Determination of the Electromagnetic Field Created by Line Current and Sheet Current Source at the Earth’s Surface

Affiliation(s)

^{1}
Mathematics and Statistics Department, Faculty of Science, King Faisal University, Al-Hassa, KSA.

^{2}
Mathematics Department, Faculty of Science, Ain Shams University, Cairo, Egypt.

ABSTRACT

The electromagnetic field that generated by line current and sheet current at the surface of the earth can be expressed in analytical form. The line current created at the earth’s surface by an infinitely long line current is given by the inverse Fourier integrals over a horizontal wave number. The sheet current can be obtained by integrating the line current expansions using a Neumann and Struve functions; these functions have known mathematical properties, including the series expansions. The series expansions are exact with neglecting the displacement currents. Assuming a uniform earth and that there is no propagation, the three nonzero field components can be expressed in terms of the Neumann and Struve functions. The integrals of line current expansions are calculated by using the numerical methods. The results represented graphically and illustrated by figures. Results can be used to evaluate numerical solutions of more complicated modeling algorithms.

The electromagnetic field that generated by line current and sheet current at the surface of the earth can be expressed in analytical form. The line current created at the earth’s surface by an infinitely long line current is given by the inverse Fourier integrals over a horizontal wave number. The sheet current can be obtained by integrating the line current expansions using a Neumann and Struve functions; these functions have known mathematical properties, including the series expansions. The series expansions are exact with neglecting the displacement currents. Assuming a uniform earth and that there is no propagation, the three nonzero field components can be expressed in terms of the Neumann and Struve functions. The integrals of line current expansions are calculated by using the numerical methods. The results represented graphically and illustrated by figures. Results can be used to evaluate numerical solutions of more complicated modeling algorithms.

Cite this paper

Sami, G. , Al-Nami, M. (2014) Determination of the Electromagnetic Field Created by Line Current and Sheet Current Source at the Earth’s Surface.*International Journal of Geosciences*, **5**, 1584-1593. doi: 10.4236/ijg.2014.513129.

Sami, G. , Al-Nami, M. (2014) Determination of the Electromagnetic Field Created by Line Current and Sheet Current Source at the Earth’s Surface.

References

[1] Keller, G.V. and Frischknecht, F.C. (1966) Electrical Methods in Geophysical Prospecting. Pergamon Press, Oxford.

[2] Annan, A.P. (1973) Radio Interferometry Depth Sounding, Part I. “Theoretical discussion”. Geophysics, 38, 557-580. http://dx.doi.org/10.1190/1.1440360

[3] Tsang, L., Brown, R., Kong, J.A. and Simmons, G. (1974) Numerical Evaluation of Electromagnetic Fields Due to Dipole Antennas in the Presence of Stratified Media. Journal of Geophysical Research, 79, 2077-2080. http://dx.doi.org/10.1029/JB079i014p02077

[4] Wait, J.R. (1971) Electromagnetic Probing in Geophysics. Golem, Boulder, Colo.

[5] Bishay, S.T. and Sami, G.M. (2003) Natural-Frequency Concept Utilized in Remote Probing of the Earth. Canadian Journal of Physics, 81, 705-712. http://dx.doi.org/10.1139/p02-067

[6] Sami, G.M. (2005) Influence of a Magnetically Permeable Surface Layer on Transient Electromagnetic Field by Using Natural-Frequency Concept Utilized in Remote Probing of the Earth. Czechoslovak Journal of Physics, 55, 555-562. http://dx.doi.org/10.1007/s10582-005-0060-8

[7] Uman, M.A. and Mc Lain, D.K. (1970) Lightning Return Stroke Current from Magnetic and Radiation Field Measurements. Journal of Geophysical Research, 75, 5143-5147.

http://dx.doi.org/10.1029/JC075i027p05143

[8] Johler, J.R. (1967) Propagation of Pulse from Nuclear Burst. IEEE Transactions on Antennas and Propagation, 15, 256-264. http://dx.doi.org/10.1109/TAP.1967.1138896

[9] Moffatt, D.L. and Mains, R.K. (1975) Detection and Discrimination of Radar Targets. IEEE Transactions on Antennas and Propagation, 23, 358-367. http://dx.doi.org/10.1109/TAP.1975.1141078

[10] De Hoop, A.T. and Frankena, H.J. (1960) Radiation of Pulses Generated by a Vertical Electric Dipole above a Plane Non-Conducting Earth. Applied Scientific Research, Section B, 8, 69-377.

[11] Langenberg, K.J. (1974) The Transient Response of a Dielectric Layer. Applied Physics, 3, 179-188. http://dx.doi.org/10.1007/BF00884494

[12] Wait, J.R. (1972) Theory of Wave Propagation Long a Thin Wire Parallel to an Interface. Radio Science, 7, 675-679. http://dx.doi.org/10.1029/RS007i006p00675

[13] Bishay, S.T., Abo-Seida, O.M. and Sami, G.M. (2001) Transient Electromagnetic Field of a Vertical Magnetic Dipole on a Two-Layer Conducting Earth. IEEE Transactions on Geoscience and Remote Sensing, 39, 894-897. http://dx.doi.org/10.1109/36.917919

[14] Sami, G.M. (2004) Influence of a Magnetically Permeable Surface Layer on Transient Electromagnetic Field of a Vertical Magnetic Dipole on a Two-Layer Conducting Earth. Czechoslovak Journal of Physics, 54, 339-348. http://dx.doi.org/10.1023/B:CJOP.0000018130.01468.df

[15] Bishay, S.T. and Sami, G.M. (2002) Time-Domain Study of Transient Fields for a Thin Circular Loop Antenna. Canadian Journal of Physics, 80, 995-1003. http://dx.doi.org/10.1139/p02-049

[16] Wait, J.R. (1972) On the Theory of Transient Electromagnetic Sounding over a Stratified Earth. Canadian Journal of Physics, 50, 1055-1061. http://dx.doi.org/10.1139/p72-146

[17] Sunde, E.D. (1968) Earth Conduction Effects in Transmission Systems. Dover Publications, New York.

[18] Wait, J.R. (1996) EM Fields of a Phased Line Current over a Conducting Half-Space. IEEE Transactions on Electromagnetic Compatibility, 38, 608-611. http://dx.doi.org/10.1109/15.544318

[19] Olsen, R.G. and Pankaskie, T.A. (1983) On the Exact, Carson and Image Theories for Wires at or above the Earth’s Interface. IEEE Transactions on Power Apparatus and Systems, PAS-102, 769-778. http://dx.doi.org/10.1109/TPAS.1983.317939

[20] Pirjola, R. (1982) Electromagnetic Induction in the Earth by a Plane Wave or by Fields of Line Current Harmonic in Time and Space. Geophysica, 18, 1-161.

[21] Arfken, G. (1985) Mathematical Methods for Physicists. 3rd Edition, Academic, San Diego, 985.

[22] I-Iermance, J.F. and Peltier, W.R. (1970) Magnetotelluric Fields of a Line Current. Journal of Geophysical Research, 75.

[1] Keller, G.V. and Frischknecht, F.C. (1966) Electrical Methods in Geophysical Prospecting. Pergamon Press, Oxford.

[2] Annan, A.P. (1973) Radio Interferometry Depth Sounding, Part I. “Theoretical discussion”. Geophysics, 38, 557-580. http://dx.doi.org/10.1190/1.1440360

[3] Tsang, L., Brown, R., Kong, J.A. and Simmons, G. (1974) Numerical Evaluation of Electromagnetic Fields Due to Dipole Antennas in the Presence of Stratified Media. Journal of Geophysical Research, 79, 2077-2080. http://dx.doi.org/10.1029/JB079i014p02077

[4] Wait, J.R. (1971) Electromagnetic Probing in Geophysics. Golem, Boulder, Colo.

[5] Bishay, S.T. and Sami, G.M. (2003) Natural-Frequency Concept Utilized in Remote Probing of the Earth. Canadian Journal of Physics, 81, 705-712. http://dx.doi.org/10.1139/p02-067

[6] Sami, G.M. (2005) Influence of a Magnetically Permeable Surface Layer on Transient Electromagnetic Field by Using Natural-Frequency Concept Utilized in Remote Probing of the Earth. Czechoslovak Journal of Physics, 55, 555-562. http://dx.doi.org/10.1007/s10582-005-0060-8

[7] Uman, M.A. and Mc Lain, D.K. (1970) Lightning Return Stroke Current from Magnetic and Radiation Field Measurements. Journal of Geophysical Research, 75, 5143-5147.

http://dx.doi.org/10.1029/JC075i027p05143

[8] Johler, J.R. (1967) Propagation of Pulse from Nuclear Burst. IEEE Transactions on Antennas and Propagation, 15, 256-264. http://dx.doi.org/10.1109/TAP.1967.1138896

[9] Moffatt, D.L. and Mains, R.K. (1975) Detection and Discrimination of Radar Targets. IEEE Transactions on Antennas and Propagation, 23, 358-367. http://dx.doi.org/10.1109/TAP.1975.1141078

[10] De Hoop, A.T. and Frankena, H.J. (1960) Radiation of Pulses Generated by a Vertical Electric Dipole above a Plane Non-Conducting Earth. Applied Scientific Research, Section B, 8, 69-377.

[11] Langenberg, K.J. (1974) The Transient Response of a Dielectric Layer. Applied Physics, 3, 179-188. http://dx.doi.org/10.1007/BF00884494

[12] Wait, J.R. (1972) Theory of Wave Propagation Long a Thin Wire Parallel to an Interface. Radio Science, 7, 675-679. http://dx.doi.org/10.1029/RS007i006p00675

[13] Bishay, S.T., Abo-Seida, O.M. and Sami, G.M. (2001) Transient Electromagnetic Field of a Vertical Magnetic Dipole on a Two-Layer Conducting Earth. IEEE Transactions on Geoscience and Remote Sensing, 39, 894-897. http://dx.doi.org/10.1109/36.917919

[14] Sami, G.M. (2004) Influence of a Magnetically Permeable Surface Layer on Transient Electromagnetic Field of a Vertical Magnetic Dipole on a Two-Layer Conducting Earth. Czechoslovak Journal of Physics, 54, 339-348. http://dx.doi.org/10.1023/B:CJOP.0000018130.01468.df

[15] Bishay, S.T. and Sami, G.M. (2002) Time-Domain Study of Transient Fields for a Thin Circular Loop Antenna. Canadian Journal of Physics, 80, 995-1003. http://dx.doi.org/10.1139/p02-049

[16] Wait, J.R. (1972) On the Theory of Transient Electromagnetic Sounding over a Stratified Earth. Canadian Journal of Physics, 50, 1055-1061. http://dx.doi.org/10.1139/p72-146

[17] Sunde, E.D. (1968) Earth Conduction Effects in Transmission Systems. Dover Publications, New York.

[18] Wait, J.R. (1996) EM Fields of a Phased Line Current over a Conducting Half-Space. IEEE Transactions on Electromagnetic Compatibility, 38, 608-611. http://dx.doi.org/10.1109/15.544318

[19] Olsen, R.G. and Pankaskie, T.A. (1983) On the Exact, Carson and Image Theories for Wires at or above the Earth’s Interface. IEEE Transactions on Power Apparatus and Systems, PAS-102, 769-778. http://dx.doi.org/10.1109/TPAS.1983.317939

[20] Pirjola, R. (1982) Electromagnetic Induction in the Earth by a Plane Wave or by Fields of Line Current Harmonic in Time and Space. Geophysica, 18, 1-161.

[21] Arfken, G. (1985) Mathematical Methods for Physicists. 3rd Edition, Academic, San Diego, 985.

[22] I-Iermance, J.F. and Peltier, W.R. (1970) Magnetotelluric Fields of a Line Current. Journal of Geophysical Research, 75.