Two theorems by Helmholtz

Fleming, H.

doi:10.1590/s1806-11172001000200003

Cited by 2 publications

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“…For more mathematical details and insights see [3][4][5][6][7]. This result is true for an extended localized source.…”

Section: -2mentioning

confidence: 89%

Gauss's law, infinite homogenous charge distributions and Helmholtz theorem

Tort

2011

Rev. Bras. Ensino Fís.

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Rediscutimos a validade da lei de Gauss no caso de distribuições discretas e contínuas de carga que se estendem por todo o espaço. Palavras-chave: eletromagnetismo clássico, eletrostática, lei de Gauss.We rediscuss the validity of Gauss's law in the case of homogenous discrete and continuous charge distributions fulfilling all space. Keywords: classical electromagnetism, electrostatics, Gauss's law.In physics, some questions that we can formulate though frequently not realizable from a practical point of view, must as a matter of principle be satisfactorily answered. Moreover, they can be pedagogical useful tools and help the teacher to introduce new topics or discuss particularly hard ones. The following interesting question belongs to this type of question and can be found in [1]: Given a uniform charge distribution ρ that fulfills all the space, what is the value of the electric field at an arbitrary point of the distribution? The answer given in [1] though correct is nevertheless intriguing. By symmetry the electric field must be zero everywhere which makes perfect sense, but this means that the electric flux through an arbitrary surface S enclosing a certain charge Q (S) is zero in conflict with the integral form of the Gauss's law that states that the flux must be equal to Q (S) / 0 . In order to keep the symmetrical solution for the field it is stated that Gauss's law is not valid in this situation because there are charges at the infinity. It is pedagogically worth to understand and rediscuss this problem and its answer and this is our aim in what follows.Let us begin by taking an alternative route to Gauss's law. Consider a point charge q placed at a distance d from a point C and a hypothetical Gaussian sphere of radius R centered at C, see Fig. 1. Consider also a point P on the surface of the Gaussian sphere. If the distance of the point charge to P is r, then the electric field at P is given by It is convenient to treat d, R, and r as vectors, then r = d + R and we can writêwhere θ plays the role of a polar angle. The electric flux through an infinitesimal area centered around P is given by d Φ = E·n da, where the outward normal vectorn at P can be set equal toê R = R/R. Therefore, the electric flux through the infinitesimal area da = R 2 sin θ dθ dϕ is

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“…For more mathematical details and insights see [3][4][5][6][7]. This result is true for an extended localized source.…”

Section: -2mentioning

confidence: 89%

Gauss's law, infinite homogenous charge distributions and Helmholtz theorem

Tort

2011

Rev. Bras. Ensino Fís.

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Ondas de choque não colisionais no espaço interplanetário

Echer

Alves

González

2006

Rev. Bras. Ensino Fís.

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Este trabalho apresenta uma introdução à propagação das ondas de choque não colisionais no espaço interplanetário. Inicialmente, são apresentados conceitos básicos envolvendo a propagação de informação em gases ordinários e em plasmas, os modos de ondas presentes num plasma e o seu escarpamento levando a formação de ondas de choque. A seguir são introduzidos alguns tipos de choques presentes no espaço interplanetário: frentes de choque planetárias, choques transientes devido às remanescentes de ejeções coronais de massa e choques corrotantes devido à interação de feixes rápido e lento do vento solar. Os choques também podem ser classificados quanto à direção de propagação - se afastando do Sol ou se propagando em direção ao Sol - e se são originários devido ao escarpamento de ondas magnetossônicas do modo lento ou rápido. Finalmente, um exemplo de observação de parâmetros de um choque transiente frontal rápido, com dados medidos no vento solar próximo a Terra e de seus efeitos na magnetosfera terrestre é apresentado.

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Two theorems by Helmholtz

Abstract: Why isitthat div and curl crop up everywhere in the study of vector fields? These theorems by Helmholtz explain.

Cited by 2 publications

References 2 publications

Gauss's law, infinite homogenous charge distributions and Helmholtz theorem

Gauss's law, infinite homogenous charge distributions and Helmholtz theorem

Ondas de choque não colisionais no espaço interplanetário

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