Estratégia de Resolução de porblemas de Ensino Médio envolvendo blocos

Para resolver problemas envolvendo blocos devemos, inicialmente desenhar um diagrama de forças para o bloco. corpo.

Se o corpo está se movendo sobre o plano, ele estará em equilíbrio estático na direção perpendicular ao plano (PESO = NORMAL) A condição para que isto ocorra, é que a soma das componentes das forças nesta direção seja nula. I

A aceleração do bloco pode ser obtida calculando-se a resultante das componentes no sentido do movimento e  aplicando a 2ª Lei de Newton. 

Fr = m.a

Conhecer todas as variáveis envolvidas na equação acima é fundamental.

A evolução contradiz a segunda lei da termodinâmica?

Se caminhamos da ordem para a desordem, como a natureza sitemas poderia ter originado sistemas vivos e tão complexos? Esse argumento vem sendo usado em muitas ocasiões para invalidar o modelo de Darwin e acho importante desmistifica-lo.

Resposta:

Em sistemas abertos, sistemas que trocam energia e entropia com o ambiente, é possível aparecer ordem (diminuição da entropia) sem violar a Segunda Lei da Termodinâmica (SLT). A SLT afirma que no sistema total, que neste caso é o sistema aberto mais o ambiente, a entropia aumenta. A SLT não impede a possibilidade de que em uma parte do grande sistema haja diminuição da entropia.

Os seres vivos podem diminuir ou manter constante sua entropia por exportar entropia para o ambiente.

Um refrigerador, para dar um exemplo de um sistema não-vivo, é um sistema que também diminui localmente a entropia e não viola a SLT.

Desta forma a SLT não invalida o darwinismo.

Interessantes refutações dos argumentos antievolucionistas encontrarás nas obras de Richard Dawkins por exemplo.

Segue em frente com as tuas idéias evolucionistas, assumindo uma posição crítica em relação a elas e às supostas refutações do darwinismo! Esta é a verdadeira atitude científica, que não se dobra para verdades reveladas e que somente aceita idéias que passem pelo crivo da racionalidade!!

Prof. Fernando Lang da Silveira - IF-UFRGS

www.if.ufrgs.br/~lang/

Leia um artigo da Amer. J. Phys. explicando em linguagem técnica essa questão.

Profissão de físico no Brasil

Sem prejuízo do exercício das mesmas atividades por outros profissionais igualmente habilitados na forma da legislação específica, o Físico poderá, entre outras atividades:

I - realizar pesquisas científicas e tecnológicas.

II - difundir conhecimentos, orientar trabalhos técnicos e científicos, ministrar palestras, seminários e cursos, organizar eventos científicos, treinar especialistas e técnicos;

III - administrar atividades de pesquisas e aplicações, planejar e executar pesquisas científicas e tecnológicas, planejar instalações, especificar equipamentos e infra-estrutura laboratorial;

IV - realizar medidas físicas no âmbito de sua especialidade;

V - projetar, desenvolver, construir, calibrar, fazer manutenção, fiscalizar, auditar e controlar dispositivos, equipamentos e sistemas no âmbito de sua especialidade;

VI - elaborar documentação técnica e científica, realizar perícias e auditorias, emitir e assinar laudos técnicos e pareceres, definir procedimentos operacionais, de segurança, de radioproteção e de análise de impacto ambiental;

VII - aplicar princípios, conceitos e métodos da Física em atividades específicas envolvendo radiação ionizante e não ionizante, estudos ambientais, análise de sistemas ecológicos, estudos na área financeira e outros;

VIII - desenvolver programas e aplicativos computacionais baseados em modelos físicos;

IX - Atuar na área médica em serviços de radioterapia, radiodiagnóstico, medicina nuclear, proteção radiológica e outros para os quais esteja especificamente habilitado pelo Conselho Federal de Física.

http://www.camara.gov.br/sileg/Prop_Detalhe.asp?id=395897

Regras para utilizar meu apoio pelo Messenger e e-mail

No meu site www.fisica.net existe muito material útil.


Aprenda a usar o Google. Se algo existe, o Google encontra. Uso ele desde 1999 e GARANTO!

Exemplo:
Quer saber o que é CALOR? Digite define:calor
Deseja algum material de qualidade, especifique que seja PDF (bla, bla filetype:pdf)
Está querendo um ppt pronto?, especifique filetype:ppt
E por aí vai. NÃO EXISTE CONTEÚDO DE ENSINO MÉDIO QUE NÃO TENHA ALGUM BOM SITE NO GOOGLE.



Meu Messenger está lotado, por isso, não é possível eu adicionar, mas posso ser adicionado.

Seja objetivo, sem ser agressivo. Não desejo fazer amigos pelo MSN. Amigos já tenho em número e qualidade aqui na minha cidade.


Orkut é para contatos rápidos. Não responderei dúvidas longas por ele. Use e-mail.







Não faço trabalhos escolares

Não resolvo exercícios, a não ser que eu me sinta atraído pelo problema.

Odeio fazer continhas. Sou físico e não matemático.



Se precisar de ajuda em algum exercício, desenvolva o SEU raciocínio e me apresente de tal forma que eu possa entender qual a sua dificuldade. É importante que diga se é estudante de ensino médio, superior , etc. Preciso adequar a linguagem.


Recebo dezenas de pedidos a cada dia. Não tenho tempo, saúde e paciência para responder a todos os e-mails.






Moro em Porto Alegre, RS.
Fiz graduação em Física, especialização em Radiações e mestrado em Física
Lecionei Física para 8a série, ensino médio e pré-vestibular por 15 anos. Agora parei.

Me dedico as minhas pesquisas na universidade e a direção de tecnologia do Grupo Universitário ( www.universitario.com.br ).

Atualmente luto para viver com uma doença cruel chamada LÚPUS (faça um curso de Medicina se desejar entender o que é), isso tem roubado mais de 50% do meu tempo.

Sobre Lúpus, leia http://imunologico.blogspot.com

Why is c the symbol for the speed of light?

"As for c, that is the speed of light in vacuum, and if you ask why c, the answer is that it is the initial letter of celeritas, the Latin word meaning speed."
Isaac Asimov in "C for Celeritas (1959)" [1]



A Short Answer

Although c is now the universal symbol for the speed of light, the most common symbol in the nineteenth century was an upper-case V which Maxwell had started using in 1865. That was the notation adopted by Einstein for his first few papers on relativity from 1905. The origins of the letter c being used for the speed of light can be traced back to a paper of 1856 by Weber and Kohlrausch [2]. They defined and measured a quantity denoted by c that they used in an electrodynamics force law equation. It became known as Weber's constant and was later shown to have a theoretical value equal to the speed of light times the square root of two. In 1894 Paul Drude modified the usage of Weber's constant so that the letter c became the symbol for the speed of electrodynamic waves [3]. In optics Drude continued to follow Maxwell in using an upper-case V for the speed of light. Progressively the c notation was used for the speed of light in all contexts as it was picked up by Max Planck, Hendrik Lorentz and other influential physicists. By 1907 when Einstein switched from V to c in his papers, it had become the standard symbol for the speed of light in vacuum for electrodynamics, optics, thermodynamics and relativity.

Weber apparently meant c to stand for "constant" in his force law, but there is evidence that physicists such as Lorentz and Einstein were accustomed to a common convention that c could be used as a variable for velocity. This usage can be traced back to the classic Latin texts in which c stood for "celeritas" meaning "speed". The uncommon English word "celerity" is still used when referring to the speed of wave propagation in fluids. The same Latin root is found in more familiar words such as acceleration and even celebrity, a word used when fame comes quickly.

Although the c symbol was adapted from Weber's constant, it was probably thought appropriate for it to represent the velocity of light later on because of this Latin interpretation. So history provides an ambiguous answer to the question "Why is c the symbol for the speed of light?", and it is reasonable to think of c as standing for either "constant" or "celeritas".





The Long Answer
In 1992 Scott Chase wrote on sci.physics that "anyone who read hundreds of books by Isaac Asimov knows that the Latin word for `speed' is `celeritas', hence the symbol `c' for the speed of light". Asimov had written an article entitled "C for Celeritas" in a sci-fi magazine in 1959 and had reprinted it in some of his later books [1]. Scott was the first editor of the Physics FAQ on Usenet and Asimov's explanation was later included in the relativity section as the "probable" answer to the question "Why is c the symbol for the speed of light?". Since then, Asimov's answer has become a factoid repeated in many articles and books. But if you go back and read his essay you discover that Asimov merely stated his case in one sentence, and made no further attempt to justify his theory for the origin of the "c" notation. So is his claim really born out by history, or was c originally introduced as a variable standing for something else? The special theory of relativity is based on the principle that the speed of light is constant; so did c stand for "constant", or did it simply appear by accident in some text where all the other likely variables for speed had already been used up? These questions have been asked repeatedly on usenet, and now after much searching through old papers and books the answers can be revealed.

A lower-case c has been consistently used to denote the speed of light in textbooks on relativity almost without exception since such books started to be written. For example, the notation was used in the earliest books on relativity by Lorentz (1909) [4], Carmichael (1913) [5], Silberstein (1914) [6], Cunningham (1915) [7], and Tolman (1917) [8]. That was not the case just a few years before. In his earliest papers on relativity from 1905--1907 Einstein began by using an upper-case V for the speed of light [9]. At that time he was also writing papers about the thermodynamics of radiation, and in those he used up upper-case L [10]. All of these papers appeared in volumes of the German periodical Annalen Der Physik. Einstein's notation changed suddenly in 1907 in a paper for the Journal Jahrbuch der Radioaktivität und Elektronik [11]. There he used the lower case c, and his most famous equation E = mc2 came into being.

It is not difficult to find where the upper case V had come from. Maxwell used it extensively in his publications on electrodynamics from as early as 1865 [12]. It was the principal symbol for the speed of light in his 1873 treatise on electrodynamics [13]. By the 1890s Maxwell's book was in wide circulation around the world and there were translations available in French and German. It is no surprise then that the upper-case V is found in use in such papers as the 1887 report of Michelson and Morley on their attempt to find seasonal variations in the speed of light [14]. That was written in the United States, but the same notation was also found across Europe, from papers by Oliver Lodge [15] and Joseph Lamor [16] in England, to the lecture notes of Poincaré in France [17], and the textbooks of Paul Drude in Germany [18] and Lorentz in the Netherlands [19]. Einstein's education at the Polytechnik in Zurich had not covered Maxwell's theory of Electrodynamics in the detail he would have liked. But he had read a number of extra textbooks on the new Electrodynamics as self study, so he would have been familiar with the standard notations. From 1905 he wrote his first papers on relativity, and there is nothing extraordinary in his choice of the symbol V for the speed of light [9].

Why then, did he change it to c in 1907? At that time he still worked as a clerk in the Bern patent office, but for the previous two years he had been in regular correspondence with eminent physicists such as Max Laue, Max Planck, Wilhelm Wien and Johannes Stark. Stark was the editor of the Jahrbuch, and had asked Einstein to write the article in which he was to first use the letter c. Einstein mentioned to Stark that it was hard for him to find the time to read published scientific articles in order to acquaint himself with all the work others have done in the field, but he had seen papers by Lorentz, Kohn, Monsegeil and Planck [20]. Lorentz and Planck in particular had been using c for the speed of light in their work. Lorentz had won the 1902 Nobel prize for physics, and it is not surprising that physicists in Germany had now taken up the same notation. It is also not surprising that Einstein, who was looking for an academic position, aligned himself to the same conventions at that time. Another reason for him to make the switch was that the letter c is simply more practical. The upper-case V would have been easily confused with the lower case v appearing in the equations of relativity for the velocity of moving bodies or frames of reference. Einstein must have found this confusion inconvenient, especially in his hand written notes.

Looking back at papers of the late 1890s, we find that Max Planck and Paul Drude in particular were using the symbol c at that time. The name of Drude is less well known to us today. He worked on relations between the physical constants and high precision measurements of their value. These were considered to be highly worthy pursuits of the time. Drude had been a student of Voigt, who himself had used a Greek ω for the speed of light when he wrote down an almost complete form of the Lorentz transformations in 1887 [43]. Voigt's ω was later used by a few other physicists [44, 45], but Drude did not use his teacher's notation. Drude first used the symbol c in 1894, and in doing so he referenced a paper by Kirchhoff [3]. As already mentioned, Paul Drude also used V. In fact he made a distinction of using V in the theory of optics for the directly-measured speed of light in vacuum, whereas he used c for the electromagnetic constant that was the theoretical speed of electromagnetic waves. This is seen especially clearly in his book "Theory of Optics" of 1900 [21], which is divided into two parts with V used in the first and c in the second part. Although Maxwell's theory of light predicted that they had the same value, it was only with the theory of relativity that these two things were established as fundamentally the same constant. Other notations vied against Drude's and Maxwell's for acceptance. Herglotz [46] opted for an elaborate script B, while Himstedt [47], Helmholtz [48] and Hertz [49] wrote the equations of electrodynamics with the letter A for the reciprocal of the speed of light. In 1899 Planck backed Drude by using c, when he wrote a paper introducing what we now call the Planck scale of units based on the constants of electrodynamics, quantum theory and gravity [22]. Drude and Planck were both editors of the prestigious journal Annalen Der Physik, so they would have had regular contact with most of the physicists of central Europe.

Lorentz was next to change notation. When he started writing about light speed in 1887 he used an upper case A [23], but then switched to Maxwell's upper case V [24]. He wrote a book in 1895 [25] that contained the equations for length contraction, and was cited by Einstein in his 1907 paper. While Drude had started to use c, Lorentz was still using V in this book. He continued to use V until 1899 [26], but by 1903 when he wrote an encyclopedia article on electrodynamics [27] he too used c. Max Abraham was another early user of the symbol c in 1902, in a paper that was seen by Einstein [28]. From Drude's original influence, followed by Planck and Lorentz, by 1907 the c symbol had become the prevailing notation in Germanic science and it made perfect sense for Einstein to adopt it too.

In France and England the electromagnetic constant was symbolised by a lower case v rather than Drude's c. This was directly due to Maxwell, who wrote up a table of experimental results for direct measurements of the speed of light on the one hand and electromagnetic experiments on the other. He used V for the former and v for the latter. Maxwell described a whole suite of possible experiments in electromagnetism to determine v. Those that had not already been done were performed one after the other in England and France over the three decades that followed [29]. In this context, lower case v was always used for the quantity measured. But using v was doomed to pass away once authors had to write relativistic equations involving moving bodies, because v was just too common a symbol for velocity. The equations were much clearer when something more distinct was used for the velocity of light to differentiate it from the velocity of moving bodies.

While Maxwell always used v in this way, he also had a minor use for the symbol c in his widely read treatise of 1873. Near the end he included a section about the German electromagnetic theory that had been an incomplete precursor to his own formulation [30]. This theory, expounded by Gauss, Neumann, Weber, and Kirchhoff, attempted to combine the laws of Coulomb and Ampère into a single action-at-a-distance force law. The first versions appeared in Gauss's notes in 1835 [31], and the complete form was published by Weber in 1846 [32]. Many physicists of the time were heavily involved in the process of defining the units of electricity. Coulomb's law of electrostatic force could be used to give one definition of the unit of charge while Ampère's force law for currents in wires gave another. The ratio between these units had the dimension of a velocity, so it became of great practical importance to measure its value. In 1856 Weber and Kohlrausch published the first accurate measurement [2]. To give a theoretical backing they rewrote Weber's force law in terms of the measured constant and used the symbol c. This c appeared in numerous subsequent papers by German physicists such as Kirchhoff, Clausius, Himstedt, and Helmholtz, who referred to it as "Weber's constant". That continued until the 1870s, when Helmholtz discredited Weber's force law on the grounds of energy conservation, and Maxwell's more complete theory of propagating waves prevailed.

Two papers using Weber's force law are of particular note. One by Kirchhoff [33] and another by Riemann [34] related Weber's constant to the velocity at which electricity propagated. They found this speed to be Weber's constant divided by the square root of two and it was very close to the measured speed of light. It was already known from experiments by Faraday that light was affected by magnetic fields, so there was already much speculation that light could be an electrodynamic phenomenon. This was the inspiration for Maxwell's work on electrodynamics, so it is natural that he finally included a discussion of the force law in his treatise [30]. The odd thing is that when Maxwell wrote down the force law, he changed the variable c so that it was smaller than Weber's constant by a factor of the square root of two. So Maxwell was probably the first to use c for a value equal to the speed of light, although he defined it as the speed of electricity through wires instead.

So c was used as Weber's constant having a value of the speed of light times the square root of two, and this can be related to the later use of c for the speed of light itself. Firstly, when Maxwell wrote Weber's force law in his treatise in 1873, he modified the scale of c in the equation so that it reduced by a factor of the square root of two. Secondly, when Drude first used c in 1894 for the speed of light [3], the paper by Kirchhoff that he cited [35] was using c for Weber's constant, so Drude had made the same adjustment as Maxwell. It is impossible to say if Drude copied the notation from Maxwell, but he did go one step further in explicitly naming his c as the velocity of electrodynamic waves which by Maxwell's theory was also the speed of light. He seems to have been the first to do so, with Lorentz, Planck, and others following suit a few years later.

So to understand why c became the symbol for the speed of light we now have to find out why Weber used it in his force law. In the paper of 1856 [2] Weber's constant was introduced with these words "and the constant c represents that relative speed, that the electrical masses e and e must have and keep, if they are not to affect each other." So it appears that c originated as a letter standing for "constant" rather than "celeritas". However, it had nothing to do with the constancy of the speed of light until much later.

Despite this, there could still be some substance to Asimov's claim that c is the initial letter of "celeritas". It is true, after all, that c is also often used for the speed of sound, and it is commonly used as the velocity constant in the wave equation. Furthermore, this usage was around before relativity.

Starting with the Latin manuscripts of the 17th century, such as Galileo's "De Motu Antiquiora" or Newton's "Principia", we find that they often use the word "celeritas" for speed. However, their writing style was very geometric and descriptive. They did not tend to write down formulae where speed is given a symbol. But an example of the letter c being used for speed can be found from the eighteenth century. In 1716 Jacob Hermann published a Latin text called Phoronomia, meaning the science of motion [36]. In it he developed Newton's mechanics in a form more familiar to us now, except for the Latin symbols. His version of the basic Newtonian equation F = ma was dc = p dt, where c stands for "celeritas" meaning speed, and p stands for "potentia", meaning force.

Apart from in relativity, the most pervasive use of c to represent a speed today is in the wave equation. In 1747 Jean d'Alembert made a mathematical study of the vibrating string and discovered the one dimensional wave equation, but he wrote it without the velocity constant. Euler generalised d'Alembert's equation to include the velocity, denoting it by the letter a [38]. The general solution is y = f(x - at) + f(x + at), representing two waves of fixed shape travelling in opposite directions with velocity a.

Euler was one of the most prolific mathematicians of all time. He wrote hundreds of manuscripts and most of them were in Latin. If anyone established a convention for using c for "celeritas", it has to have been Euler. In 1759 he studied the vibrations of a drum, and moved on to the 2-dimensional wave equation. This he wrote in the form we are looking for with c now the velocity constant [39].

The wave equation became a subject of much discussion, being investigated by all the great mathematicians of the époque including Lagrange, Fourier, Laplace, and Bernoulli. Through their works, Euler's form of the wave equation with c for the speed of wave propagation was carved in stone for good. To a first approximation, sound waves are also governed by the same wave equation in three dimensions, so it is not surprising that the speed of sound also came to be denoted by the symbol c. This predates relativity and can be found, for example, in Lord Rayleigh's classic text "Theory of Sound" [40]. Physicists of the nineteenth century would have read the classic Latin texts on physics, and would have been aware that c could stand for "celeritas". As an example, Lorentz used c in 1899 for the speed of the Earth through the ether [41]. We even know that Einstein used it for speed outside relativity, because in a letter to a friend about a patent for a flying machine, he used c for the speed of air flowing at a mere 4.9 m/s [42].

In conclusion, although we can trace c back to Weber's force law where it most likely stood for "constant", it is possible that its use persisted because c could stand for "celeritas" and had therefore become a conventional symbol for speed. We cannot tell for sure how Drude, Lorentz, Planck or Einstein thought about their notation, so there can be no definitive answer for what it stood for then. The only logical answer is that when you use the symbol c, it stands for whatever possibility you prefer.





References
[1] Isaac Asimov "C for Celeritas" in "The Magazine of Fantasy and Science Fiction", Nov-59 (1959), reprinted in "Of Time, Space, and Other Things", Discus (1975), and "Asimov On Physics", Doubleday (1976)

[2] R. Kohlrausch and W.E. Weber, "Ueber die Elektricitätsmenge, welche bei galvanischen Strömen durch den Querschnitt der Kette fliesst", Annalen der Physik, 99, pg 10 (1856)

[3] P. Drude, "Zum Studium des elektrischen Resonators", Göttingen Nachrichten (1894), pgs 189--223

[4] H.A. Lorentz, "The theory of Electrons and its applications to the phenomena of light and radiant heat". A course of lectures delivered in Columbia University, New York, in March and April 1906, Leiden (1909)

[5] R.D. Carmichael, "The Theory of Relativity", John Wiley & Sons (1913)

[6] L. Silberstein, "The Theory of Relativity", Macmillan (1914)

[7] E. Cunningham, "The Principle of Relativity", Cambridge University Press (1914)

[8] R.C. Tolman, "The Theory of the Relativity of Motion", University of California Press (1917)

[9] A. Einstein, From "The Collected Papers, Vol 2, The Swiss Years: Writings, 1900--1909", English Translation, he wrote five papers using V, e.g. "On the Electrodynamics of Moving Bodies", Annalen Der Physik 17, pgs 891--921 (1905), "On the Inertia of Energy Required by the Relativity Principle", Annalen Der Physik 23, pgs 371--384 (1907)

[10] A. Einstein, e.g. "On the Theory of Light Production and Light Absorption", Annalen Der Physik, 20, pgs 199--206 (1906)

[11] A. Einstein, "On the Relativity Principle and the Conclusions Drawn From It", Jahrbuch der Radioaktivität und Elektronik 4, pgs 411--462 (1907)

[12] J. Clerk Maxwell, "A dynamical theory of the electromagnetic field", Philos. Trans. Roy. Soc. 155, pgs 459--512 (1865). Abstract: Proceedings of the Royal Society of London 13, pgs 531--536 (1864)

[13] J. Clerk Maxwell, "A Treatise on Electricity and Magnetism", Oxford Clarendon Press (1873)

[14] A.A. Michelson and E.W. Morley, "On the Relative Motion of the Earth and the Luminiferous Ether", Amer. J. Sci. 34, pgs 333--345 (1887), Philos. Mag. 24, pgs 449--463 (1887)

[15] O. Lodge, "Aberration Problems", Phil. Trans. Roy. Soc. 184, pgs 729--804 (1893)

[16] J. Larmor, "A Dynamical Theory of the Electric and Luminiferous Medium I", Phil. Trans. Roy. Soc. 185, pgs 719--822 (1894)

[17] H. Poincaré, "Cours de physique mathématique. Electricité et optique. La lumière et les théories électrodynamiques" (1900)

[18] P. Drude, "Physik des Äthers auf elektromagnetischer Grundlage", Verlag F. Enke, Stuttgart (1894)

[19] H. Lorentz, "Versuch einer Theorie der elektrischen und optischen Erscheinungen in bewegten Körpern", Leiden (1895)

[20] A. Einstein, from "The Collected Papers, Vol 5, The Swiss Years: Correspondence, 1902--1914", English Translation, Doc 58.

[21] P. Drude, "The theory of optics", translated from German by C.R. Mann and R.A. Millikan, New York, Longmans, Green, and Co. (1902)

[22] M. Planck, "Uber irreversible Strahlungsvorgange", Verl. d. Kgl. Akad. d. Wiss. (1899)

[23] H.A. Lorentz, "De l'Influence du Mouvement de la Terre sur les Phenomenes Lumineux", Arch. Neerl. 21, pg 103 (1887)

[24] H.A. Lorentz, "On the Reflection of Light by Moving Bodies", Versl. Kon. Akad. Wetensch Amsterdam I, 74 (1892)

[25] H.A. Lorentz, "Versuch einer Theorie der elektrischen und optischen Erscheinungen in bewegten Körpern", Leiden (1895)

[26] H. A. Lorentz, "Théorie simplifiée des phenomènes electriques et optiques dans des corps en mouvement", Proc. Roy. Acad. Amsterdam I 427 (1899)

[27] H.A. Lorentz, "Maxwells elektromagnetische Theorie" Encyclopädie der Mathematischen Wissenschaften. Leipzig, Teubner (1903)

[28] M. Abraham, "Prinzipien der Dynamik des Elektrons", Annalen der Physik 10, pgs 105--179 (1903)

[29] e.g. J.J. Thomson and G.F.C. Searle, "A Determination of `v', the Ratio of the Electromagnetic Unit of Electricity to the Electrostatic Unit", Proc. Roy. Soc. Lond. 181, pg 583 (1890), M. Hurmuzescu, "Nouvelle determination du rapport v entre les unites electrostatiques et electromagnetiques", Ann. de Chim. et de Phys., 7a serie T. X April 1897, pg 433. (1897)

[30] J. Clerk Maxwell, "A Treatise on Electricity and Magnetism", Oxford Clarendon Press, Vol II; Chapter 23, section 849 (1873)

[31] K.F. Gauss, "Zur mathematischen Theorie der elektrodynamischen Wirkung" (1835), in "Werke", Göttingen 1867; Vol. V, pg 602

[32] W. Weber, "Elektrodynamische Maassbestimmingen uber ein allgemeines Grundgesetz der elektrischen Wirkung", Abh. Leibnizens Ges., Leipzig (1846)

[33] G. Kirchhoff, "Ueber die Bewegung der Elektricität in Leitern" Ann. Phys. Chem. 102, 529--544 (1857)

[34] G.F.B. Riemann, "Ein Beitrag zur Elektrodynamik", Annalen der Physik und Chemie, pg 131 (1867)

[35] G. Kirchhoff, "Zur Theorie der Entladung einer Leydener Flasche", Pogg. Ann. 121 (1864)

[36] J. Hermann, "Phoronomia", Amsterdam, Wetsten, (1716)

[37] J. d'Alembert, "Recherches sur les cordes vibrantes", L’Académie Royal des Sciences (1747)

[38] L. Euler, "De La Propagation Du Son" Memoires de l'acadamie des sciences de Berlin [15] (1759), 1766, pgs 185--209, in "Opera physica miscellanea epistolae. Volumen primum", pg 432

[39] L. Euler, "Eclaircissemens Plus Detailles Sur La Generation et La Propagation Du Son Et Sur La Formation De L'Echo", "Memoires de l'acadamie des sciences de Berlin" [21] (1765), 1767, pgs 335--363 in "Opera physica miscellanea epistolae. Volumen primum", pg 540

[40] J.W. Strutt, "Theory of Sound" Vol 1, pg 251, McMillan and Co. (1877)

[41] H.A. Lorentz, "Stokes' Theory of Aberration in the Supposition of a Variable Density of the Aether", Proc. Roy. Acad. Amsterdam I, pg 443 (1899)

[42] A. Einstein, "The Collected Papers, Vol 5, The Swiss Years: Correspondence, 1902--1914", English Translation, Doc 86 (1907)

[43] W. Voigt, "Ueber das Doppler'sche Princip", Goett. Nachr. 2, pg 41 (1887)

[44] E. Cohn, "Zur Elektrodynamik bewegter Systeme. II", Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, der physikalisch-mathematischen Classe (1904)

[45] M. Brillouin, "Le mouvement de la Terre et la vitesse de la lumière", comptes rendu 140, pg 1674 (1905)


[46] G. Herglotz, "Zur Elektronentheorie", Nachrichten von der Gesellschaft 6, pg 357 (1903)

[47] F. Himstedt, "Ueber die Schwingungen eines Magneten unter dem dämpfenden Einfluß einer Kupferkugel", Nachrichten von der Gesellschaft 11, pg 308 (1875)

[48] H. Helmholtz, Berlin: Verl. d. Kgl. Akad. d. Wiss. (1892)

[49] H. Hertz, "Electric Waves", Macmillan (1893)


Fonte: Usenet Physics FAQ

Beautifully strange - The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius

The list of famous Bristolians is an illustrious one. The Victorian engineer Isambard Kingdom Brunel, for example, is recognized everywhere in Bristol for his many iconic structures, even though he was not born, bred or even resident in the city. Another well-known son of the city is the Hollywood legend Cary Grant, born as Archie Leach in the suburb of Horfield and now commemorated with a striking bronze statue outside Bristol’s hands-on science museum. The physicist Paul Dirac actually went to the same elementary school as Grant/ Leach, and the abstract sculpture dedicated to him stands just a stone’s throw away from Grant’s bronze likeness. Dirac also has a building named after him: Dirac House, the headquarters of IOP Publishing (which publishes Physics World).

Yet in spite of these efforts to publicize Dirac’s many contributions to science, his city of birth and (until recently) the school where he was educated seemed almost unaware that in Dirac, Bristol produced one of the great minds of the last century, and arguably the greatest British physicist since Isaac Newton. Part of this lack of knowledge among both Bristolians and the general public is Dirac’s legendary reticence, literal-mindedness and almost total inability to communicate with anyone — except, possibly, his immediate family.

All of this makes Dirac a very difficult subject for the sort of sympathetic biography that Graham Farmelo has produced in The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius. The book represents years of careful research and conversations with family and friends who knew Dirac and his work. In it, Farmelo, a science communicator and senior research fellow at the London Science Museum, describes the life and work of this profoundly brilliant man, exploring the origins of his near-pathological reticence and in the last chapter proposing a possible explanation. I doubt whether a better biography will appear in most of our lifetimes.

Dirac’s parents Charles and Florence were married in 1899 and lived for a time at 42 Cotham Road, probably in rented rooms, where Dirac’s older brother Felix was born. Shortly afterwards, Charles bought a small terraced house in Monk Road and Paul Adrien Maurice Dirac, the second son, was born in 1902. His sister Betty was born in 1906, so Flo certainly had her hands full with a young family and the ever-increasing and apparently irrational demands of her husband.

These demands included Charles’ insistence that only French be spoken at the family dining table. As a result, Flo, Felix and Betty ate in the kitchen, while Paul — whose French was just passable — was allowed to sit with his Swiss-born father. In later life, Dirac acknowledged that his difficulty in communicating with others may have stemmed from this period, poignantly explaining to Kurt Hofer — an Austrian- born cell biologist who became a close friend — that “since I found that I couldn’t express myself in French, it was better for me to stay silent than to talk in English”.

Time and again, Farmelo returns to the difficult personal relations that plagued Dirac’s family. Although in today’s parlance the Diracs were upwardly mobile — they soon moved to a larger semi-detached house in Julius Road, a more salubrious part of Bristol — Charles was also a serial tax evader. His crimes only came to light after his death, however, leaving Flo with an unwelcome tax bill. At one stage in the relationship she appears to have sought separation from her husband due to suggestions that he was having an extramarital affair, and their oldest child Felix committed suicide when Dirac was 23. But despite all of these traumas, Dirac is said to have wept only once in his life: in 1955, when he heard of the death of his hero, Einstein.

Given this background, it is hardly surprising that in his later life it was only with some unhappiness and after pleading from his mother that Dirac could be persuaded to visit Bristol. Instead, St John’s College, Cambridge, became the place he regarded as his true home. While there, Dirac made his most important breakthrough: he succeeded in welding together special relativity and quantum mechanics to produce what is often and rightly regarded as one of the great equations in physics. He became the Lucasian Professor of Mathematics there in 1932, and in 1933 his famous equation won him a Nobel prize (shared with Schrödinger) “for the discovery of new productive forms of atomic theory”.

Master of the equation: Paul Dirac.

Credit: Science Source/Science Photo Library

The conclusions of the Dirac equation were highly controversial when they were first described in 1928, but in a curious way, the criticisms appeared to simply bounce off Dirac — a consequence, perhaps, of his deeply private personality. The idea of negative energy states and the consequent hole theory was finally resolved by the discovery of the positron in 1932. The equation also showed that spin was a natural consequence of relativity and quantum mechanics, and not simply an add-on to explain atomic spectra. Recognizing this, it is only just and fair that the unique characteristics of electrons that make such devices as transistors, mobile phones and solid-state lasers possible are known as Fermi–Dirac statistics.

Farmelo takes the reader through difficult physics in a masterly manner — a consequence, no doubt, of his vast experience in science communication. The author also describes some aspects of Dirac’s work of which even professional physicists may not be aware. For example, in 1933 Dirac started an experimental study with Peter Kapitza on the possibility of bending a beam of electrons with light. He also developed an experiment to separate isotopes — much to the approval of Ernest Rutherford, who thought that it “augurs well for theoretical physics that the Lucasian Professor is soiling his hands in the laboratory”. As a result, Dirac became peripherally involved in the Manhattan Project, performing theoretical investigations of the “separation power” of uranium-enriching devices, although he declined a fulltime position.

Dirac’s life changed dramatically during a sabbatical at Princeton University in 1934 when he met Margit Wigner, a Hungarian divorcee and mother of two children, Gabriel and Judy. Margit, the sister of nuclear physicist Eugene Wigner, was known to friends and family as Manci. She was the opposite in nearly every sense to Dirac, but their affection turned to love and they were married in January 1937. Manci had to spend some time in Budapest after the honeymoon and as a result, Dirac penned “the first love letter I have ever written”. Until then, Dirac had replied to questions from Manci in tabular form!

The marriage did experience some strains (often arising from Manci’s dislike of Cambridge), but Dirac was a loving husband and stepfather to Manci’s children and to the two daughters of the marriage, Mary and Monica. Within the family, Dirac appears to have been far more communicative than he was with outsiders. At the opening of Dirac House in 1997, I remember Monica describing how his scientific approach to vegetable gardening caused much amusement in the family, which Dirac took in good humour.

One feels a sense of anticlimax as the book nears its end. Dirac fell out with the Cambridge hierarchy over what seems a rather trivial dispute about car parking, and by the mid- 1960s he spent most of the week working at home. Meanwhile, Manci had set her heart on escaping from Cambridge, and in 1971, having seen their children well settled (except for Dirac’s stepdaughter Judy, who had disappeared in 1968 and was by then presumed to be dead), the couple finally emigrated from the UK to Florida, where Dirac died in 1984.

Physicists remain divided over the legacy of Dirac’s later years. Was his opposition to the success of quantum electrodynamics justified on the grounds that the theory lacked beauty? Do monopoles really exist? Can his large-number hypothesis — which suggests that fundamental constants change with time — ever be reconciled with general relativity? But all physicists agree that the towering achievement of the Dirac equation will, as Farmelo makes clear, set Dirac apart and place him in a league with Newton and Einstein.

Perhaps the most controversial part of the book is its last chapter, in which Farmelo explores the possibility that Dirac’s pathological reticence was in fact undiagnosed autism or Asperger’s syndrome. Autism covers a wide spectrum of behaviour, and as the writer and doctor Milo Keynes points out in The Notes and Records of the Royal Society (2008 62 289), it has become something of a catch-all phrase for behaviour that departs significantly from the norm: “In the past 10 years it has been firmly claimed that Newton must have shown the development disorder of Asperger’s syndrome, a disorder that has been posthumously assigned to Michelangelo, Henry Cavendish, Albert Einstein, Marie Curie, Ludwig Wittgenstein and Paul Dirac.” Clearly, Dirac joins a long and distinguished list of retrospectively diagnosed luminaries.

For what it is worth, my guess is that Dirac was by nature a shy individual and that this shyness was reinforced by a difficult early home environment. Farmelo is correctly very cautious in what he has written, and regardless of the conclusions he draws about Dirac’s personality, it is clear that writing about him has been a labour of love. I most warmly recommend this book both to professional physicists and to laypersons interested in fundamental physics, as well as to anyone who finds the interaction between personality and intellectual endeavour fascinating.
About the author

FONTE: PhysicsWorld
Sir John Enderby is professor emeritus of physics at Bristol University and past president of the Institute of Physics

Big gains for physics as Obama signs stimulus bill

Science fared well in the $787bn package to stimulate the US economy that President Barack Obama signed into law today. The “recovery and reinvestment bill” includes $21.5bn for research and development (R&D), the bulk of which — some $18bn — will go directly to researchers. The remaining $3.5bn is allocated for facilities and equipment.

Politicians have been bickering over the bill since it was fist unveiled on 15 January. American legislation takes a circuitous course on its way to the President. Typically, the House of Representatives and the Senate approve different versions of a bill, and then appoint negotiators to agree on compromise legislation that both houses must approve again before sending to the President.

"These prudent investments lay the necessary foundation for long-term economic growth and prosperity for our country"
Cherry Murray, American Physical Society

The $838bn Senate bill on 10 February included significantly less funding for physical science than the $825bn House bill on 29 January. Even though the Senate bill may have higher priority, most of the cuts to the physical sciences were reversed in the final $787bn bill agreed on 14 February. Indeed, physicists have welcomed the $21.5bn for science, with more than $10 bn of it going to government agencies responsible for funding the physical sciences.


NSF is a winner

The National Science Foundation (NSF) will receive $3.0 bn in stimulus funding on top of its $6.0bn budget for 2009. This will include $1bn for research infrastructure and construction and $2 bn for “other research and related activities.”

The DOE’s Office of Science, meanwhile, will get $1.6 bn in funding beyond its 2009 budget of $4.0 bn. Two other DOE programmes: energy efficiency and renewables, and fossil energy will receive $2.5 bn and $1.0 bn respectively, which is almost twice as much as their 2009 budget allocations of $1.2 bn and $576 m.

With a budget of $737 m for this year, the National Institute of Standards and Technology will receive an extra $580 m, of which $360 m will go on building research facilities. NASA will receive an extra $1.1 bn beyond its current budget of $17.2 bn with $400 m going towards its science and exploration programme.

"The good news is that there is a lot of money for infrastructure. The big challenge is how to spend it"
Kei Koizumi, American Association for the Advancement of Science

“The surprise is how much money there is for science in the final bill,” says Kei Koizumi, budget analyst at the American Association for the Advancement of Science. “The good news is that there is a lot of money for infrastructure. The big challenge is how to spend it.”


“These critical investments will not only benefit American science and innovation, but they will put thousands of Americans back to work through construction and manufacturing projects,” American Physical Society president Cherry Murray said in a statement. “Furthermore, these prudent investments lay the necessary foundation for long-term economic growth and prosperity for our country.”

The fresh funding has implications for US science beyond the current financial year. It puts back on track the goal of doubling federal government support for physical science — an ambition of the America COMPETES Act of 2007 that had fallen behind schedule.

Further evidence of the Obama administration’s ambitions for science will become clear later this month, with the release of its revised budget for the 2009 financial year, which started on the 1st of October last year.

FONTE: Peter Gwynne is North America correspondent for Physics World

Esticando o tempo e voltando ao passado

Por Roberto Belisário

Boa notícia para os atrasados e os apressados: é possível “dilatar” o tempo, de forma a transformar um dia em dez dias, e também viajar ao passado e depois voltar ao presente para contar a história!

Não é ficção: trata-se de possibilidades teóricas previstas pela física moderna. A dilatação do tempo acontece corriqueiramente na física subatômica; viagens para o futuro acontecem de forma natural e automática em viagens aéreas, ainda que através de intervalos de tempo minúsculos. Viagens para o passado são ainda apenas previsões teóricas, mas há quem aposte que aparecerão espontaneamente casos microscópicos e raros de “máquinas do tempo” naturais dentro do LCH, um acelerador de partículas que entrará em operação em maio de 2008.

É possível ousar mais e imaginar se, no futuro, não será possível implementar esses fenômenos em uma tecnologia cotidiana que ajude a diminuir um efeito nocivo que o próprio avanço tecnológico, indiretamente, produz: a falta de tempo, a cultura da pressa, o acúmulo de tarefas. Na verdade, a tecnologia está o tempo todo sendo usada para esse fim: máquinas para fazer serviços domésticos, computadores, automóveis e aviões, todas essas inovações têm a função principal de permitir concentrar mais tarefas em menos tempo e com menos esforço. Mas o que se pergunta aqui é se é possível usar os efeitos descritos anteriormente para manipular diretamente o tempo de modo a, ao invés de diminuir a demora das atividades humanas, “esticar” o próprio tempo, ou mesmo “andar para trás no tempo”, de modo a termos mais horas disponíveis para nossos caprichos.

A resposta não é animadora: o uso prático de tais fenômenos na vida cotidiana requer quantidades tão grandes de energia e aglomerados de matéria tão densos – trilhões de vezes maior do que a das rochas mais duras – que é virtualmente impossível com a atual tecnologia à disposição. Mas nada impede que alguma idéia brilhante e nova possa eliminar algumas limitações, como tantas vezes já aconteceu. Algumas já foram imaginadas e aumentaram um pouco as possibilidades.

Como transformar um dia em duas semanas

A dilatação do tempo chega a ser banal: acontece naturalmente e de forma automática quando se compara o ritmo do “passar do tempo” para duas pessoas deslocando-se uma em relação à outra. Trata-se de um efeito previsto pela teoria da relatividade especial, em 1905, produzida por Albert Einstein, Henri Poincaré, Hendrik Lorentz e outros, e que substituiu a mecânica de Newton, inconteste pelos 300 anos anteriores. Até então, imaginava-se o tempo como uma entidade absoluta, cujo “fluir” seria o mesmo para todos os observadores. Algo bem diferente de outros conceitos que sabemos que dependem de um referencial, como a velocidade. A rigor, a especificação da velocidade só é completa quando se diz em relação a quê. Um carro pode estar a 100 km/h em relação à estrada; mas, em relação ao Sol, estará a 180 mil km/h, acompanhando o movimento de translação da Terra ao redor dele.

O que a teoria da relatividade mostrou sobre o tempo foi que ele é tão dependente de um referencial quanto a velocidade. Se eu olho para o relógio de alguém deslocando-se em relação à mim, vejo-o andar mais lentamente que o meu; os seus batimentos cardíacos parecem (e estão) mais vagarosos; suas palavras chegam até mim mais espaçadas e sua voz mais grave, vejo-a envelhecer mais devagar e assim por diante.

Neste ponto, para os atrasados e apressados, há uma boa notícia e uma má. A má é que seria necessário que dois observadores se afastassem a 42% da velocidade da luz para produzir um atraso de apenas 10% no correr do tempo entre um e outro. Portanto, é muito difícil e caro produzir uma dilatação do tempo útil, que transforme horas em mais horas, de forma que eu não precisasse mais escrever este texto para amanhã, mas só para daqui a duas semanas. Além disso, trata-se de um efeito que acontece entre observadores. Se eu permaneço parado em relação ao editor desta revista, meu tempo flui necessariamente da mesma forma que o dele. A não ser que alguém tenha alguma idéia brilhante...

Na verdade, alguém teve uma idéia brilhante: Einstein e o matemático francês Pierre Langevin. E esta é a “notícia boa” prometida acima. Trata-se de um efeito chamado “paradoxo dos gêmeos”; Einstein previu-o em 1905 e, em 1911, Langevin colocou-o nos termos dramáticos seguintes. Imaginemos dois irmãos gêmeos, Ulisses e Penélope, sendo que Ulisses realiza uma viagem espacial em alta velocidade e volta anos depois. Segundo a previsão da relatividade, durante a viagem, Penélope observaria daqui da Terra todos os fenômenos relacionados a Ulisses mais lentos que o normal, desde as batidas do seu coração até a velocidade do seu caminhar. E isso inclui o ritmo do seu envelhecimento. De forma que, quando Ulisses retornar à Terra, estará alguns dias mais novo que Penélope.

Esse efeito já foi demonstrado substituindo-se os gêmeos por relógios atômicos, que são relógios de altíssima precisão. Em 1971, os físicos J. C. Hafele e R. E. Keating sincronizaram dois desses relógios e embarcou-se um deles em um vôo comercial ao redor do mundo, enquanto o outro permaneceu no Observatório Naval dos Estados Unidos, em Washington. O experimento foi feito duas vezes, uma num vôo de oeste para leste e outra de leste para oeste. Na primeira, o relógio atrasou-se 59 bilionésimos de segundo (59 nanossegundos) e, na segunda, adiantou 273 nanossegundos. Os resultados foram compatíveis com as previsões das equações da relatividade.

Viagens ao futuro e ao passado

A idéia de Einstein e Langevin foi boa o suficiente para se poder observar a dilatação do tempo sem grande dificuldade com velocidades perfeitamente acessíveis e vencer completamente a limitação do referencial entre eu e meu editor. Mas ela não é boa o suficiente para resolver as urgências do dia-a-dia – a velocidade envolvida teria que ser colossal. Há, porém, projetos de naves espaciais que talvez possam alcançar tais velocidades acelerando constantemente durante um longo tempo – para isso, usam como fonte de energia os raios cósmicos, que existem em qualquer lugar do espaço.

Mas, se o problema é a velocidade, a teoria da relatividade geral, feita também por Einstein e por David Hilbert entre 1905 e 1916, sugere uma forma de dilatar o tempo com os dois observadores parados: com o auxílio de campos gravitacionais intensos. Pela teoria, um observador no espaço interestelar (praticamente sem gravidade) vê tudo o que acontece quando outro observador na superfície da Terra correr mais devagar, da mesma forma que no caso de observadores em movimento. O campo gravitacional tem um efeito sobre o tempo. Mas é necessário um campo muito intenso para produzir uma diferença sensível e, para produzi-lo, seria necessário dispor de um acúmulo de matéria muito grande. Em 1976, foi medida a dilatação do tempo gravitacional entre um relógio atômico na Terra e um outro em um foguete lançado a 10 mil quilômetros de altura pelo Observatório Astrofísico Smithsonian, em Cambridge, nos EUA. O resultado foi de 4,5 partes em 10 bilhões (um desvio de apenas 0,01% em relação à previsão da teoria). Para produzir uma diferença de 10%, um objeto do tamanho da Terra teria que ter 10 elevado à 26ª potência (“1” seguido de 26 zeros) vezes o peso do nosso planeta, o que significa uma densidade só superada por um buraco negro.

E com relação à viagem no tempo? A coisa interessante com a idéia brilhante de Einstein e Langevin é que o paradoxo dos gêmeos não é apenas uma dilatação no tempo: é um deslocamento através do tempo. Podemos dizer que Ulisses, em seu périplo, viajou em direção ao futuro. Na verdade, sempre que alguém se desloca pelo espaço, desloca-se também no tempo, em relação a observadores que permanecem parados. Isso significa que estamos viajando para o futuro o tempo todo. O efeito é evidentemente diminuto demais em situações cotidianas – mas existe.

Já a volta ao passado, que seria útil para as urgências da modernidade (se não cair em mãos erradas!), é de implementação muito mais difícil. Muitas vezes, tem-se a idéia popular de que uma viagem ao passado implicaria em uma velocidade superior à da luz. Não é verdade: pode-se fazer tais viagens sem ultrapassá-la. O que se precisa nesse caso, segundo a relatividade geral, é de um campo gravitacional de formato muito exótico – o que implica em uma porção de matéria com formato igualmente exótico e extremamente densa. Esse campo produziria uma espécie de “túnel” no espaço-tempo chamado “buraco de verme” ou “buraco de minhoca”. Ele permitiria deslocamentos em grandes distâncias e/ou através do tempo. Seriam necessárias também enormes quantidades de energia para impedir que esse túnel colapsasse e se fechasse quase instantaneamente. Foi esse efeito que inspirou a “velocidade warp” da série Jornada nas Estrelas e tantos outros saltos espaciais e temporais em filmes de ficção científica.

Se assim aconteceu, assim acontecerá

Além disso, a teoria aparentemente não prevê a possibilidade de mudar o passado. Não seria possível, por exemplo, um homem voltar algumas décadas e matar a própria mãe antes de ele próprio nascer, pois esse filho não só estaria alterando um evento que já aconteceu (seu nascimento), como também impossibilitando a própria alteração. Mas as equações não impedem que essa pessoa volte no tempo e ajude sua mãe a conhecer seu pai, de modo que o nascimento ocorra. Tem-se aqui uma típica situação circular no tempo (tecnicamente, chamada “curva tipo tempo fechada” ou CTC), mas que quase não apresenta contradições lógicas (“quase” porque em algumas situações parece ser possível produzir informação a partir do nada). A ficção científica também explorou casos semelhantes, como no filme O Exterminador do Futuro, de 1984, e na sua continuação, de 1991, dirigidos por James Cameron.

De qualquer forma, “buracos de minhoca” podem ser, teoricamente, produzidos em condições extremamente energéticas, como as que acontecem nas colisões subatômicas em aceleradores de partículas. Há cientistas que acreditam que eles poderão aparecer, ainda que raramente e microscópicos, no acelerador LHC, que está sendo construído no Centro Europeu de Pesquisas (CERN), perto Genebra, na fronteira entre França e Suíça. Tais buracos de minhoca, porém, seriam demasiadamente pequenos, da ordem de um “comprimento de Planck” (ou seja, de um centésimo de quintilionésimo do diâmetro de um próton). Além disso para transformar um deles em uma máquina do tempo boa para seres humanos, seria preciso colocar uma de suas pontas em um campo gravitacional extremamente intenso, como o de uma estrela de nêutrons – e aí volta o problema de se arrumar uma porção de matéria extremamente densa.

Nada impede, porém, que alguma outra idéia brilhante e nova possa contornar algumas dessas limitações e permitir a produção de algum aparelho capaz de distorcer o tempo de forma a transformar 50 minutos em 60 minutos. Mas não terá essa possibilidade o mesmo destino dos outros “sucessos” da tecnologia no aumento do tempo livre das pessoas? Afinal, mesmo com todas essas máquinas, continuamos com falta de tempo! As pessoas continuam correndo, executivos trabalham com laptops em viagens à noite, empresários bem-sucedidos ficam em atividade 14 horas por dia. Se tivéssemos um dia de 48 horas, provavelmente a jornada de trabalho pularia para 32 horas diárias... O que pode melhorar o problema de gerenciamento do tempo não são novas inovações tecnológicas, mas uma mudança de atitude para com o trabalho e a vida.

Ao leitor interessado, o livro A evolução da física, escrito pelo próprio Einstein e por Leopold Infeld, explica as teorias da relatividade especial e geral de forma bastante compreensível para não-físicos. Uma quantidade de experimentos possíveis sobre relatividade especial utilizando material caseiro aparece no site da Feira de Ciências, de Luiz Ferraz Netto. Um especial sobre o tempo, incluindo artigos sobre física – e um de Paul Davies sobre como construir uma máquina do tempo –, apareceu na Scientific American Brasil de outubro de 2002, que foi republicada neste mês de outubro de 2007. Uma abordagem sobre viagens no tempo mais extensa e acessível a não-fisicos está no livro Máquina do tempo – um olhar científico, do físico brasileiro Mário Novello.


Fonte: Roberto Belisário é doutor em física, professor de eletrônica digital e física nas Faculdades Integradas Pedro Leopoldo (MG).

Tuning up for teleportation

H-bars

A new technique for controlling the speed of “teleportation” in quantum systems has been created by physicists in the US and the UK. The researchers have demonstrated a way of “tuning” beams of light to distribute quantum information to specific points in space and time. Manipulating and storing data in this way is an important step towards developing new communication devices and eventually a quantum computer, say the researchers.

In quantum teleportation, the sender (Alice) instantaneously transfers the quantum state of a particle to a receiver (Bob). In 1997 physicists captured public attention by teleporting quantum states between “entangled” photons for the first time. Entanglement is a feature of quantum mechanics that allows particles with two distinct quantum states to share a much closer relationship than classical physics allows.

Over the intervening 12 years teleportation has been demonstrated over increasing distances and between larger particles.

Now, Alberto Marino and colleagues have addressed a different challenge of quantum computing – the need to control the flow of quantum information. In the experiment, two beams of light were “entangled” then slowed down in a controlled manner as they passed through a cloud of hot rubidium vapour (Nature:2009.10.1038).

“In classic computing, information needs to arrive at the processor just at the right time. In quantum computing, exactly the same is true,” says Marino, a quantum-information researcher at the University of Maryland.

Harnessing the random

Until now researchers have sought to develop quantum memory for long-term data storage. Unfortunately, these systems have been highly inefficient, losing at least 80 % of the data. By slowing the speed of quantum data flow, Marino and colleagues have created a short-term memory device that is, according to the researchers, significantly more reliable.

Firstly the team split a laser beam into two before firing the it at a cloud of hot rubidium gas. Rubidium atoms have just one loosely bound electron in the outer shell, leading to a gas that is highly nonlinear in the way it interacts with light. Within the gas the incoming laser beams become entangled in a process known as “four-wave mixing”.

Quantum information is then carried in the form of fluctuations in the phase and intensity of the beams. Initially, the information travels at the speed of light but is then slowed in a controlled way in the atomic vapour.

“This type of delay will be essential for the realization of quantum networks,” says Hans-Albert Bachor, a quantum-computing researcher at the Australian National University.

Applications?

Using this mechanism, detection of quantum information was delayed for up to 27 nanoseconds. “Our quantum ‘images’ are the equivalent of the data buses in digital computers,” says Marino. The reason this delay could not be even longer is that the longer data are stored, the more noise is introduced. “Our next challenge is to preserve the quantum correlations while maintaining their quality,” said Vincent Boyer, also at the University of Maryland.

In Bachor's opinion, it is too early to consider applications for this system, but this is "limited only by our imagination”.

“Short term applications might include quantum sensors; these could work with only short fractional delays,” said Boris Blinov, a quantum systems researcher at the University of Washington.

Fonte: James Dacey is a reporter for physicsworld.com

Electricity unplugged

In the near future, wireless electricity could replace the ubiquitous power cable. Aristeidis Karalis looks at a revolutionary new way of transmitting power without wires

The judge was driving back late one cold winter night. Entering the garage, the battery-charging indicator in his wirelessly powered electric car came on. “Home at last,” crossed his mind. He swiped his personal smartcard on the front-door detector to be let in. He heard a “charging” beep from his mobile phone. The blinking cursor on the half-finished e-mail on the laptop had been waiting all day on the side table. He picked the computer up and walked towards his desk. “Good evening, your honour. Your wirelessly heated robe,” said the butler-robot as it approached from the kitchen. Putting on the electric garment, he sat on the medical desk chair. His artificial heart was now beating faster.


Science fiction usually expresses society’s impeding desires and sense of anticipation for certain technological miracles to happen. A society without power cables is pretty much a given in most science fiction. Indeed, today we do live in the “wireless age”, in which the air that we breathe probably contains more information than oxygen. However, this is also an age where mobile phones, MP3 players, laptop computers and domestic robots exist alongside old-fashioned power wires and bulky batteries. Unlike information, electrical energy is still physically confined to these borderline anachronistic appliances. Overcoming these last obstacles would finally make this a truly wireless world. Science? Yes. Fiction? Not anymore.

It all started a few years ago when Marin Soljačić, a physicist at the Massachusetts Institute of Technology (MIT) in the US, was driving back home one cold winter night and he heard an unfriendly beep from his mobile phone. It was the annoying reminder that the battery was running out, once again. It then suddenly occurred to Soljačić how great it would be if the mobile phone could take care of its own charging. The next morning, he returned to his office at the MIT determined to find a solution to the problem.

An exhaustive literature search soon revealed that wireless transmission of power was not an original idea. Back in the 1890s Nikola Tesla, one of great pioneers of electromagnetism, was the first to envisage that electricity, then a newly found form of energy, should be delivered to every house, in every city, in every country on the planet. However, Tesla did not foresee that people would be willing to drag wires around the entire globe to use electricity. Instead, he dreamed of a way of transferring electrical energy wirelessly over long distances. This would be achieved using big, coupled electromagnetic resonators able to generate very large electric fields, which were meant to propagate most likely either via conduction through the ionosphere (presumably including gigantic sparks) or through the Earth (possibly via intermediate coupling to the Earth’s charge resonances, so-called Schumann resonances). The epitome of Tesla’s efforts to achieve his goal was Wardenclyffe Tower, a 57 m high structure in Long Island that was meant to deliver electricity to the entire planet. The construction was interrupted in about 1905, not because the method was considered impractical or dangerous, but because the funder, the famed financier and banker J P Morgan, was concerned that there would be no way to bill remote electricity users. Nowadays, more than a century after Tesla, electricity reaches nearly every home through a global electrical grid. Nevertheless, J P Morgan’s objections meant a premature end to the first attempt at wireless electricity.
No wires attached

Today, we know of a variety of methods to transmit power without wires. The simplest example is electromagnetic radiation, such as radio waves. Omni-directional radiative antennas are one of the most widely used technologies, which are utilized in the provision of wireless Internet services, mobile telecoms, and radio and TV broadcasting. These antennas typically operate in the high-MHz/low-GHz frequency regimes. Even though such antennas are highly robust and suitable for use with mobile receivers, since they can operate in all directions and do not require a line of sight to the receiver, they are highly inefficient. Only a tiny portion of the radiated power in the direction of the receiver is actually picked up, since the vast majority of the radiation is lost in all the other directions. The use of a highly directional antenna, such as a microwave-beam antenna, in principle solves this problem and achieves a high efficiency in power transmission even over long distances (i.e. kilometres). On the other hand, this type of antenna does require an uninterrupted line of sight, which in itself requires a complicated device-tracking and beam-steering mechanism. Also, high-power focused beams may constitute a safety hazard.

An alternative approach to antennas is the use of an inductive transformer, a device commonly used in power circuits and electromechanical motors (for example electrical toothbrushes and chargers). A transformer typically operates up to mid-kHz frequencies. It essentially transfers electrical energy from one circuit to another via induction: the time-varying magnetic flux produced by a primary coil crosses a secondary coil and induces in it a voltage. The primary and the secondary coils are not physically connected, hence the method is wireless. Transformers can be very efficient but the distance between the coils must be very small (typically a few millimetres). For distances a few times the size of the coils, the efficiency drops significantly.

Part of the underlying physics for most of the existing methods for the wireless transfer of electricity is the fundamental principle of resonance: the property of certain physical systems to oscillate with maximum amplitudes at certain frequencies. It follows that, for any type of excitation (mechanical, acoustic, electromagnetic, nuclear) with a given frequency, a receiver will pick up the transmitted energy efficiently only when designed to resonate at the excitation frequency. Only then do successive excitations after each oscillation period add coherently in phase and lead to a build up of energy within the receiver.

To illustrate, consider 100 glasses filled with wine at different levels so that they support acoustic resonances at different frequencies. Now let an electric-guitar player produce and sustain a very well-defined note. Only one of the glasses, the one resonant with the frequency of this note, will respond to the excitation, to the extent that it may even break, while the rest will remain unaffected. Similarly, we tune the electromagnetic antenna of a radio to be resonant with the frequency of the station we want to listen to. Many transformers used in power circuitry and elsewhere are also designed to employ resonance to enhance the power transmission.
Cutting the cord at MIT

Since these days electricity is delivered to pretty much every single house in the world, it is not necessary anymore to transmit electricity over large distances à la Wardenclyffe Tower. Transmitting electricity within a room, namely over distances a few times greater than the size of the receiving devices themselves (what engineers define as mid-range distances), is sufficient for most modern applications. Achieving this goal with satisfactory efficiency, safety and low cost remains an unsolved problem. That was the challenge for Soljačić and his collaborators at the MIT labs: John Joannopoulos, Peter Fisher, Andre Kurs, Robert Moffatt and me.

Power up

The “quality factor”, Q, of a coil (see main text for details) that can be used for wireless power transmission at frequencies of about 10⁷ Hz when Q is at its peak, which is when the combined losses due to resistive absorption (green) and radiation (blue) are slowest.

Revisiting the fundamental principle of resonance, we posed the question of which physical conditions maximize the efficiency of energy transfer between two resonant objects. The energy of any resonator naturally decays due to intrinsic energy-loss mechanisms (friction for mechanical resonances, radiation and resistive absorption for electromagnetic resonances, collisions with phonons and spontaneous emission for atomic resonances). Losses are typically quantified by the number of oscillation periods that it takes for the energy to decay by a factor of 2.72. This number, represented by the “quality factor” Q, is an intrinsic property of resonators and depends on the strength of the loss mechanisms. (As a simple analogue, water inside a bucket with a hole will leak out at a rate that depends on the size of the hole.)

If two equal resonators exchange energy, it also takes a characteristic number of oscillation periods to transfer the energy from resonator A to resonator B, which is proportional to a constant that quantifies the strength of the coupling between the resonators, Qk. (If water is pumped from one bucket to another via a hose, then the transfer time depends on the strength of the pump.) Clearly, for energy transfer to be efficient, Q needs to be much larger than Qk, i.e. the rate at which energy is being transferred needs to far exceed the rate at which energy is being lost. (Water will be efficiently transferred between two leaking buckets if the pump is faster then the leaks from the holes.) The efficiency of the system can then be characterized by Q/Qk. The transfer of energy is efficient only when this ratio is larger than one, the so-called strong-coupling regime.

For our wireless method, we used one of the most basic electric circuits as a resonator: the LC circuit. This circuit is an electromagnetic resonant circuit that consists of an inductor (L), made by a wire coil, and a capacitor (C). Two such wire coils transfer energy via induction, like a transformer device, and the Qk clearly depends on the distance between the coils. For mid-range distances and long enough wavelengths, the spatial-decay rate of the magnetic field means that Qk is roughly proportional to the cube of the ratio of the distance between the coils, D, and the size of each coil, d, while showing little dependency on the frequency and the geometry of the coils. This means that, for mid-range distances, Qk will be large and the coupling very weak.

As a result, the best way to maximize the efficiency is to engineer the resonators to have the highest possible value of Q (try to seal the holes in the buckets). The resonance frequency of each coil (which has to be the same for both coils) can be tuned by varying the capacitance (and tuning a circuit element is exactly what the knob is tuning in a radio antenna). Q varies with the tuneable frequency, and this variation is shown in the figure above for a coil with a diameter of 60 cm made of copper pipe with a radius of 2 cm. It can be seen that, for high-MHz frequencies, the resonator loses energy fast (low Q, often even less than 10) due to radiation. This is exactly how an antenna is designed to work. Similarly, for mid-kHz frequencies, it loses energy fast (Q less than 100) via resistive absorption, which is typical of transformers. This explains why both omni-directional antennas and transformers fail to be efficient power transmitters at mid-range distances: the transfer-time measure Qk is large because of D, and Q is small. On the other hand, in the intermediate, low-MHz regime, much longer loss-times are observed, with Q often larger than 1000. That was our chosen regime.

Based on our theory, we started experiments in late 2006. The main challenges consisted of designing a driving circuit that would operate in our desired low-MHz regime and constructing coils that would resonate with a high enough value of Q. After a trial-and-error phase, we realized that a simple coil design without a separate capacitor, but using the coil’s self-capacitance to achieve resonance, was the best option in terms of Q.

We made two copper-pipe coils with 60 cm diameters and with five turns, such that they resonate at 10 MHz and have Q = 1000. A 60 W light bulb was our chosen device, since it operates at the tested frequencies (and what can be a clearer sign of the functionality of a system than the switch on of a light bulb?). We suspended the coils from the ceiling with fishing wire, at a distance of 2 m from each other, tuned them up, turned them on and…there was light. At an efficiency of 45%, this was, to our knowledge, the first-ever demonstration of midrange efficient wireless energy transfer.
On the safe side

The selective property of resonance means that almost all of the source power will be transmitted to the destined device and not to anywhere else. This is because any random object, including a biological organism, is almost always a non-resonant structure. Even if an object happens to be resonant, say a mobile-phone antenna, its resonance will be very different from the precise source-resonator frequency (just like those 99 wine glasses). Furthermore, even in the extremely unlikely case of it having the same resonance frequency, its Q value would be so low that no significant amount of power would be transmitted to it.

Wireless innovators

Martin Soljačić (left), the current author (middle) and John Joannopoulos from the Massachusetts Institute of Technology, along with a lab demonstration of their technology — used here to light a 60 W bulb. (Credit: Donna Coveney/MIT)

In our long-wavelength regime of operation (30 m wavelength at 10 MHz compared with 60 cm coils), power is transmitted from one object to another by spreading away from the source resonator and then “focusing” back into the device resonator. In contrast to higher frequencies, where power would be radiated across as a focused beam with a much smaller cross-sectional area, the former mechanism implies that, in our system, the power density locally and thus the fields will be considerably smaller at all points, except perhaps those too close to the coils. Smaller fields obviously imply safer performance.

Furthermore, our wireless-electricity method uses magnetic, rather than electric, fields to transfer energy. From the point of view of magnetic fields, most poor conductors, like wood, bricks, plastics and people, look a lot like air. On the other hand, electric fields do pose health hazards, because they can interact with biological organisms. With our method, these electric fields are confined to the capacitor inside our resonator. This method is quite similar to induction hobs on cookers, whereby a hob may transmit kilowatts of power to a metallic pot via induction, but it is safe to touch with our non-conducting hands. Note also that even the “large” magnetic fields in our system actually have tiny strength, approximately 10–4 T near the coils for 60 W of transmitted power, about the order of the time-invariant magnetic field of the Earth. It is the high-Q resonance that magically converts this tiny field into considerable usable power.
Wireless mobility

Long-wavelength fields naturally wrap and redistribute themselves around random objects in their vicinity or those standing between the source and mobile receiver. Therefore, while a radiated beam would immediately be interrupted by obstacles, our method stays robust and does not require an uninterrupted line of sight to the source. Sources can be hidden under floors, behind walls or inside furniture, and the receiving devices do not find shade while roaming freely behind random objects or when integrated inside other systems.

The near field produced by a resonant source coil spreads out quite uniformly in all directions, in contrast to a directed radiation beam. Thus, appropriate placement of one or more device coils can guarantee omni-directional coverage with low system complexity and thus cost.

The response of the system to dynamic variations of its parameters due to variable interaction with its environment during motion can be as fast as within 0.1 ms, based on the available frequency bandwidth of the sharp MHz resonances. This is good enough for the changes associated with daily motion.

Ray Bradbury, the prolific science-fiction writer, once said that “Anything you dream is fiction, and anything you accomplish is science.” If our innovation is successfully commercialized, then the concept of a completely wireless world could soon leap from dream to widespread accomplishment. We will forget charging our mobile phones, laptops and other personal digital devices. The maze of cables behind every home or office apparatus will disappear. Cars will drive on electricity for much longer and more cheaply. Robots will completely forget about returning to their charging stations. Micro-robots will forever hide inside electronic chips. Battery-powered sensors buried underground will never die. And the story of the judge will soon belong to history.

“Dad, I found a lamp in the basement, but it doesn’t work, see?” said the 10 year old, while ascending the stairs. “It does my son,” replied the judge, “but it connects to a wall plug and our new house does not have any of those.”
About the author

Aristeidis Karalis is at the Massachusetts Institute of Technology in the US

Fonte: PhysicsWorld

The greatest equations ever

Maxwell's equations of electromagnetism and the Euler equation top a poll to find the greatest equations of all time. Robert P Crease discusses the results of his reader survey

Earlier this year I asked readers to send me their shortlists of great equations. I also asked them to explain why their nominations belonged on the list and why, if at all, the topic matters (Physics World May p19).

James Clerk Maxwell

I received about 120 responses -- including single candidates as well as lists -- proposing about 50 different equations. They ranged from obvious classics to "overlooked" candidates, personal favourites and equations invented by the respondents themselves.

Several people inquired about the difference between formulae, theorems and equations -- and which I meant. Generally, I think of a formula as something that obeys the rules of a syntax. In this sense, E = mc² is a formula, but so is E = mc³. A theorem, in contrast, is a conclusion derived from more basic principles -- Pythagoras's theorem being a good example. An equation proper is generally a formula that states observed facts and is thus empirically true. The equation that describes the Balmer series of lines in the visible spectrum is a good example, as are chemical equations that embody observations about reactions seen in a laboratory.

However, these distinctions are not really so neat. Many classic physics equations -- including E = mc² and Schrödinger's equation -- were not conclusions drawn from statements about observations. Rather, they were conclusions based on reasoning from other equations and information; they are therefore more like theorems. And theorems can be equation-like for their strong empirical content and value.

It thus makes sense to classify both kinds as equations, which is exactly what respondent David Walton from the University of Manchester did. He distinguished between equations (such as F = ma) that comprise axiomatic models that "define the interrelationships between various observables for all circumstances" and equations that are approximate models (such as Hooke's law), which define "the interrelationships between the various observables over a defined range and within a defined accuracy". I therefore interpreted the term "equation" loosely.

Simplicity

Respondents had many different criteria for greatness in equations. Half a dozen people were so impressed with simplicity that they proposed 1 + 1 = 2.

"I know that other equations have done more, express greater power [and have a] broader understanding of the universe," wrote Richard Harrison from Calgary in Canada, "but there's something to be said for the beauty of the simplest things of their kind." He then recalled how 1 + 1 = 2 was the first equation he taught his son. "I remember [him] holding up the index finger of each hand as he learned the expression, and the moment of wonder when he saw that the two fingers, separated by his whole body, could be joined in a single concept in his mind."

Neil Blackie also voted for 1 + 1 = 2. "For this equation to come into being there had to be the invention of a method for representing a physical reality, quantities had to be given names and symbols," he argued. "There had to be a system to show how these quantities could be grouped together or taken apart. The writing down of this equation gave us the ability to present ideas, to discuss concepts, which led to an ever-expanding sphere of knowledge."

Other simple equations that were proposed included v = H₀d, which Edwin Hubble composed in 1929 to describe the fact that the galaxies are moving away from us at a speed, v, that is proportional to their distance, d, where H₀ is the Hubble constant. Balagoj Petrusev, an undergraduate student at the Institute of Physics in Skopje, Macedonia, suggested the Hamiltonian variational principle in the form δS = 0. A proper selection of the form of SPhysics World last month (September p64). articulates "a universal principle that stands true in classical mechanics, classical electrodynamics, relativistic mechanics, non-relativistic quantum mechanics and so on". In fact, Andy Hone from the University of Kent wrote a eulogy to this equation in

The unifying power of a great equation is not as simple a criterion as it sounds. A great equation does more than set out a fundamental property of the universe, delivering information like a signpost, but works hard to wrest something from nature. As Michael Berry from Bristol University once said of the Dirac equation for the electron: "Any great physical theory gives back more than is put into it, in the sense that as well as solving the problem that inspired its construction, it explains more and predicts new things" (Physics World February 1998 p38).

Great equations change the way we perceive the world. They reorchestrate the world -- transforming and reintegrating our perception by redefining what belongs together with what. Light and waves. Energy and mass. Probability and position. And they do so in a way that often seems unexpected and even strange.

For this reason, several respondents proposed equations that linked two or more disparate concepts, concrete and abstract things, the visible and the invisible. They included Boltzmann's equation S = k InW.

It relates entropy, S, which emerged as a concept during the development of thermodynamics early in the 19th century, and a purely abstract quantity, W, that emerged from the statistical treatment of systems with many degrees of freedom. Bragg's equation (nλ = 2dsinϑ)wrote another respondent, "links diffraction spots (visible reality) with the underlying crystal structure (invisible reality) and can be easily visualized with a standard textbook picture."

One of the most frequently mentioned equations was Euler's equation, e^iπ + 1 = 0. Respondents called it "the most profound mathematical statement ever written"; "uncanny and sublime"; "filled with cosmic beauty"; and "mind-blowing". Another asked: "What could be more mystical than an imaginary number interacting with real numbers to produce nothing?" The equation contains nine basic concepts of mathematics -- once and only once -- in a single expression. These are: e (the base of natural logarithms); the exponent operation;π; plus (or minus, depending on how you write it); multiplication; imaginary numbers; equals; one; and zero.

Practicality

Many respondents were impressed by equations that have a practical influence on human life. These included: the compound-interest equation, the implications of which from the Renaissance to the present are "obvious, staggering and unwelcome"; income-tax formulae; the simple ratio a/b = c/d, which is basic to construction, surveying and so forth; simple electrical equations, such as V = IR; basic mechanical equations, such as work done = force x distance; Shannon's capacity equation, which relates to the modern world through the Internet and digital communication; and, last but not least, Pythagoras's theorem.

Roger Bailey nominated the "sunrise equation" cos(time) = -tan(lat) x tan(dec), which identifies the time of sunrise or sunset as a function of latitude and solar declination. This, he pointed out, is "fundamental to our sense of time" and it "fits on a T-shirt". Engineer John Wilcher suggested the ideal-gas law, PV = nRT, pointing out that "the relation of pressure, volume and temperature is relevant to almost everything we do", including common but often overlooked uses such as car tyres, angioplasty procedures and oil drilling.

Meanwhile, Iain Christison, an emeritus professor of animal agriculture at the University of Saskatchewan, Canada, suggested E_M = H + P -- metabolized energy equals heat plus product. It describes the fact that "all of the useful energy consumed by animals, including people, is released as heat or stored as product". The equation, he added, "carries within it an intricate balance of cause and effect that influences all of us with every mouthful and with every step".

Historical relevance

Some respondents proposed equations that played key roles in the history of science. For example, Alan Denham proposed the Balmer series 1/λ = R (1/n₁² + 1/n₂²). The long history of this equation stretches from Fraunhofer's studies of the spectrum of sunlight in 1814, to Kirchhoff's suggestion in 1859 that each atomic species has a unique spectrum, to Angstrom's publication of the wavelengths of a thousand Fraunhofer lines in 1868, to schoolteacher Johann Balmer, who in 1885 noticed that the frequencies of light emitted by hydrogen atoms were mathematically related. The equation's history was continued by Lyman's observations in the ultraviolet region and by others in the infrared, by Rydberg (who gave his name to the constant R in the equation), and by Bohr, whose work in 1913 explained the equation. "As soon as I saw Balmer's formula the whole thing was immediately clear to me," Bohr once said.

These equations are listed in order of the number of people who proposed them. The first two received about 20 mentions each out of a total of about 120; the rest received between two and 10 each. Equations are given, where appropriate, in their most common form.

"Thus this century-long story," Denham wrote, "involving the theoretical and practical investigation of science by some of its most distinguished practitioners, would be incomplete without giving due honour to the contribution of a secondary-schoolmaster who spotted that the published scientific data conformed to a pattern that none of the scientists of his day were aware of."

Maxwell's equations

The responses suggested that there is no single criterion for greatness, and that a truly great equation ranks high in each of the above criteria. However, most votes were given to Euler's equation and to Maxwell's equations, which describe how an electromagnetic field varies in space and time. Although Maxwell's equations are relatively simple, they daringly reorganize our perception of nature, unifying electricity and magnetism and linking geometry, topology and physics. They are essential to understanding the surrounding world. And as the first field equations, they not only showed scientists a new way of approaching physics but also took them on the first step towards a unification of the fundamental forces of nature. A firm called Ocean Optics in Florida even sells T-shirts with Maxwell's equations on.

Tony Watkins recalled how he learned the equations during his second year as an undergraduate at Southampton University almost 20 years ago. "I still vividly remember the day I was introduced to Maxwell's equations in vector notation," he wrote. "That these four equations should describe so much was extraordinary...For the first time I understood what people meant when they talked about elegance and beauty in mathematics or physics. It was spine-tingling and a turning point in my undergraduate career. After a year of rapidly dwindling interest in physics (and rapidly decreasing results!), my passion was reignited by four lines of symbols." He even renamed his next bicycle Maxwell in honour of the great man, having previously ridden on his Carnot Cycle. Sadly for him, he never got round to learning tensors to see Maxwell's equations expressed even more simply.

The critical point

Nobody accepted my invitation to discuss why greatness in equations matters, which leaves me free to address the topic myself. Debating the issue has drawbacks, for it can foster the idea that equations are independent tools rather than embedded in networks of other equations, practices and information. Nevertheless, it helps us to recollect, among other things, what Richard Harrison called that "moment of wonder" that was apparent in his son's contemplation of 1 + 1 = 2.

As adults, we lose that wonder. We come to think of equations as just another set of tools that lie about ready-at-hand in the world. We lose our appreciation for their origin, thinking that they are not really of human origin: on the eighth day, God created equations as the blueprint for His recent work. As Galileo wrote -- disingenuously, polemically -- the Book of Nature is written in mathematical symbols. That's untrue, of course. We write, and continually rewrite, the book of nature.

As the philosopher Immanuel Kant once wrote: "When we discover that two or more heterogeneous empirical laws of nature can be unified under one principle that comprises them both, the discovery does give rise to a noticeable pleasure...even an admiration that does not cease when we have become fairly familiar with its object". This delight is more than having our expectations fulfilled or surprised, more than about the domination and control of nature, more than a biological product. The pleasure, Kant continued, is a feature of the exercise of the human intellect. "Even the commonest experience would be impossible without it," he wrote, which is why we "gradually come to mix it with mere cognition and no longer take any special notice of it."

In reawakening that sense of wonder, debating what makes equations great therefore re-educates us about the fundamental nature of science, and knowledge, itself.

About the author

Robert P Crease is in the Department of Philosophy, State University of New York at Stony Brook, and historian at the Brookhaven National Laboratory.

Fonte: PhysicsWorld