Table of contents

We give a very simple derivation of the Lie algebra of the rotation group relative to a general curvilinear basis. The derivation is independent of any matrix representation of the group generators.

We give a simple rule to associate a pair of complex numbers to a spinor. This association rule is suggested by an analysis of the link between Jones vectors and the Poincaré sphere, which are tools commonly used in the description of polarized light, and allows the basic properties of spinors, such as the effect of rotations, to be derived in a simple way. In our treatment, we give a purely mathematical description of spinors, without using the physical properties of the spin, so that the theory of spinors can be introduced independently from its application to physics.

Riassunto.In questo lavoro forniamo una semplice regola per associare una coppia di numeri complessi ad uno spinore. Tale regola è suggerita da un'analisi del legame che sussiste tra i vettori di Jones e la sfera di Poincaré, strumenti comunemente usati nella descrizione della luce polarizzata. Essa consente di derivare in modo semplice le proprietà di base degli spinori, come ad esempio l'effetto delle rotazioni. Nella nostra analisi, gli spinori sono descritti da un punto di vista puramente matematico, senza fare uso delle proprietà fisiche dello spin; in tal modo la teoria degli spinori può essere sviluppata indipendentemente dalla sua applicazione alla fisica.

We show that knowing the addition law of parallel velocities, which can be derived directly from the first postulate, we are able to derive the Lorentz transformations for the space and time coordinates, for mass and momentum, for the frequency and wave vector as well as for the normal components of the electric and magnetic fields.

Zusammenfassung.Man zeigt, dass durch dass Kennen des Gesetzes der Zusammensetzung der parallelen Geschwindigkeiten, welches direkt aus dem ersten Postulat hervorgeht, die Lorentz Transformation für Raum und Zeit Koordinaten, die Masse und Impuls, Frequenz und Wellenzahl sowie die normalen Komponenten des elektrischen und magnetischen Feldes bestimmt werden können.

It is noted that certain well known formulae of thermodynamics are no longer valid when systems whose entropies are not extensive (that is, not homogeneous of degree one) are under consideration. Attention is drawn to the fact that the accepted form for the entropy of a black hole falls into this category and so great care must be exercised when using black holes to generate added interest in undergraduate thermodynamics courses.

The term `ideal' is currently employed for thermodynamic systems of different natures (gases, elastic systems, magnetic systems, etc). However, different meanings are usually given to this term. Joule's law, which establishes that the internal energy of a system depends exclusively on temperature, is considered here as a general criterion for an ideal system in classical thermodynamics. The conditions that a system should satisfy to obey Joule's law and the implications of these conditions on the equation of state are discussed. Some similarities and differences between ideal systems of different natures are also emphasized.

Resumen.El término `ideal' se emplea en la actualidad para sistemas termodinámicos de naturaleza distinta (gases, sistemas elásticos, sistemas magnéticos, etc). Sin embargo, es habitual dar a este término distintos significados. La ley de Joule, que establece que la energía interna de un sistema depende sólo de la temperatura, se considera aquí como un criterio general para la condición de sistema ideal. Se discuten las condiciones que un sistema debe satisfacer para verificar la ley de Joule, así como las implicaciones de estas condiciones sobre la ecuación de estado. Se subrayan también las similitudes y diferencias entre sistemas ideales de distinta naturaleza.

The general expressions of energy, heat capacity and Bose - Einstein condensation (BEC) temperature are given for an n-dimensional ideal Bose gas with a dispersion relation of , with which the possibility of occurrence of BEC can be determined. The curve of against temperature of the gas with is continuous at , while that of the gas with has a finite jump . These are two different kinds of phase transitions.

Microcomputer-based laboratory (MBL) tools have been developed which interface with a great variety of computers. Students use these tools to collect physical data in real time which can later be manipulated and analysed. This new investigative method together with a high standard of precision enables students to investigate many principles of physics that have heretofore not been feasible. In this paper, we describe some examples of experiments designed for college students in an introductory physics course, in which energy pulses were transferred to an LC circuit. We mainly show the response of the circuit to low-frequency external energy pulses and the influence of the distance between the source and the absorbing LC circuit on the energy absorbed by the circuit in its resonant state. The data obtained were analysed by taking advantage of the wide range of facilities in the Explorer MBL.

Resumen.Instrumentos de Laboratorio basado en Microcomputadoras (LBM) han sido desarrollados en conjunción con una gran variedad de computadoras. Estudiantes hacen uso de tales instrumentos para registrar los resultados de experimentos físicos durante la realización del experimento, y luego analizarlos en distintas formas. Este nuevo método de investigación, junto con altas normas de precisión, permite que los estudiantes investiguen muchos principios físicos que hasta ahora no tenían la posibilidad de investigar. En este artículo, describimos algunos ejemplos de experimentos proyectados para estudiantes universitarios en cursos de introducción a la física, en los cuales pulsos energéticos son transferidos a un circuito eléctrico de tipo LC. Principalmente se analiza la reacción del circuito al recibir pulsos energéticos de baja frecuencia, y la influencia de la distancia entre el circuito originario de los pulsos y el circuito receptor sobre la energía absorbida por el circuito receptor en estado de resonancia. Los resultados obtenidos fueron analizados aprovechando la extensa gama de facilidades que ofrece el sistema Explorer LBM.

In this work we use the unusual behaviour of a light beam in a diffusion cell as an experimental introduction for university students of physics and engineering to the optical - mechanical analogy. We have found that the lecture demonstration is a key element in the development of the presentation and subsequent use of the analogy.

Riassunto.In questo lavoro si utilizza l'inusuale comportamento di un fascio luminoso in una cella di diffusione come introduzione sperimentale, per studenti di fisica e di ingegneria, all'analogia tra il comportamento di una particella materiale ed un fascio luminoso (Analogia di Hamilton). La dimostrazione didattica rappresenta un elemento chiave nello sviluppo della presentazione e del successivo utilizzo dell'analogia.

When a moving magnetic dipole enters a strong electric field the velocity of the magnet changes by a tiny amount. When a polarized, coherent beam of magnetic particles is split, the two parts passed through strong electric fields of opposite sign, then rejoined, interference patterns are observed due to the phase shift between the de Broglie waves of the two branches. Two recent, extremely delicate experiments, one using neutron interferometry and the second using F nuclei in TlF molecules, are reviewed and interpreted in this way. The fractional velocity change was 1 part in for the neutrons and 1 in for the TlF molecules. This explains the phase shift discussed by Aharonov and Casher and, contrary to their statement, is not equivalent to an Aharonov - Bohm effect. The latter requires that a de Broglie wave phase shift be produced by a vector potential in the absence of a field.

A simple derivation of the formula for the flux of light at a point from cosmic light sources is given. Both the obscuration of distant sources by nearby ones and the expansion of the universe are taken into account. The explanation of the dark night sky is mainly that the distance travelled by a light particle is much smaller than the mean free path for a photon. Both the obscuration and the expansion are relatively unimportant. We live in a young universe with little energy.

Sammendrag.En enkel utledning av formelen for lysfluks i et punkt fra de kosmiske lyskilder blir gitt. B de at de næ rliggende lyskilder vil skygge for de fjerne lyskilder og universets ekspansjon blir tatt i betraktning. Forklaringen på den mørke nattehimmelen er i hovedsak at avstanden som en lyspartikkel har beveget seg, er mye mindre enn midlere fri veilengde for et foton. Bå de skygge-effekten og ekspansjonen er nokså lite viktige. Vi lever i et ungt univers med lite energi.

Incoherent averaging over orientations of refracted beams of rays can generate a `fake caustic', namely an intensity pattern with a geometrical singularity identical to that of a caustic produced by focusing of a family of rays (i.e. its envelope). However, the diffraction at a fake caustic is very different from the Airy fringes near a genuine caustic. We calculate this unusual type of diffraction and demonstrate its existence experimentally with a rotating prism.

Zusammenfassung.Inkoherente Mittelung über die Orientierungen von gebrochenen Strahlenbündeln kann eine `falsche Kaustik' erzeugen, nämlich ein Intensitätsmuster mit einer geometrischen Singularität identisch zu der einer Kaustik, die durch Fokusierung einer Familie von Strahlen (d.h. ihre Einhuellende) hervorgebracht wurde. Die Beugung in einer falschen Kaustik aber ist sehr verschieden von den Airy Streifen nahe einer echten Kaustik. Wir berechnen diesen ungewöhnlichen Typ einer Beugung und zeigen ihre Existenz experimentell mit einem rotierenden Prisma.

Diamagnetic objects are repelled by magnetic fields. If the fields are strong enough, this repulsion can balance gravity, and objects levitated in this way can be held in stable equilibrium, apparently violating Earnshaw's theorem. In fact Earnshaw's theorem does not apply to induced magnetism, and it is possible for the total energy (gravitational + magnetic) to possess a minimum. General stability conditions are derived, and it is shown that stable zones always exist on the axis of a field with rotational symmetry, and include the inflection point of the magnitude of the field. For the field inside a solenoid, the zone is calculated in detail; if the solenoid is long, the zone is centred on the top end, and its vertical extent is about half the radius of the solenoid. The theory explains recent experiments by Geim et al, in which a variety of objects (one of which was a living frog) was levitated in a field of about 16 T. Similar ideas explain the stability of a spinning magnet above a magnetized base plate. Stable levitation of paramagnets is impossible.

Samenvatting.Magnetische velden stoten diamagnetische voorwerpen af. Zulke velden kunnen zo sterk zijn dat zij de zwaartekracht opheffen. Het is op deze wijze mogelijk zulke voorwerpen te laten zweven. Dit vormt een stabiel evenwicht, wat in tegenspraak schijnt te zijn met Earnshaw's Theorema. Echter Earnshaw's Theorema is niet langer geldig als het magnetisme veld geinduceerd is. De totale energie (bevattende bijdragen van het magnetisme en de zwaartekracht) kan toch een lokaal minimum vertonen. Algemene criteria voor zo'n minimum zullen worden opgesteld. Verder zal worden aangetoond dat voor een cilindrisch symmetrisch veld, langs zijn symmetrie as altijd een zone gevonden kan worden waarin een stabiel evenwicht bestaat. Voor het veld binnen een solenoïde zal deze zone in detail bepaald worden. Als deze spoel voldoende land is bevindt deze zone zich aan het uiteinde van de spoel. De lengte van deze zone langs de symmetrie as van het veld is ongeveer de helft van de straal van de spoel. Deze theorie geeft een goede verklaring voor de experimenten van Geim et al. In deze experimenten werden een grote verscheidenheid aan verschillende voorwerpen (waaronder een levende kikker) tot zweven gebracht in velden van ongeveer 16 T. Analoge theoriën verklaren de stabiliteit van een roterend permanent magneetje boven een magneetische grondplaat. Het is onmogelijk om paramagnetische voorwerpen stabiel te doen zweven.