Table of contents

Describes experiments suitable for undergraduate laboratory work, using low-pass transmission lines composed of discrete inductors and capacitors to simulate several aspects of linear lattice vibrations. Advantages compared with the use of mechanical systems of masses and springs are suggested.

An undergraduate laboratory experiment on the statistical fluctuations of nuclear reactions is described. The aim is to familiarise students with the use of solid-state nuclear track detectors using Cf²⁵² as a source of spontaneous fission fragments and ordinary microscope slide glass (soda glass) as a detector, used in a geometry <2 pi . A simple method for exposure of the detector is given. The results indicate that the observed distribution of tracks, counted using a graticule fitted in the eyepiece of the microscope, is in good agreement with the expected Poisson distribution. However, for larger average values of the counts the Poisson and Gaussian distributions tend to coalesce.

The electromagnetic induction associated with periods of changing current in a stationary circuit is discussed with particular reference to the presence and the fundamental role of electric fields in this situation.

Theoretical investigations into the stability of electrified liquid drops take as their starting point a result due to Rayleigh. It was felt that this result is important enough to warrant a more complete derivation than that originally given by Rayleigh in the reference which is usually quoted. A literature search revealed a second paper of Rayleigh's which is never quoted, but which contains important steps, not given elsewhere, towards his original calculation. Even when both papers are taken together, however, the calculation still seems incomplete. For this reason the entire calculation is presented using a modern formalism which in particular recognises the effects of electrostriction. To complement this the complete calculation as originally conceived by Rayleigh is presented in an appendix.

Describes what a shock wave is when the dispersion law is linear. Then studies the general case using the method of 'wave packets'. When the dispersion relation is omega =akⁿ, one obtains the following results: no shock wave when n>1; shock wave of constant angle when n<1. The case n=1 is the usual case (linear dispersion relation).

The authors discuss a simple model for the relaxation of two interacting dynamical systems, based on the Ehrenfest's wind-tree model. It is still exactly solvable if Boltzmann's Stosszahlansatz is assumed to hold.

In any irreversible process the amount of energy made unavailable to perform work equals T Delta S, where T is the temperature of the lowest temperature reservoir available and Delta S is the entropy change of the closed system. That the application of this concept is not restricted to processes involving heat flow or temperature differences is here illustrated by considering the diffusive mixing of ideal gases.

The authors study the wavefronts (i.e. the surfaces of constant phase) of the wave discussed by Aharonov and Bohm, representing a beam of particles with charge q scattered by an impenetrable cylinder of radius R containing magnetic flux Phi . Defining the quantum flux parameter by alpha =q Phi /h, they show that for the case R=0 the wave psi _AB possesses a wavefront dislocation on the flux line, whose strength (i.e. the number of wave crests ending on the dislocation) equals the nearest integer to alpha . When alpha passes through half-integer values, the strength changes, by wavefronts unlinking and reconnecting along a nodal surface. In quantum mechanics this phase structure is unobservable, but they devise an analogue where surface waves on water encounter an irrotational 'bathtub' vortex; in this case alpha depends on the frequency of the waves and the circulation of the vortex. Experiments show dislocation structures agreeing with those predicted. psi _AB is an unusual function in which incident and scattered waves cannot be clearly separated in all asymptotic directions; they discuss its properties using a new asymptotic method.

Non-linear partial differential equations in physics are discussed using water waves in a channel as an illustration. Weakly non-linear waves, wave breaking shocks, and solitary waves are considered. Although these topics have been understood for many years, they are unfamiliar to the majority of physicists. They are also the background to the recent advances in non-linear physics which will be discussed in a companion paper.

The authors recall a well-known difficulty arising in quantum mechanics when insufficient care is taken with unbounded operators. They illustrate it in the particular case of the second-derivative operator defined on a half line. They show how it must be modified to satisfy the symmetry relation (f mod H mod g)=(g mod H mod f)* for arbitrary functions f and g. As an application of the result, they derive very easily hypervirial theorems as well as sum rules involving the behaviour of the wavefunction at the origin for the one-dimensional, even-parity Schrodinger operator.

The authors present a method whereby functional relationships between experimental variables may be obtained using dimensional and regression analysis. Starting from the results furnished by the pi theorem of dimensional analysis, they approach the determination of unknown functions appearing when there is more than the pi monomial, by means of statistically obtained partial functional dependence relationships. Such an approach may be of interest or utility to advanced physics laboratory students.

The use of two-dimensional vector algebra is illustrated in the mathematical treatment of hydrodynamic plane flow and horizontal atmospheric winds. These are just two possible examples of the simplification obtainable in mathematical formulae and their derivations, as well as in their physical interpretation, by the use of this algebra. For this purpose some properties derived from three-dimensional algebra are needed, which could be equally well applied to other physical phenomena. Two-dimensional vector algebra is therefore of notable pedagogical interest.

A more instructive presentation of electromagnetism is possible when the magnetic field is treated as a bivector quantity rather than as an axial vector quantity. The connection between the magnetic field and currents as its sources is more transparent. The Lorentz-Abraham analogy between the field quantities is more complete.

The equation for the frequency shift of light as measured locally by emitter and receiver is written out, so that the frequency shift is given as a function of the position and velocity of emitter and receiver, and the direction of motion of the light at their positions. The given form of the equation is generally valid in time-independent metrics. The following special cases are considered: the Minkowski metric source and absorber at rest in an arbitrary stationary metric, uniform gravitational field, rotating reference frame and the Schwarzschild metric.