Analytical solution of electromagnetic radiation by a vertical electric dipole inside the earth and the effect of atmospheric electrical conductivity inhomogeneity Taha Mosayebidorcheh, Fahimeh Hosseinibalam ⇑ , Smaeyl Hassanzadeh Department of Physics, University of Isfahan, Isfahan 81746-73441, Iran Received 4 January 2017; received in revised form 22 July 2017; accepted 24 July 2017 Abstract In this paper, the effect of atmospheric electrical conductivity on the electromagnetic waves radiated by a vertical electric dipole located in the earth, near the surface of the earth, is investigated. As far as electrical conductivity is concerned, the atmosphere is divided into three areas, in which the electrical conductivity changes with altitude. The Maxwell equations in these areas are investigated as well. Using the differential transform method, the differential equation is solved in a way that atmospheric electrical conductivity is variable. Solving the problem in these areas indicates that electrical conductivity in the middle and lower areas of atmosphere may be ignored. However, in the upper areas of atmosphere, the magnitude of the magnetic field in the ionosphere at a frequency of 10 kHz at night is five times smaller when electrical conductivity is considered compared to when it is neglected. Ó 2017 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Vertical electric dipole; Electric conductivity of atmosphere; Differential transform method 1. Introduction Electromagnetic waves radiation in the atmosphere is critically important for investigation of space weather haz- ards (Samimi and Simpson, 2015), satellite communications, radar, remote-sensing, and earthquake precursors (Baba, 2000; Adachi et al., 2001). Electromagnetic wave propaga- tion radiated by a horizontal electric dipole (HED) or a ver- tical electric dipole (VED), above or inside the ground, has been under study for many years (Wait, 1953, 1956, 1957a, 1957b, 1990a, 1990b; King, 1991; King et al., 1992; King and Sandler, 1994a, 1994b). HED or VED is a horizontal or vertical delta-function current, respectively. There are two methods to solve this problem; the first one is to solve Maxwell equations assuming that earth and atmospheric layers are relatively flat. The other method accounts for the spherical shape of the earth and the atmosphere, and therefore, the sum of spherical harmonics are obtained. In this case, tending the radius of the earth to infinity, leads to the first method. The spherical shape of the earth can be neglected for waves radiated by VED or HED with wave- lengths much smaller than the radius of the earth. Zenneck, carried out the first analytical solution for the propagation of electromagnetic waves along the planar boundary between the atmosphere and the earth in 1907 (Zenneck, 1907). But at that time, the existence and signif- icance of the ionosphere was not known. Wait (Wait, 1953, 1956, 1957a, 1957b, 1970, 1990a, 1990b) and King (King, 1991; King et al., 1992; King and Sandler, 1994a, 1994b), have made the subsequent developments on the electromagnetic field of a dipole source in the multilayered region. Specifically, extensive studies have been carried out for two-layered or multilayered regions using surface- http://dx.doi.org/10.1016/j.asr.2017.07.034 0273-1177/Ó 2017 COSPAR. Published by Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail addresses: t.mosayebi@sci.ui.ac.ir (T. Mosayebidorcheh), fhb@sci.ui.ac.ir (F. Hosseinibalam), shz@phys.ui.ac.ir (S. Hassanzadeh). www.elsevier.com/locate/asr Available online at www.sciencedirect.com ScienceDirect Advances in Space Research xxx (2017) xxx–xxx Please cite this article in press as: Mosayebidorcheh, T., et al. Analytical solution of electromagnetic radiation by a vertical electric dipole inside the earth and the effect of atmospheric electrical conductivity inhomogeneity. Adv. Space Res. (2017), http://dx.doi.org/10.1016/j.asr.2017.07.034