact objects such as atoms and stars, as well as the separation of galaxies. One usually hears that bound systems do not take part in the general expansion, but a much more subtle question is whether bound systems expand partially. In this paper, a definitive answer is given for a very simple system: a classical “atom” bound by electrical attraction. With a mathematical description appropriate for undergraduate physics majors, we show that this bound system either completely follows the cosmological expansion, or, after initial transients, completely ignores it. This all-or-nothing behavior can be understood using analysis techniques used in junior-level mechanics. We also demonstrate that this simple description is a justifiable approximation of the relativistically correct formulation of the problem.
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A better presentation of Planck’s radiation law American Journal of Physics -- May 2012 -- Volume 80, Issue 5, pp. 399
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Introductory physics and astronomy courses commonly use Wien’s displacement law to explain the colors of blackbodies, including the Sun and other stars, in terms of their temperatures. We argue here that focusing on the peak of the blackbody spectrum is misleading for three reasons. First, the Planck curve is too broad for an individual spectral color to stand out. Second, the location of the peak of the Planck curve depends on the choice of the independent variable in the plot. And third, Wien’s displacement law is seldom used in actual practice to find a temperature and direct fitting to the Planck function is preferable. We discuss these flaws and argue that, at the introductory level, presentation of blackbody radiation in terms of photon statistics would be more effective pedagogically. The average energy of the emitted photons would then be presented in place of Wien’s displacement law, and discussion of the Stefan-Boltzmann law would include the total number of photons emitted per second. Finally, we suggest that the Planck spectrum is most appropriately plotted as a “spectral energy density per fractional bandwidth distribution,” using a logarithmic scale for the wavelength or frequency.
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The creation and propagation of radiation: Fields inside and outside of sources American Journal of Physics -- April 2012 -- Volume 80, Issue 4, pp. 321
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We present an algorithm for computing the electromagnetic fields due to currents inside and outside of finite sources with a high degree of spatial symmetry for arbitrary time-dependent currents. The solutions for these fields do not involve the time derivatives of the currents but involve only the currents and their time integrals. We give solutions for moving planar sheets of charge, and a rotating spherical shell carrying a uniform charge density. We show that the general solutions reduce to the standard expressions for magnetic dipole radiation for slow time variations of the currents. If the currents are turned on very quickly, the general solutions show that the amount of energy radiated equals the magnetic energy stored in the static fields a long time after current creation. We give three problems which can be used in undergraduate courses and one problem suitable for graduate courses. These problems illustrate that because the generation of radiation depends on what has happened in the past, a system of currents can radiate even during time intervals when the currents are constant due to radiation associated with earlier acceleration.
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THE TRANSIT OF VENUS: JUNE 5/6, 2012 American Journal of Physics -- May 2012 -- Volume 80, Issue 5, pp. 361
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Simple model to estimate the contribution of atmospheric CO2 to the Earth’s greenhouse effect American Journal of Physics -- April 2012 -- Volume 80, Issue 4, pp. 306
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We show how the CO2 contribution to the Earth’s greenhouse effect can be estimated from relatively simple physical considerations and readily available spectroscopic data. In particular, we present a calculation of the “climate sensitivity” (that is, the increase in temperature caused by a doubling of the concentration of CO2) in the absence of feedbacks. Our treatment highlights the important role played by the frequency dependence of the CO2 absorption spectrum. For pedagogical purposes, we provide two simple models to visualize different ways in which the atmosphere might return infrared radiation back to the Earth. The more physically realistic model, based on the Schwarzschild radiative transfer equations, uses as input an approximate form of the atmosphere’s temperature profile, and thus includes implicitly the effect of heat transfer mechanisms other than radiation.
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Topics in quantum physics with origins in astronomy: Two examples American Journal of Physics -- May 2012 -- Volume 80, Issue 5, pp. 406
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Astronomy has provided the inspiration for several investigations in quantum physics. These topics can serve as pedagogical vehicles for undergraduate courses in physics and astronomy. Two examples are considered, atoms in strong magnetic fields and the negative ion of hydrogen. Both are fundamental problems of quantum physics which involve basic principles and techniques and are of practical interest and historically important. They also provide, in a form accessible to undergraduates, analogs of exotic topics such as the stability at a saddle of a potential surface, supersymmetry, dimensional reduction, and models for fundamental constants.
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New perspective on the optical theorem of classical electrodynamics American Journal of Physics -- April 2012 -- Volume 80, Issue 4, pp. 329
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A proof of the optical theorem (also known as the optical cross-section theorem) is presented, which reveals the intimate connection between the forward scattering amplitude and the absorption-plus-scattering of the incident wave within the scatterer. The oscillating charges and currents as well as the electric and magnetic dipoles of the scatterer, driven by an incident plane wave, extract energy from the incident beam. The same oscillators radiate electromagnetic energy into the far field, thus giving rise to well-defined scattering amplitudes along various directions. The essence of the proof presented here is that the extinction cross-section of an object can be related to its forward scattering amplitude using the induced oscillations within the object without knowledge of the form assumed by these oscillations.
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Aerodynamics in the classroom and at the ball park American Journal of Physics -- April 2012 -- Volume 80, Issue 4, pp. 289
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Experiments suitable for classroom projects or demonstrations are described concerning the aerodynamics of polystyrene balls. A light ball with sufficient backspin can curve vertically upward through the air, defying gravity and providing a dramatic visual demonstration of the Magnus effect. A ball projected with backspin can also curve downward with a vertical acceleration greater than that due to gravity if the Magnus force is negative. These effects were investigated by filming the flight of balls projected in an approximately horizontal direction so that the lift and drag forces could be easily measured. The balls were also fitted with artificial raised seams and projected with backspin toward a vertical target in order to measure the sideways deflection over a known horizontal distance. It was found that (a) a ball with a seam on one side can deflect either left or right depending on its launch speed and (b) a ball with a baseball seam can also deflect sideways even when there is no sideways component of the drag or lift forces acting on the ball. Depending on the orientations of the seam and the spin axis, a sideways force on a baseball can arise either if there is rough patch on one side of the ball or if there is a smooth patch. A scuff ball with a rough patch on one side is illegal in baseball. The effect of a smooth patch is a surprising new observation.
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Stellar temperatures by Wien’s law: Not so simple American Journal of Physics -- May 2012 -- Volume 80, Issue 5, pp. 391
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A star’s surface temperature is among the most important features that can be deduced from its light. We have made measurements to see how reliably we could determine the surface temperatures of some A to K stars using Wien’s Displacement Law. We took spectra, corrected them for atmospheric extinction and instrumental response, found the wavelengths of their intensity maxima, and then from Wien’s law found the surface temperatures of the observed stars. For F to early K stars, our results agree with temperatures determined in other ways. For A and later K stars, the agreement is poor because the spectra are appreciably different from ideal blackbody spectra and because our equipment responds poorly to the deep red and blue wavelengths where the spectra of these stars have their peak intensities. This paper points out several interesting concepts in and outside the astrophysical domain that can be instructive for undergraduate students.
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Computing accurate age and distance factors in cosmology American Journal of Physics -- May 2012 -- Volume 80, Issue 5, pp. 367
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As the universe expands astronomical observables such as brightness and angular size on the sky change in ways that differ from our simple Cartesian expectation. We show how observed quantities depend on the expansion of space and demonstrate how to calculate such quantities using the Friedmann equations. The general solution to the Friedmann equations requires a numerical solution, which is easily coded in any computing language (including excel). We use these numerical calculations in four projects that help students build their understanding of high-redshift phenomena and cosmology. Instructions for these projects are available as supplementary materials.
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