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Da Força Ao Tensor: Evolução Do Conceito Físico E Da Representação Matemática Do Campo Eletromagnético

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Da Força Ao Tensor: Evolução Do Conceito Físico E Da Representação Matemática Do Campo Eletromagnético DA FORÇA AO TENSOR: EVOLUÇÃO DO CONCEITO FÍSICO E DA REPRESENTAÇÃO MATEMÁTICA DO CAMPO ELETROMAGNÉTICO Cibelle Celestino Silva Orientador: Prof. Dr. Roberto de Andrade Martins Tese apresentada ao Instituto de Física Gleb Wataghin - UNICAMP para obtenção do grau de Doutor em Ciências. Campinas, 11 de outubro de 2002. Para Francisca e Francisco, minhas raízes e frutos. iii AGRADECIMENTOS Ao Professor Roberto Martins pela confiança, dedicação, paciência, estímulo e por tudo que tenho aprendido nesses anos de convivência. Ao Luciano pelo seu amor e companheirismo. À minha família por todo o amor e alegria. A todos os amigos e colegas que de alguma maneira ou outra estiveram ao meu lado nesta caminhada. E finalmente agradeço à FAPESP pelo apoio financeiro. iv Resumo: O objetivo deste trabalho é estudar o desenvolvimento paralelo dos conceitos físicos e do formalismo matemático usado para descrever o conhecimento eletromagnético no século XIX e início do século XX. Este estudo considera questões como os modelos de éter usados para descrever o campo eletromagnético, a simetria associada com os campos, as dimensões das grandezas físicas. Estudamos também a escolha entre o cálculo de quatérnions e vetores e o desenvolvimento do formalismo quadridimensional como a melhor maneira de descrever os fenômenos eletromagnéticos. Abstract: The aim of this work is to study the parallel development of the physical concepts and the mathematical formalism used to describe the electromagnetic knowledge in nineteenth and early twentieth century. This study takes into account issues such as the ether models used to describe the electromagnetic field, the symmetries associated with the fields, the dimensions of physical quantities. We also study the choice between quaternions and vectors and the development of the four-dimensional formalism as the best way to describe the electromagnetic phenomena. v ÍNDICE: 1. 1. INTRODUÇÃO ........................................................................................................ 1 2. INTERPRETAÇÃO MECÂNICA DAS GRANDEZAS ELETROMAGNÉTICAS 12 2.1 INTRODUÇÃO......................................................................................................... 12 2.2 MICHAEL FARADAY .............................................................................................. 15 2.3 A ANALOGIA ENTRE ELETRICIDADE E O FLUXO DE CALOR DE WILLIAM THOMSON 21 2.4 A ANALOGIA COM UM MEIO ELÁSTICO .................................................................. 25 2.5 JAMES CLERK MAXWELL ...................................................................................... 28 2.6 MODELOS MECÂNICOS DE ÉTER............................................................................. 43 2.7 A FORMULAÇÃO LAGRANGEANA DO ELETROMAGNETISMO ................................... 48 2.8 CONCLUSÃO .......................................................................................................... 52 3. O DESENVOLVIMENTO DO FORMALISMO VETORIAL A PARTIR DO CÁLCULO DE QUATÉRNIONS..................................................................................... 55 3.1 INTRODUÇÃO......................................................................................................... 55 3.2 CRONOLOGIA E PRINCIPAIS TRABALHOS................................................................ 57 3.3 POR QUE OS QUATÉRNIONS SÃO COMPOSTOS POR QUATRO NÚMEROS.................... 58 3.4 INTERPRETAÇÃO GEOMÉTRICA DE UM QUATÉRNION.............................................. 63 3.5 VERSORES E VETORES UNITÁRIOS.......................................................................... 65 3.6 QUATÉRNION COMO QUOCIENTE DE DOIS VETORES ............................................... 67 3.7 O OPERADOR NABLA ............................................................................................. 69 3.8 TAIT E OS QUATÉRNIONS ....................................................................................... 70 3.9 MAXWELL E OS QUATÉRNIONS .............................................................................. 75 3.10 O SISTEMA DE GRASSMANN .................................................................................. 81 3.11 O SISTEMA DE CLIFFORD....................................................................................... 85 3.12 O SISTEMA VETORIAL DE GIBBS ............................................................................ 86 3.13 HEAVISIDE E A ANÁLISE VETORIAL........................................................................ 95 3.14 A CONTROVÉRSIA................................................................................................ 106 3.15 CONCLUSÃO ........................................................................................................ 114 4. A SIMETRIA DAS GRANDEZAS ELETROMAGNÉTICAS ........................... 119 4.1 INTRODUÇÃO....................................................................................................... 119 4.2 OS VÁRIOS TIPOS DE GRANDEZAS FÍSICAS DEFINIDAS POR MAXWELL ................. 120 4.3 ARGUMENTOS INTUITIVOS DE SIMETRIA.............................................................. 125 4.4 TIPOS DE SIMETRIA.............................................................................................. 129 4.5 VETORES POLARES E AXIAIS ................................................................................ 131 4.6 O PRINCÍPIO DE SIMETRIA DE PIERRE CURIE........................................................ 133 4.7 A EXPERIÊNCIA DE ØRSTED E O PRINCÍPIO DE CURIE........................................... 137 4.8 AS CONTRIBUIÇÕES DE LANGEVIN ...................................................................... 146 vi 4.8.1 Escalares e pseudo-escalares .....................................................................147 4.8.2 Vetores polares, vetores axiais e tensores ........................................................148 4.9 UM PARADOXO INTERESSANTE ............................................................................151 4.10 CONCLUSÃO.........................................................................................................155 5. DIMENSÃO DAS GRANDEZAS ELETROMAGNÉTICAS ..............................158 5.1 INTRODUÇÃO .......................................................................................................158 5.2 ANÁLISE DIMENSIONAL........................................................................................160 5.3 MAXWELL E AS DIMENSÕES DAS GRANDEZAS ELETROMAGNÉTICAS ....................162 5.4 ANÁLISE DIMENSIONAL E AS DIFERENTES TEORIAS ELETROMAGNÉTICAS ............166 5.4.1 Arthur Rücker .............................................................................................169 5.5 O ENFOQUE GEOMÉTRICO DA ANÁLISE DIMENSIONAL..........................................171 5.6 A ANÁLISE DIMENSIONAL E A NATUREZA DOS FENÔMENOS ELETROMAGNÉTICOS173 5.7 CONCLUSÃO.........................................................................................................182 6. O FORMALISMO QUADRIDIMENSIONAL......................................................185 6.1 INTRODUÇÃO .......................................................................................................185 6.2 LORENTZ E A ELETRODINÂMICA DOS CORPOS EM MOVIMENTO ............................188 6.3 AS CONTRIBUIÇÕES DE HENRI POINCARÉ.............................................................190 6.4 O FORMALISMO QUADRIDIMENSIONAL DE MINKOWSKI .......................................192 6.4.1 As transformações de Lorentz escritas no formalismo de Minkowski........193 6.4.2 As equações da eletrodinâmica escritas por Minkowski............................197 6.4.3 Vetores do tipo I e vetores do tipo II ..........................................................203 6.5 A ÊNFASE NOS ASPECTOS MATEMÁTICOS DE MINKOWSKI....................................206 6.6 A ACEITAÇÃO DO TRABALHO DE MINKOWSKI......................................................209 6.7 EINSTEIN E O FORMALISMO QUADRIDIMENSIONAL...............................................210 6.8 MINKOWSKI E POINCARÉ .....................................................................................211 6.9 A RELATIVIDADE ESPECIAL NA FORMA DE QUATÉRNIONS....................................213 6.10 OS ARTIGOS DE SILBERSTEIN ...............................................................................214 6.10.1 As equações eletromagnéticas no vácuo na forma de quatérnions............218 6.11 CONCLUSÃO.........................................................................................................221 7. CONCLUSÃO...........................................................................................................226 8. BIBLIOGRAFIA ......................................................................................................239 vii 1. 1. INTRODUÇÃO O objetivo deste trabalho é estudar o desenvolvimento paralelo dos conceitos físicos e do formalismo matemático usado para descrever os conhecimentos eletromagnéticos, do início do século XIX e começo do século XX (antes do desenvolvimento da eletrodinâmica quântica). No contexto da teoria eletromagnética relativística o campo eletromagnético é representado por um tensor quadridimensional antissimétrico Fαβ de segunda ordem (α,β = 0, 1, 2, 3), cujas componentes estão associadas às componentes do campo elétrico E e da indução magnética B: 1 0 − E − E − E x y z − Ex 0 Bz By Fµν = − Ey Bz 0 Bx − Ez By Bx 0 Essa notação condensa um grande número de propriedades dos campos elétrico e magnético,
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