It is natural to consider the density of electric charges distribution in the body, the electric current density  in the body, as well as the body magnetization and polarization as sources of electromagnetic field. Indeed, the Maxwell equations enable us to represent the strength , and flux density , of electromagnetic field in the entire space as linear integral operators of , , and specified in the domain occupied by the body. These parameters are sources or origins of the body electromagnetic effect on the substance of this body itself as well as other bodies in the vicinity.

The above-mentioned fact as well as the relationships of substance susceptibility to electromagnetic effect such as Ohm’s law, Poisson law of magnetization proportionality for total magnetic field as well as its equivalent – the law of electric polarization lead to the formulation of the main electrodynamic problems using integral theory equations where the sources are considered as unknown values.

This concept of sources giving rise to electromagnetic process originates from the studies of S. Poisson [1, 2] giving integral representation of the of static electric field strength and magnetic field strength as the density of electric charges and magnetization, respectively, and the problem of their determination is reduced to the solution of integral equations. This concept of sources was successfully employed by О. V. Тоzоni and I. D. Mayergoiz [3, 4] and other representatives of the Kiev school in electric engineering theory for solving the problems of low-frequency electrodynamics using the method of secondary sources.

In this paper the concept of sources is considered in application to the stationary electrodynamics problems where, for a number of reasons, a different principle has been adopted and widely used [5, 6]. In accordance with this principle the integral equations are derived using field strength and flux density conditions satisfied by at the body boundaries.


For solving the formulated task let us employ the Maxwell equations in the complex form [5], represented as follows:


assuming that outside the area V occupied by the body we have :


while at the body boundary S the following conditions are satisfied:


where the subscript «n» indicates normal components and the subscript «s» indicates tangential components of vector functions with respect to S; the superscripts «+» and «-» indicate their limit values at S, respectively, on the inner and outer sides;  — surface density of electric charge.

For the interest of space and convenience let us merge the sources and into one source :


assuming that and are specified independently of each other.

In this case equation (1) is substituted by a symmetric system:


and boundary conditions (2) are replaced by :


The general solution of the equation system (4) with the boundary conditions (5) is obtained by assuming successively that and .

1. At let us assume that:

,            (6)

then it follows from the second equation of the equation system (4) that:

.        (7)

Substitution of equations (6) and (7) into the equation system (4) at results in the situation when for the validity of eq. (4) the electromagnetic potentials and shall satisfy the equations:


at the following condition shall be satisfied :

.            (9)

The unique solution (8) satisfying the conditions (9), boundary conditions (5) and zero conditions at will have the form:


where  — distance between space point where the values of potentials and integration variable , running across the area . Using the Gauss theorem the second of the equations (10) can be written as:


where the prime at gradient operation means that the operation is performed with respect to integration variable coordinates.

The result is that in accordance with (6) and (7), the strength of electromagnetic field can be expressed as:


which satisfies the equations (4) and boundary conditions (5). It is easily demonstrated by direct substitution.

These expressions can be considered as the unique solution of equations (4), (5) because these equations have non-zero solutions only, which turns into zero at infinity .

2. Let us assume
in (4), (5) and further assume

,            (12)



And electrodynamic potentials and shall satisfy the equations:


and condition:

.        (15)

Solution of equations (14) give:


which define the unique solution of equations (4), (5) in the case under consideration:


The solution of equation system (4) with conditions (5) in a general case at specified vector functions and will be superposition of solutions (11) and (17):


where subscripts «Q» and «J» indicate solutions of equations (11) and (17).


The analysis of equations (11), (17) reveals that if any of the electromagnetic field sources or represents the potential vector function whose potential at the boundary of body S is zero, then the strength and flux density produced by this electromagnetic field is zero outside this body, but the electromagnetic potentials in this case are not zero. Let us indicate the multitude of potential vector functions in the area V by the symbol.

Let us assume , i.e.

.        (19)

Taking into account the identical equation:

the right-hand side of which is zero according to (19), it can be concluded that in all points of the Euclidean space is . In this case, it follows from the first of the equations (4) that:

At the same time from (10) it follows that:


where at ; at the expression is in the most simple way derived from condition (9).

Let us assume that. In this case , and the simplest example of electric charge flow producing zero electromagnetic field strength outside the body is the current in the sphere of radius , which satisfies the following conditions in the Cartesian coordinate system with the origin in the sphere center:

In this case:

i.e. the electric charge is flowing from the inside of the sphere to its surface and back, so that the total charge of the sphere is zero. This example is illustrating the nature of variations in polarization giving no rise to external field.

Similarly, at

we obtain:


Considering the relation between the field strength and induction defined by the expressions:


It is easy to see that in the case when and belong to , outside the body we always have:

and inside the body:


The obtained results are valid for limiting cases corresponding to electric and magnetic statics, zero field strength and induction outside the body being observed in static conditions also in other cases. In case of current, when  is solenoidal vector function, whose vector potential is solenoidal vector function and turns into zero at the body boundary :


In case of body magnetization and polarization zero field strength is achieved, in addition, when the body magnetization and polarization are also solenoidal vector functions whose normal component to is zero:


and in this case flux densities are different from zero and equal to:


These facts were mentioned earlier by the author in [7].


Formulae (11) and (17) can be used to define the sources of electromagnetic field in the diffraction type problems. In the most simple way it can be done in case of weak electromagnetic effect on the body in the range of not very high frequencies when there is a linear relationship – generalized Ohm law and the Poisson law of induced magnetization:


where  — are dielectric and magnetic susceptibility, respectively,  — conductivity;


 — strength of electromagnetic field action on the body,  — field strength defined by formulae (11) and (17):


Using formulae (11) — (17) the equations (23) are transformed into a system of linear integral equations with respect to at the specified external electromagnetic effect .

Let us consider the properties of these equations using only Ohm’s law example (). In this case equations (23) are reduced to:


Solution of this equation requires considerable effort even at low frequencies when it is possible to assume that within the body. These difficulties are caused by the skin effect resulting in near-surface concentration of current.

For overcoming the problems of this kind one can use the source properties following from equation (4), which can be described by:

,    (25)

where .

The validity of (25) allows us to employ the identical equation [5]:


By using operation «rot» to (26) with subsequent vector multiplication of the result by the normal to and assuming , we obtain the integral equation for S:


Solution of this equation with respect to allows us to use the following representation:

where  — linear operator at , and, in accordance with (26), the current density in the area can be expressed only by its values for .

As a result the equation (24) can be transformed into the equation of this form:


at it is reduced to the Fredholm equation of the second kind with respect to surface current density . This equation can also be used to express the field strength produced by the electromagnetic field current in the entire space.



1 Poisson S. Mem. de l’Inst. de France, 1811, p.1163

2 Poisson S. Mem. de l’Acad. Roy. des Scince t.5.1826

3 Tozoni О.V. Calculation of electromagnetic fields using computers. Kiev. Teknika, 1967 p.231

4 Tozoni О.V. and Mayergoiz I. D.  Calculation of three-dimensional electromagnetic fields Kiev. Teknika, 1979.

5 Stratton J., Theory of Electromagnetism M. – L. GITTLE, 1948, p.539

6 Colton D., Cross R. Methods of integral equations in the scattering theory. M. Mir, 1987, p.391

7 Krasnov I.P. On magnetization of an arbitrary body and electric current concentrations containing no external magnetic field. Proceedings of International Shipbuilding Conference (ISC’98), 24 – 26 November 1998, St.Petersburg, Section Е, p.3 – 9.


1 коммент.

  1. Perov @ Апрель 13th, 2010

    Благодарю. Появилась умная мысль, но нуждается в подробной реорганизации старой идеи, займусь на днях. И сразу поделюсь информацией с читателями блога!

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