Volume 57, Issue 7, July 1989
 Letters To The Editor



Are deliberate mistakes a valid teaching tool?
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The chart of nuclides
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Beats as moving interference patterns
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Guest Comment: Are there lessons from history?
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 Papers


American Association of Physics Teachers 1989 Oersted Medalist: Anthony P. French
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A wave‐packet description of the motion of a charged particle in a uniform magnetic field
View Description Hide DescriptionA discussion is given of the construction of wave packets to describe the motion of a charged particle in a uniform magnetic field. Examples are given of the construction of wave packets from eigenstates of the Hamiltonian for both the Landau gauge in Cartesian coordinates and the cylindrically symmetric gauge in cylindrical coordinates.

Spatial geometry in a uniformly accelerating reference frame
View Description Hide DescriptionThe meaning of spatial geometry in a reference frame is carefully analyzed. It is shown that in a uniformly accelerating reference frame spatial geometry is Euclidean if distance is measured with measuring rods and non‐Euclidean if distance is measured with light signals. The distance function and the square of the line element associated with each mode of measurement are obtained.

Applications of the residue theorem to two‐dimensional electrostatic and magnetostatic situations
View Description Hide DescriptionAmpere’s circuital law for magnetostatics and two‐dimensional analogs of Gauss’ law for electrostatics and magnetostatics are obtained by constructing appropriate complex functions and applying the residue theorem. The approach is motivated by the formal similarities between the residue theorem and Ampere’s circuital law for magnetostatics.

The magnetic field of a circular turn
View Description Hide DescriptionThis article presents the numerical solution for the magnetic field of a current‐carrying circular turn at any point. Textbook solutions usually confine themselves to the solution on the axis of the turn because the on‐axis Biot–Savart law integration is easy to do in closed form. As the knowledge and use of hand‐held calculators and computers has increased, the numerical solution of this problem for off‐axis points has become possible by students in the introductory calculus‐based course. A good way to do this problem is in the physics laboratory where actual measurements can be compared with the values obtained from a numerical solution.

Energy balance and the Abraham–Lorentz equation
View Description Hide DescriptionIt is shown that the energy balance usually assumed for the derivation of the Abraham–Lorentz equation is incomplete because it does not take the change in the energy of the bound self‐field into account. Starting from the energy balance implied by Poynting’s theorem, a new derivation of the Abraham–Lorentz equation is presented. The energy balance is applied first to an extended charge distribution and then the point‐charge limit is taken. This approach neither requires neglecting the so‐called Schott energy term nor any further hypothesis. The calculations point to the dragged self‐field as the origin of the ‘‘radiation’’ damping term.

On the magnetic field generated by a short segment of current
View Description Hide DescriptionThis article describes a series of simple experiments with which students can discover the well‐known formula for the magnetic field generated by an infinitesimal current segment. No such procedure seems to have been described previously in the literature.

Self‐similarity and long‐tailed distributions in the generation of thermal light
View Description Hide DescriptionTwo counterintuitive phenomena are studied. (1) It is well known that a thermal electromagnetic field has a Bose–Einstein (geometric) distribution of photons within a coherence volume. This arises because of the photon clumping characteristic of a thermal Boson field. On the other hand, the distribution of the number of atoms emitting photons through spontaneous emission must be Poisson if emissions are truly independent. (2) The average time between atomic decays is finite, being just the inverse of the total decay rate of the atoms. However, it is shown that in a coherence volume or in a single mode of the resulting Gaussian electromagnetic field, the average photon interarrival time is infinite. Hence, on average, an infinite length of time must pass before 〈N〉 photons arrive in the field. These apparent paradoxes are discussed, showing how both arise from random interference of Boson fields. The infinite waiting time is seen to be one manifestation of a long‐tailed distribution. Such distributions are increasingly important by virtue of their relation to self‐similarity and fractals, e.g., strange attractors in the description of deterministic chaos; therefore, it is of interest to understand their counterintuitive properties and see how they arise naturally even in more traditional analyses.

Applications of Bohr’s correspondence principle
View Description Hide DescriptionThe Bohr correspondence‐principle (cp) formula d E/d n=ℏω is presented (ω is the classical angular frequency) and its predicted energy levels E _{ n } are compared to those given by the stationary state solutions of the Schrödinger equation, first for several examples in one dimension (1D), including the ‘‘quantum bouncer,’’ and then for several examples in three dimensions (3D), including the hydrogen atom and the isotropic harmonic oscillator. For the 3‐D cases, the cp predictions based on classical circular orbits are compared with the ‘‘circlelike’’ Schrödinger solutions (those with the lowest energyeigenvalue for a given l) and the cp predictions based on classical ‘‘needle’’ orbits (having zero angular momentum) with the Schrödinger l=0 solutions. For the H atom and the isotropic oscillator, the cp prediction does not depend on the classical orbit chosen because of a ‘‘degeneracy’’: the fact that for these systems ω is independent of the orbit. As a more stringent test of the cp, analogous nondegenerate systems V=−k/r ^{3} ^{/} ^{2} in place of the H‐atom potential V=−e ^{2}/r and V=k r ^{4} in place of the oscillator potential V=(1/2)mω^{2} r ^{2} are therefore considered. An interesting anomaly that occurs for the harmonic oscillator and its nondegenerate analog V=k r ^{4} is encountered (but not for the H atom nor its nondegenerate analog V=−k/r ^{3} ^{/} ^{2}), wherein half of the states predicted by application of the cp to the needle orbits are ‘‘spurious’’ in that there are no corresponding Schrödinger l=0 states. The assumption that generated the spurious cp states is uncovered—a plausible, but erroneous factor of 2 in calculating the classical frequency—and thus the spurious states are eliminated.

Relativistic phase velocity transformation
View Description Hide DescriptionBy utilizing the appropriate Doppler formulas, it is shown that phase velocities in both Galilean and relativistic kinematics transform according to rules different from the ordinary velocity addition laws. As corollaries, it is demonstrated that (i) phase velocities parallel to the inertial frames’ relative velocity transform according to the Lorentz velocity addition rule; (ii) regardless of its direction, the velocity of light in vacuum transforms as a relativistic particle velocity; and (iii) the relativistic phase velocity transformation rule reduces to the Galilean rule in the nonrelativistic limit.

A direct laboratory approach to the study of capacitors
View Description Hide DescriptionThe construction of a device called a charge pump is described. This charge pump can be built for less than $10 and, with the aid of a voltmeter and stopwatch, allows the student of introductory physics to measure the capacitance of a capacitor and the charge on a capacitor. Four typical experiments that utilize the charge pump are outlined. Errors of the order of 1% are obtained by students when the experiments are performed.

Elementary calculation of soil damping depth
View Description Hide DescriptionAn important problem in biophysics, calculation of the damping depth in relation to penetration of temperature ‘‘waves’’ through soil, is solved by adopting an approximate method of solving Fourier’s partial differential equation. The method illustrates a way in which students of physics majoring in life sciences, with limited mathematical experience, can come to terms with quantitative analysis of problems that would otherwise be beyond the scope of usual courses.

Electric field and radiation by a pointlike source moving in an isotropic medium
View Description Hide DescriptionThe electric field in space and time of a source moving in a lossless, isotropic, and dispersive medium is obtained from the inhomogeneous wave equation combined with the charge continuity equation. The Fourier transform method is employed together with the theory of functions of complex variables to express the causality condition. The existence of two retarded times is pointed out for the case of a pointlike source moving in a nondispersive medium faster than the propagation velocity of the electromagnetic field. The Čerenkov as well as the polarization radiation is obtained for a dispersive isotropic medium.

On the spring constant of a close‐coiled helical spring
View Description Hide DescriptionThe relationship between the spring constant of a close‐coiled helical spring, its geometry, and the shear modulus of its material is examined in a way suitable for presentation in physics textbooks. Contributions from both torsion and pure shear are considered.
