^{1}

^{*}

^{1}

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.

In the last few decades, there has been increased interest in the behavior of electromagnetic systems which are operated near a conducting earth. Many of these systems have been successfully used to measure the electromagnetic properties of the earth. That is, if the field of a transmitting source can be calculated in terms of idealized earth models, then measurements of actual fields near a real earth can be interpreted in terms of these models. This method is known as the induction method of geophysical exploration [

Electromagnetic boundary conditions at the surface of the wire were leading to a characteristic equation that determines the propagation constant along the wire [

The main objective of paper is to determine the electromagnetic field due to a line current and sheet current. Sheet current having a finite width implies a more realistic model. It can be obtained by integrating the line current expansions using a Neumann and Struve functions. These functions have known mathematical properties [

In the Cartesian coordinate system (x, y, z), we assume that the earth’s surface is the xy plane, and the line current J flows is parallel to the y-axis at a height z = −d. The line current oscillating harmonically in time

A model containing an infinitely long line current parallel to the surface between two half-spaces is frequently used in electromagnetic applications extending from geophysics. The medium in which the line current lies is nonconducting (the air), and the other half-space is conducting (the earth). The line current flows parallel to the y-axis at a height Z = ‒d. Neglecting displacement current effects, and assuming that

The magnetic field has only the magnetic field component

R is the reflection coefficient at the earth’s surface is depending on the conductivity structure of the earth, and is given by,

and J gives the magnitude of the line current oscillating harmonically in time

The components of the electric fields are expressed in terms of

Equation (7) can be written in Terms of Struve function of the first order

The Neumann function

where

The function

where

and

The gamma function Г satisfies,

The expansions of the Neumann and Struve functions Equation (13) and (14) with (16) may now be substituted into (12) to obtain a series representation for the magnetic field

where

and

By applying the series expression of

To determine the electromagnetic field created by the sheet current source at the surface, a sheet current can be lies in non-conducting (the air), and the other half-space is conducting (the earth). The sheet current having a finite width implies a more realistic model. It can be obtained by integrating the line current expansions using Neumann and Struve functions. We now consider an infinitely long sheet current on surface of the Earth. Let

Applying [(7)-(9)] for each line current

The electric field by (28) and (29) is expressible in terms of the magnetic field (28)

Rewriting (20) with

where

and

where

and if we assume that the wave number

Let us define the functions

and

We know that

and

A straightforward algebraic manipulation now allows for expressing (32) as

the quantities

and

where

We now consider a uniform sheet located between

Assuming

where

and

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

Figures 1. (a)-(d) represent the imaginary part of the magnetic field component Hy due to a line current of 1 MA with different values of distance x at the periods 10 s, 50 s, 100 s, and 700 s, respectively. (e)-(h) represent real and imaginary parts of the electric field component Ez, and (j)-(l) represent real and imaginary parts of the electric field component Ex, at the earth’s surface due to a line current of 1 MA with different values of x at the periods 10 s, 50 s, 100 s, and 700 s, respectively. The continuous and dots curves correspond to the real and imaginary part of the electric field, respectively.

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

Figures 2. (a)-(d) represent the imaginary part of the magnetic field component Hy due to sheet current of 1 MA with different values of distance x at the periods 10 s, 50 s, 100 s, and 700 s, respectively. (e)-(h) represent real and imaginary parts of the electric field component component Ez, and (j)-(l) represent real and imaginary parts of the electric field component Ex, at the earth’s surface due to sheet current of 1 MA with different values of x at the periods 10 s, 50 s, 100 s, and 700 s, respectively. The continuous and dots curves correspond to the real and imaginary part of the electric field, respectively.

Numerical computation of the electromagnetic field that generated by line current and sheet current at the surface of the earth can be easily performed, based on the Equations [(12), (24) and (25)] and [(44), (49) and (50)], respectively. The following parameters have been chosen to draw the curves shown by Figures 2(a)-(l):

The electromagnetic field defined as a function of x coordinate at the surface of the earth due to line current and sheet current and plotted against the distance x.

For line current, Figures 1(a)-(d) represent the imaginary part of magnetic field component

For sheet current, Figures 2(a)-(d) represent the imaginary part of magnetic field component

The magnetic and electric fields were treated as complex quantities, the real part of the right-hand side of Equations (3)-(19) is ignored. It is seen that the magnetic field component

The results we obtained that with increasing in the period indicate to increase in the imaginary part of magnetic field component

We compared the magnetic and electric fields at the surface of a uniform earth with those produced by line current and sheet current.

In the sheet current, the values of the imaginary part of the magnetic field component

From a physical point of view, this means that the series expansions including the propagation constant

We conclude that the electromagnetic field at the earth’s surface created by line current can be expressed in terms of the Neumann and Struve functions, and take the conductivity of the earth into consideration.

The magnetic and electric fields were derived in new series expansions and computations. Also we present the magnetic and electric fields due to sheet current obtained by integrating the line current expansion. The results are represented graphically. We compared the results of the magnetic and electric fields values at the surface of a uniform earth with those produced by line current with sheet current.