Abstract
A system of N coupled linear boundary Fredholm integral equations of the second kind is derived to describe the electric current system and the magnetic field distribution in space for an infinite plane electrical conducting sheet with N non-overlapping insertions, permeated by a uniform parallel electric field. The cases of one or two insertions obtained earlier are recovered. The system for three insertions is derived and solved numerically to provide solutions for new problems involving elliptic or square insertions. The lines of current are plotted and the results for each case are discussed to assess the efficiency of the numerical method. The solved problems provide a detailed study of the complex behaviour of harmonic functions in space and in the plane, represented, respectively, by the magnetic scalar potential and the current function, at lines of discontinuity of the electrical conductivity. It is noted that the case of holes can be treated equally well. The method and the obtained results allow to evaluate the magnetic field on the Earth’s surface on and around a multitude of islands. They may be useful in non-destructive testing of current sheets through magnetic field measurement. The tackled problems display different types of discontinuities of the magnetic field components. The numerical results indicate that the case of three or more insertions may differ qualitatively from that of one or two insertions, namely the possibility of existence of current loops.
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Abbreviations
- \(C_m\) :
-
Boundary of region \(D_m, m=1,2, \ldots , N\)
- \({\mathbf {E}}\) :
-
Initial electric field parallel to the sheet
- \(F_m\) :
-
Unknown functions entering in the expressions for the magnetic field components. These functions are determined as solutions of a system of coupled integral equations.
- \(\mathbf {i,~j,~k}\) :
-
Unit base vectors in the directions of increasing of the coordinates x, y, z, respectively
- I :
-
Undisturbed electric current intensity
- U :
-
Magnetic scalar potential
- \(V_{m}^{0}(x.y)\) :
-
electric potential required to maintain the undisturbed current flow in region \(D_m, m=0,1,2,\ldots ,N\)
- \(V_{m}\) :
-
Additional electric potential in region \(D_{m} ,m=0,1,2,\ldots ,N\)
- \(V_{m}^{C_m}\) :
-
Limiting value of \(V_{m}\) on the boundary \(C_m\)
- \(V_{m}^{^{\prime }}\) :
-
total electric potential in region \(D_{m} ,m=0,1,2,\ldots ,N\)
- \(\sigma _{m}\) :
-
Electric conductivity of m-th insertion, \(m=1,2, \cdots , N\)
- \(\sigma _{0}\) :
-
Electric conductivity of the host medium
- \(\psi ^{0}(x,y)\) :
-
Current function of the undisturbed current flow if the whole sheet were of integrated conductivity \(\sigma _{0}\)
- \(\Psi _{m}\) :
-
Additional current function in region \(D_{m} ,m=0,1,2,\ldots ,N\)
- \(\psi _{m}^{^{\prime }}\) :
-
Total current function in region \(D_{m} ,m=0,1,2,\ldots ,N\)
References
Abou-Dina, M.S., Ashour, A.A.: 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. Il Nuovo Cimento 12C(5), 304–311 (1989)
Abou-Dina, M.S., Ashour, A.A.: Exact solutions for certain two-dimensional problems of induction in thin non-uniform sheets. Geophys. Res. 95, 17547–17553 (1990)
Ashour, A.A.: Electromagnetic induction in finite thin sheets. Quart. J. Mech. Appl. Math. 18, 69–82 (1965)
Ashour, A.A.: The magnetic field of plane current sheet with different uniform conductivities in different parts with results for elliptic area. Geophys. J. R. Astrnom. Soc. 22, 401–416 (1971)
Ashour, A.A.: The magnetic field of an infinite plane current sheet uniform except for two circular insertions of different uniform conductivities. Geophys. J. R. Astron. Soc. 83, 127–142 (1985)
Ashour, A.A., Chapman, S.: The magnetic field of electric currents in an unbounded plane sheet, uniform except for a circular area of different uniform conductivity. Geophys. J. R. Astron. Soc. 10, 31–44 (1965)
Berdichevskiy, M.N., Dmitriev, V.I.: Distortion of magnetic and electrical fields by near-surface lateral inhomogeneities. Acta Geod. Geophys. Montan. Acad. Sci. Hung. 2(3–4), 447–483 (1976)
Chapman, S., Ashour, A.A.: Meteor geomagnetic effects. Smithonian Contrib. Astrophys. 8, 181–197 (1965)
Dastjerdi, M.H., Rubesam, M., Ruter, D., Himmel, J., Kanoun, O.: Non destructive testing for cracks in perforated sheet metals. In: 8-th International Multi-conference on Systems, Signals and Devices, pp. 1–5 (2011)
De Carli, L., Hudson, S.M.: Geometric remarks on the level curves of harmonic functions. Bull. Lond. Math. Soc. 42(1), 83–95 (2010)
Dong, S.J., Kelkar, G.P., Zhou, Y.: Electrode sticking during micro-resistance welding of thin metal sheets. IEEE Trans. Electron. Manag. Packag. 25(4), 355–361 (2002)
El-Sakout, D.M., Ghaleb, A.F., Abou-Dina, M.S., Ashour, A.A.: The magnetic field of electrical conducting sheets with two non-overlapping insertions. Appl. Math. Inf. Sci. 9(3), 1213–1223 (2015)
Erdélyi, A.: Higher transcendental functions, vol. 2. McGraw-Hill, New York (1953)
Flatto, L., Newman, D.J., Shapiro, H.S.: The level curves of harmonic functions. Trans. Am. Math. Soc 123, 425–436 (1966)
Flatto, L.: A theorem on level curves of harmonic functions. J. Lond. Math. Soc. 2(1), 470–472 (1969)
Gopalakrishnan, K., Goldberg, K., Bone, G.M., Zaluzec, M.J., Koganti, R., Pearson, R., Deneszczuk, P.A.: Unilateral fixtures for sheet-metal parts with holes. IEEE Trans. Autom. Sci. Eng. 1(2), 110–120 (2004)
Madle, P.J., Robinson, W.A.: In situ method for measuring the permeability and resistivity of metal sheets. IEEE Trans. Instrum. Meas. 24(4), 300–305 (1975)
Millar, R.F.: The location of singularities of two-dimensional harmonic functions. I: theory. SIAM J. Math. Anal. 1(3), 333–344 (1970a)
Millar, R.F.: The location of singularities of two-dimensional harmonic functions. II: applications. SIAM J. Math. Anal. 1(3), 345–353 (1970b)
Price, A.T.: The induction of electric currents in non-uniform thin sheets and shells. Quart. J. Mech. Appl. Math. 2(3), 283–310 (1949)
Reed, J.A., Byrne, D.M.: Using periodicity to control spectral characteristics of an array of narrow slots. Antennas Propag. Soc. 4, 2376–2379 (1997)
Schieber, D.: Transient eddy currents in thin metal sheets. IEEE Trans. Magnet. 8(4), 355–361 (1972)
Weidelt, P.: The electromagnetic induction in two thin half-sheets. Z. Geophys. Res. 37, 649–665 (1971)
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Al-Mutairi, I.A., Rawy, E.K., Abou-Dina, M.S. et al. The magnetic field of a plane current sheet with piecewise uniform electrical conductivity, with results for three insertions. Z. Angew. Math. Phys. 69, 134 (2018). https://doi.org/10.1007/s00033-018-1027-5
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DOI: https://doi.org/10.1007/s00033-018-1027-5