Two-Layered Conducting Earth : A Part from the Book Chapter : Exact Solution of Far-Region Scattering Electromagnetic Fields by a Dipole Antenna Located on a Planar Multi-Layered Earth

The geometric model under consideration is established in The Cartesian coordinate system (x,y,z) with z-axis directed vertically down is regarded, additionally the cylindrical polar coordinate system (ρ,φ,z) is employed for formulate the basic field equations. The source is a VMD placed directly on the earth’s surface (a planar two-layered conducting earth), whose magnetic moment is IdA (I is the current flowing in a loop of area dA).

The ε_i  ,μ_0  ,σ_i  ,k_i denote permittivity, permeability, conductivity, and propagation constant, where i = 0, 1, 2  denotes the air (layered-0 ,-∞≤z≤0),  the upper earth layered (layered-1, 0≤z≤d) with uniform thickness d, and the lower earth layered (layered-2,  d≤z≤∞), respectively. The two earths’ layered are homogeneous, and the magnetic permeability value is taken to be identical to that of free space throughout, also use is made of the harmonic time variation  e^iωt .

Author(s) Details:

 Sh. Shoeib
Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt and El-Gazeera Higher Institute for Engineering and Technology, Egypt.

Also See : Urdari Lake, the one considered in the present study : A Part from the Book Chapter : Ecological Approach on Investigating the Open Pit Lakes Developed in Oltenia Mining Region: A Case Study of Urdari Lake, Romania

Recent Global Research Developments in Electromagnetic Fields (EM fields) in the Context of a Multi-Layered Spherical ‘Earth-Ionosphere’ Model

ELF-EM Fields in the Multi-Layer Spherical ‘Earth-Ionosphere’ Model Based on WKB

  • This recent research, published in the Journal of Geophysics and Engineering, investigates the EM fields emitted by a horizontal electric dipole (HED) in a spherical ‘Earth-ionosphere’ model.
  • Previous research based on a planar model ignored the curvature of the Earth. In this work, the spherical model is considered to extend the coverage of the wireless electromagnetic method (WEM).
  • The study transfers EM fields from a vertical electric dipole (VED) and a vertical magnetic dipole (VMD) in the multi-layered Earth medium.
  • The correctness of the proposed method is verified by comparing it with approximate analytical formulas and previous work.
  • The results reveal interference effects different from those observed in the planar model, emphasizing the importance of considering Earth curvature in EM field studies.

Fast and Accurate Solution for the Surface Fields Scattering of a Dipole Antenna on a Planar Multi-Layered Earth

  • In this research, a fast and accurate solution is derived for EM scattering from a dipole antenna placed on a planar multi-layered Earth.
  • The derivation involves three steps:
  • Reducing field integrals to combinations of recognized Sommerfeld integrals (SIs).
  • Expressing the solution as an infinite series.
  • This approach is applicable irrespective of the operating frequency and saves computation time compared to other techniques.

Electromagnetic Fields in Planarly Layered Anisotropic Media 

  • Although not directly related to the spherical model, this paper presents a method for calculating EM fields from a dipole source in stratified media with general anisotropy.
  • The formulation can be applied to geophysical applications such as ground-penetrating radar and marine controlled source electromagnetic (CSEM) methods.

Exact Solution of Far-Region Scattering Electromagnetic Fields by a Dipole Antenna Located on a Planar Multi-Layered Earth

  • This study focuses on the exact solution of three-dimensional EM scattering from a vertical magnetic dipole (VMD) antenna placed on a planar multi-layered Earth.
  • The solution accounts for the layered Earth structure and provides insights into the scattering behavior.


  1. Ya Gao, Qing-Yun Di, Chang-Min Fu, Yi-Lang Zhang, ELF-EM fields in the multi-layer spherical ‘Earth-ionosphere’ model based on WKB, Journal of Geophysics and Engineering, Volume 20, Issue 2, April 2023, Pages 400–410,
  2. Shoeib, H. S. (2021). Fast and accurate solution for the surface fields scattering of a dipole antenna placed on planar multi-layered earth. Radio Science, 56(7), 1-7.
  3. O. Løseth, B. Ursin, Electromagnetic fields in planarly layered anisotropic media, Geophysical Journal International, Volume 170, Issue 1, July 2007, Pages 44–80,

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