Penetration of Electric Field into Hollow Dielectric Bodies

Determination of electric field distribution inside dielectric bodies mostly requires solving the Laplace’s equation. In many cases physical systems are so complex that analytical solutions are difficult or even impossible to obtain, as is mentioned in [1]. Nowadays, using computer, it is possible to exploit simple numerical approximation methods and in a few minutes to solve complex field problems. A widely used method for numerical electrostatic problems solving is the Charge simulation method (CSM), described in [2]. Basic idea of this method is replacing the existing system by fictitious charges (FCs) chosen in certain way and placing them inside the respective system volumes. Unknown quantities are the charges values and their coordinates. The question remains how to form system of equations and to find these values? Several possibilities exist. One of the simplest options is to use the point matching method as shown in this paper. Other options exist, such as least – square method, variation method or boundary conditions method. Using the point matching method a system of non – linear equations is formed in a way to satisfy boundary conditions for the system. Choice of initial values in iterative process of the solution, based on the linearization of the non – linear equations, can be difficult. In order to eliminate this problem it is convenient to use only the fictitious charges (FCs) as the unknowns. In [3, 4] authors suggest that FCs positions should be defined in advance. The electric field strength can be calculated from the system of linear equations using well known formulas for the electrostatic potential. The problem of a dielectric hollow body placed in the homogeneous transverse steady electrostatic field, x E ˆ 0 0 E  , as shown in Fig. 1, is considered in [5]. In this paper the CSM will be applied for the electric field calculation in an arbitrary shaped cavity of an arbitrary shaped dielectric body. Theoretically, the precision of the solution depends on the number and positions of the FCs. Higher precision can be achieved by increasing the number of FCs. The results of the electric field calculation based on the CSM are verified using the results from a commercial FEM program [6].


Introduction
Determination of electric field distribution inside dielectric bodies mostly requires solving the Laplace's equation.In many cases physical systems are so complex that analytical solutions are difficult or even impossible to obtain, as is mentioned in [1].Nowadays, using computer, it is possible to exploit simple numerical approximation methods and in a few minutes to solve complex field problems.
A widely used method for numerical electrostatic problems solving is the Charge simulation method (CSM), described in [2].Basic idea of this method is replacing the existing system by fictitious charges (FCs) chosen in certain way and placing them inside the respective system volumes.Unknown quantities are the charges values and their coordinates.The question remains how to form system of equations and to find these values?Several possibilities exist.One of the simplest options is to use the point matching method as shown in this paper.Other options exist, such as least -square method, variation method or boundary conditions method.
Using the point matching method a system of nonlinear equations is formed in a way to satisfy boundary conditions for the system.Choice of initial values in iterative process of the solution, based on the linearization of the non -linear equations, can be difficult.In order to eliminate this problem it is convenient to use only the fictitious charges (FCs) as the unknowns.In [3,4] authors suggest that FCs positions should be defined in advance.The electric field strength can be calculated from the system of linear equations using well known formulas for the electrostatic potential.
The problem of a dielectric hollow body placed in the homogeneous transverse steady electrostatic field, x E 0 0 E  , as shown in Fig. 1, is considered in [5].In this paper the CSM will be applied for the electric field calculation in an arbitrary shaped cavity of an arbitrary shaped dielectric body.
Theoretically, the precision of the solution depends on the number and positions of the FCs.Higher precision can be achieved by increasing the number of FCs.
The results of the electric field calculation based on the CSM are verified using the results from a commercial FEM program [6].

CSM procedure
The dielectric system was divided into three respective FCs systems.The FCs were taken as the line or point charges placed at the cylindrical or spherical surfaces, depending on the problem geometry.
For determination of the potential outside the dielectric body, a total of 1 N FCs were placed inside the body (Fig. 2).

Fig. 2. Internal system of the FCs
For determination of the potential inside the dielectric body, a total of 2 N FCs were placed outside the body and a total of 3 N FCs inside the cavity (Fig. 3).
For determination of the potential inside the cavity, a total of 4 N FCs were placed inside the body, around the cavity (Fig. 4).The electrical potential is given by where j q represents the (unknown) magnitude of the j th charge and ) , ( y x p j is the coefficient that depends only on the type of the distribution and position of the observation point ) , M( y x .The intensities of the FCs can be determined after solving the system of linear equations satisfying the boundary conditions on the system surface and on the surface which separates the layers (the dielectric layers and the cavity).
In order to demonstrate the application of the CSM procedure, several examples are presented.

Dielectric circular cylinder with an eccentric circular cavity
A dielectric circular cylinder with an eccentric circular cavity placed in a transverse electrostatic field, is analysed in [7].The cylinder has radius a , eccentricity c , and radius of the cavity b ( Fig. 5).

Fig. 5. Dielectric cylinder with circular cavity
Applying the CSM, the whole system was divided into three systems with FCs, according to the procedure described in the previous section.The FCs are line charges placed at the cylindrical surfaces with circular cross-section.The radius of the cylindrical surfaces was determined by coefficients ).
1 N fictitious charges were placed inside the body's volume, at points The electric potential outside the dielectric cylinder can be expressed as ) are the unknown line charges placed at the cylindrical surface.The coordinates ( j x 1 , are their positions and the constant . For determination of the potential inside the dielectric body and inside the cavity, the second system was formed.The potential inside this system was where ) are unknown line charges placed at the cylindrical surfaces and ( j x 2 , j y 2 ) and ( k x 3 , k y 3 ) are their positions and For third system, the electrical potential inside the cavity was where ) are unknown line charges placed at the cylindrical surface and ( j x 4 , j y 4 ) are their coor- A system of the linear equations was formed using the point matching method in a way to satisfy boundary conditions for the electrical potential and the normal component of the electric field at the system boundaries.The total number of unknowns was ) was the total number of the fictive charges per unit length in each system).After solving this system, the unknown values of the FCs were determined and the electric field could be calculated.
The electrostatic field vector components outside the body are: Inside the body the electrostatic field vector components are: The electrostatic field vector components inside the cavity are:

Results for dielectric circular cylinder with an eccentric circular cavity
A convergence of the electric field results for a different number of FCs is shown in Table 1, where . The results are given for the two characteristic observation points: the centre of the body with coordinates ( 0 , 0 ) and the centre of the cavity with normalized coordinates ( 0 , 45 .0 ).The input data are: . As the number of FCs increases, convergence of the results becomes better.But, for 150 e  N , the system of the linear equations becomes irregular because the FCs come very close to each other and a problem of division by zero appears.When 120 e  N , the results converge at 4 to 5 decimal places.
The electric field components, x E and y E , along the x -axis in the cavity, are presented in Fig. 6.From Fig. 6 it can be seen that the electric field vector is homogeneous in the cavity.The x E component is almost constant, while the value of the y E component is negligible.10 show how the boundary conditions were satisfied for the tangential components of the electric field vector at the separation surfaces air -dielectric and dielectric -cavity.
In the ideal case these components should be equal.Fig. 7 -Fig.10  and the different number of the FCs.A good satisfaction of the boundary conditions was achieved with an increased number of the FCs.However, better satisfaction of the boundary conditions was obtained when the FCs were a little further from the separating surfaces (Fig. 8 and Fig. 10).In that case the obtained accuracy was 10 -6 -10 -9 .
In Table 2, the CSM results for the electric field at the observation point M(0.4,0.0) are compared to the results obtained from a commercial FEM program [6] and COMSOL [8].The external electric field was taken to be V/m 100 0  E .The results were found to be in a very good agreement (relative error was less than 2%).

Determination of FCs optimal positions
It was evident that the positions of the FCs had a great influence on the CSM results.If the FCs were very close or too far from the cylindrical surfaces the obtained error was higher.Lately a large number of optimisation methods have been developed, such as genetic algorithm, presented in [9].In this paper, optimal positions of the FCs were determined for one simple example.An analytical solutions for the problem of a cavity in a dielectric medium in a homogeneous transverse electrostatic field, Fig. 11, obtained after solving Laplace's equation, is given in [10].The expression for the electric field inside the cavity is . Applying the CSM, the FCs will be the line charges placed at the cylindrical surfaces around the circular cross-section. .Boundary conditions for the potential and the electric field should be satisfied at the boundary surfaces as previously described.In that way a system of linear equations was formed.Solving this system, the potential and the electric field strength could be determined.Optimal positions of the FCs can be determined varying the positions of the FCs.The best results were determined by comparison with the analytical results.The electric field values for the different number of the FCs are shown in Table 3.The coefficients i f are known: , with a step 1 .0 .For every value of the coefficients i f the electric field has been calculated and compared with the analytical value.In Table 4 the optimal values of the coefficients i f are shown, when the number of the FCs was 70  N . The discrete electric field values inside the cavity for the different values of coefficients 1 f and 2 f and the different number of FCs, N , are shown in Fig. 12 and Fig. 13.plane in Fig. 13.It can be seen that for an increased number of the FCs, the electric field values are grouping around the analytical value.In Table 5 the results obtained using CSM have been compared with the FEMM and COMSOL results as well as with the analytical solution.A very good agreement was found (an error less than 1.6%).
An application of the CSM for calculation of the electrostatic field inside an arbitrary shaped hollow dielectric body is presented.The body was placed in the homogeneous transverse electrostatic field.Described procedure can be applied for calculation of the penetrated electric field in some human tissues in living and working environments.The correct choice of the FCs was found very important.The most commonly used FCs, depending on the problem geometry, are point, line and ring charges.The most important questions were: How to place the FCs, which is their optimal number and how to choose the matching points?The answers on these questions depend on the experience of the investigators.

Conclusions
The obtained results showed that the precision of the solution depends on the number of the FCs.Higher precision can be achieved by increasing the number of the FCs.The number of the FCs can't be too large because the system of the equations could be ill conditioned.In the examples presented here, a good convergence was obtained for N=(50÷80) FCs per system.A good satisfaction of the boundary conditions was accomplished with an increased number of the FCs.Also, better results were obtained when the FCs were a little further from the boundaries.In that case the obtained accuracy was 10 -6 -10 -9 .The obtained CSM results have been compared with the FEMM and COMSOL results.The results were in a good agreement, because the error was less than 2%.

Fig. 7 -
Fig. 6.Distribution of x E (a) and y E (b) electric field

Fig. 11 .
Fig. 11.Cylindrical cavity in a homogeneous dielectric medium Cylindrical surfaces radii were defined by the coefficients ) 2 , 1 (  i f i. Boundary conditions for the potential and the electric field should be satisfied at the boundary surfaces as previously described.In that way a system of linear equations was formed.Solving this system, the potential and the electric field strength could be determined.Optimal positions of the FCs can be determined varying the positions of the FCs.The best results were determined by comparison with the analytical results.The electric field values for the different number of the FCs are shown in Table3.The coefficients i

Fig. 12 .
Fig. 12. Normalized electric field discrete values inside the cavity In Fig. 12, a 3D presentation of the electric field for the different FCs number is shown.For better view the obtained solutions are presented in the 2 f E 

Fig. 13 .
Fig. 13.Normalized electric field discrete values inside the cavity

Table 2 .
Results of comparison

Table 3 .
Normalized electric field strength,

Table 4 .
FCs optimal positions, for

Table 5 .
Results of the comparison Applied method