Il Nuovo Cimento C

, Volume 12, Issue 5, pp 523–540 | Cite as

A general method for evaluating the current system and its magnetic field of a plane current sheet, uniform except for a certain area of different uniform conductivity, with results for a square area

  • M. S. Abou-Dina
  • A. A. Ashour
Article

Summary

We derive a linear Fredholm integral equation of the second kind on an arbitrary closed plane contour which divides an infinite plane current sheet into two regions of different uniform integrated conductivities. This integral equation is satisfield along the above-described contour by a certain combination of the limiting values of the electric potential at both sides of the boundary. This electric potential is due to the currents created in the sheet when a uniform electric field is applied to it. The derived integral equation admits exact solutions in closed form for the cases of circular and elliptical insertions. These solutions are identical with those previously obtained, by other methods, for the same cases. A general method is given for the numerical solution of the integral equation. As an illustration, this method is applied to the case of a square insertion where we used the results of Ashour to obtain numerical estimation for the results of the additional magnetic field on and around the square insertion.

PACS 91.25

Geomagnetism and paleomagnetism geoelectricity 

Riassunto

Si deriva un’equazione lineare integrale di Fredholm del secondo tipo su un contorno di un piano chiuso arbitrario che divide una lamina di corrente piana infinita in due regioni di diverse conduttività integrate uniformi. Questa equazione integrale è soddisfatta lungo il contorno descritto sopra da una certa combinazione dei valori limitanti del potenziale elettrico in entrambi i lati del confine. Questo potenziale elettrico è dovuto alle correnti create nella lamina quando vi si applica un campo elettrico uniforme. L’equazione integrale dedotta ammette soluzioni esatte in forma chiusa per i casi di inserzioni circolari ed ellittiche. Queste soluzioni sono identiche a quelle ottenute precedentemente con altri metodi per gli stessi casi. Si dà un metodo generale per la soluzione numerica dell’equazione integrale. Come esempio si applica questo metodo al caso di un’inserzione quadratica dove si sono usati i risultati di Ashour per ottenere una stima numerica per i risultati del campo magnetico aggiunto sopra e intorno all’inserzione quadratica.

Резюме

Мы выводим линейное интегральное уравнение фредгольма второго рода на произвольном замкнутом плоском контуре, который делит бесконечный плоский токовый слой на две области, имеющие различные постоянные проводимости. Некоторая комбинация предельных значений электрического потенциала на обеих сторонах границы удовлетворяет этому уравнению вдоль описанного контура. Этот электрический потенциал обусловлен токами в слое, когда постоянное электрическое поле приложено к слою. Выведенное интегральное уравнение допускает точные решения в замкнутой форме в случаях циркулярного и эллитического вводов. Эти решения идентичны ранее полученным решениям с помощью других методов. Предлагается общий метод численного решения интегрального уравнения. Как иллюстрация, предложенный метод применяется к случаю квадратного ввода.

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References

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Copyright information

© Società Italiana di Fisica 1989

Authors and Affiliations

  • M. S. Abou-Dina
    • 1
  • A. A. Ashour
    • 1
  1. 1.Mathematics Department, Faculty of ScienceCairo UniversityGizaEgypt

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