## Abstract

A mathematical model is developed to determine the steady-state electric current flow through in non-homogeneous isotropic conductor whose shape has a three-dimensional hollow body. The equations of the Maxwell's theory of electric current flow in a non-homogeneous isotropic solid conductor body are used to formulate the corresponding electric boundary value problem. The determination of the steady motion of charges is based on the concept of the electrical conductance. The derivation of the upper and lower bound formulae for the electrical conductance is based on Cauchy-Schwarz inequality. Two numerical examples illustrate the applications of the derived upper and lower bound formulae.

## 1 Introduction

Electrical resistance of an electrical conductor is a measure of the difficulty to pass a steady electric current through the conductor. The well-known elementary form of Ohm's law states that when the conductor carries a current *I* from a point *P*_{1} at potential *U*_{1} to a point *P*_{2} at potential *U*_{2} then *U*_{1}–*U*_{2} = *RI*, where *R* is the resistance of the conductor between points *P*_{1} and *P*_{2}, it depends only on the shape and temperature and the material of the conductor. The inverse of electric resistance is the electric conductance *G = 1/R.* This paper deals with the electric resistance of a three-dimensional non-homogeneous conductor body. Examination of non-homogeneous structural elements is a very important task. Maróti's study [1] deals with the bending vibration of axially non-homogeneous beams. The buckling problem of axially functionally graded beams is considered in paper [2]. For prescribed frequency and buckling loads Maróti and Elishakoff [2] determined the Young's modulus in axial direction as a function of axial coordinate. The non-homogeneous isotropic hollow conductor is bounded by two closed surfaces

*V*and its boundary surfaces are

*U*on the boundary surfaces

*I*. The determination of

*I*is based on the next equation

**is the outer normal unit vector of the inner boundary surface**

*n**R*and the conductance

*G*of the hollow conductor is defined as [6, 8],

There are several approximation methods to get the solution of the boundary-value problem Eq. (5), most of which use the results of variational calculus for example as Ritz method, finite element method [8, 9]. Other methods are also known and they used, for example finite difference methods, method of weighted residuals, boundary element method [10]. It must be mentioned that, many numerical-analytical method are used *R*-functions to solve the boundary value problems of electrodynamics [11–13]. The efficiency of the *R*-Function Method (RFM) to solving the boundary value problems of electrostatics in very complicated domain is illustrated in paper by Kravchenko and Basarab [14]. They considered a boundary-value problem of electrodynamics in the fractal regions of the Sierpiski carpet and the Koch island types [14]. Iványi solved a number of two-dimensional boundary value problems of static and stationary electromagnetisms by variational method connecting of them with the use of R-functions [12, 13, 15, 16]. It is not the aim of this paper is to give a detailed list of different analytical and numerical methods, which are used widespread in electrical engineering calculations.

## 2 Upper bound for *G* and lower bound for *R*

If the function

The proof of inequality (14)_{3} can be derived by the Cauchy-Schwarz inequality relation (15),

The combination of the inequality relation (15) with Eq. (16) and using formula (9) gives (14). A brief discussion shows that the sign of equality in relation (14) is valid only if

## 3 Lower bound for *G*, upper bound for *R*

Let

In lower bound formula (18) equality is reached only if

The proof of lower bound formula (18) is based on the Cauchy-Schwarz inequality relation (19)

From inequality relation (22) the proof of lower bound formula, (18) can be obtained immediately.

Let

*G*

## 4 Numerical examples

In the numerical examples the spherical coordinate system is used. The connection between the Cartesian coordinates

Meridian section of hollow spherical domain

Citation: Pollack Periodica 18, 1; 10.1556/606.2022.00621

Meridian section of hollow spherical domain

Citation: Pollack Periodica 18, 1; 10.1556/606.2022.00621

Meridian section of hollow spherical domain

Citation: Pollack Periodica 18, 1; 10.1556/606.2022.00621

Figure 3 shows the upper and the lower bounds of the conductance *G* as a function of *α* for

Upper and lower bounds for the conductance of non-homogeneous spherical conductor as a function of *α* for

Citation: Pollack Periodica 18, 1; 10.1556/606.2022.00621

Upper and lower bounds for the conductance of non-homogeneous spherical conductor as a function of *α* for

Citation: Pollack Periodica 18, 1; 10.1556/606.2022.00621

Upper and lower bounds for the conductance of non-homogeneous spherical conductor as a function of *α* for

Citation: Pollack Periodica 18, 1; 10.1556/606.2022.00621

Figure 4 shows the plot of function

Comparison of exact solution, approximate solution and FE approximation

Citation: Pollack Periodica 18, 1; 10.1556/606.2022.00621

Comparison of exact solution, approximate solution and FE approximation

Citation: Pollack Periodica 18, 1; 10.1556/606.2022.00621

Comparison of exact solution, approximate solution and FE approximation

Citation: Pollack Periodica 18, 1; 10.1556/606.2022.00621

The meridian section of axisymmetric hollow domain bounded by two spherical surfaces as it is shown in Fig. 5. The following data are used

The meridian section of axisymmetric hollow domain bounded by two spherical surfaces with different centre points

Citation: Pollack Periodica 18, 1; 10.1556/606.2022.00621

The meridian section of axisymmetric hollow domain bounded by two spherical surfaces with different centre points

Citation: Pollack Periodica 18, 1; 10.1556/606.2022.00621

The meridian section of axisymmetric hollow domain bounded by two spherical surfaces with different centre points

Citation: Pollack Periodica 18, 1; 10.1556/606.2022.00621

Let

Figure 6 shows the upper and the lower bounds as a function of power index for

Upper and lower bounds for the conductance as a function of power index *n*

Citation: Pollack Periodica 18, 1; 10.1556/606.2022.00621

Upper and lower bounds for the conductance as a function of power index *n*

Citation: Pollack Periodica 18, 1; 10.1556/606.2022.00621

Upper and lower bounds for the conductance as a function of power index *n*

Citation: Pollack Periodica 18, 1; 10.1556/606.2022.00621

## 5 Conclusions

A mathematical model is developed to determine the steady-state electric current flow through in non-homogeneous isotropic conductor whose shape is a three-dimensional hollow body. The hollow body considered is bounded by two closed surfaces which have no common points. The derivation of the upper and lower bound formulae for the electrical conductance is based on the two types of Cauchy-Schwarz inequality. Two numerical examples illustrate the applications of the derived upper and lower bounds for the conductance. The derived upper and lower bound formulae of electric conductance can be used to check the results of numerical computations obtained by finite element method, boundary element method and by any other numerical methods.

## References

- [1]↑
Gy. Maróti, “Finding closed-form solutions of beam vibration,”

, vol. 6, no. 1, pp. 141–154, 2011.*Pollack Period.* - [2]↑
Gy. Maróti and I. Elishakoff, “On buckling of axially functionally graded beams,”

, vol. 7, no. 1, pp. 3–13, 2012.*Pollack Period.* - [3]↑
I. Ecsedi, Á. J. Lengyel, A. Baksa, and D. Gönczi, “Bounds for the electrical resistance for homogenous conducting body of rotation,”

, vol. 11, no. 5, pp. 104–122, 2021.*Multidiszciplináris tudományok* - [4]↑
I. Ecsedi and Á. J. Lengyel, “Bounding formulae for capacitance of cylindrical capacitor with non-homogeneous material,”

, vol. 13, pp. 125–131, 2021.*WSEAS Trans. Electron.* - [5]↑
I. Ecsedi, Á. J. Lengyel, and D. Gönczi, “Bounds for the thermal conductance of body of rotation,”

, vol. 13, no. 3, pp. 185–193, 2020.*Int. Rev. Model. Simulations* - [7]
P. P. Silvester and R. L. Ferrari,

. Cambridge Univeristy Press, 1983.*Finite Elements for Electrical Engineers* - [9]↑
G. A. Korn and T. M. Korn,

. New York: D. von Nosrand, 1961.*Handbook for Scientists and Engineers* - [10]↑
A. P. Boresi, K. P. Chong, and S. Saigal,

. John Willey & Sons, Inc, 2003.*Approximate Solution Methods in Engineering Mechanics* - [11]↑
V. L. Rvachev and T. I. Sheiko, “R-functions in boundary value problems in mechanics,”

, vol. 48, no. 4, pp. 151–188, 1995.*Appl. Mech. Rev.* - [12]↑
A. Iványi,

. Budapest: Akadémiai Kiadó, 2003.*Continuous and Discrete Simulations in Electrodynamics*(in Hungarian) - [13]↑
A. Iványi, “

*R-functions in electromagnetism*,” Technical Report No. TUB-TR-93-EE08, Budapest, 1993. - [14]↑
V. F. Kravchenko and M. A. Basarab, “Solving the boundary value problems of electrodynamics in the regions of fractal geometry by the method of R-functions” (in Russian),

, vol. 29, no. 12, pp. 1055–1057, 2003.*Tech. Phys. Lett.* - [15]↑
A. Iványi, “Variational methods for static electric field” (in Hungarian),

, vol. 71, pp. 21–25, 1978.*Elektrotechnika* - [16]↑
A. Iványi, “Determination of static and stationary electromagnetic fields by variational calculus,”

, vol. 23, pp. 201–208, 1979.*Period. Polytech. Electr. Eng.* - [17]↑
E. Masina, “

*A review on the exponential-integral special function and other strictly related special functions*,” Lecture Notes, University of Bolona, Italy, 2019. - [18]↑
The exponential integral,

*Wolfram Mathworld*. [Online]. Available: https://mathworld.wolfram.com/ExponentialIntegral.html. Accessed: Jan. 21, 2021.