Solution for the electric potential distribution produced by sphere-plane electrodes using the method of images

Solução para a distribuição do potencial elétrico produzido por eletrodos esfera-plano usando o método das imagens

PRODUTOS E MATERIAIS DIDÁTICOS

Solution for the electric potential distribution produced by sphere-plane electrodes using the method of images

Solução para a distribuição do potencial elétrico produzido por eletrodos esfera-plano usando o método das imagens

Fernando F. Dall'Agnol¹; Victor P. Mammana

Centro de Tecnologia da Informação Renato Archer, Rod. Dom Pedro I, Campinas, SP, Brasil

ABSTRACT

The solution for the potential distribution between a spherical and planar electrode is briefly reviewed and derived in a simple way using the method of images. An analysis of the convergence of the potential, as a function of the number of image charges accounted, is also performed. Once the potential is obtained, the evaluation of the electric field, capacitance, energy of the system and the force between the electrodes is straightforward. The solution is illustrated with pictures, schematic representations and numerical examples.

Keywords: sphere-plane potential, electric potential, method of images, sphere-plane electrodes.

RESUMO

A solução para a distribuição do potencial em uma configuração de eletrodos esfera-plano é brevemente revisada e é obtida em detalhes usando-se o método das imagens. Uma vez que o potencial é obtido, o campo elétrico, a capacitância, a energia do sistema e a força entre os eletrodos podem ser deduzidos prontamente. E feita uma análise da convergência do potencial, como função do número de imagens consideradas no cálculo. A solução é ilustrada com figuras, representações esquemáticas e exemplos numéricos.

Palavras-chave: potencial esfera-plano, potencial elétrico, método das imagens, eletrodos esfera-plano.

1. Introduction

The solution of the potential distribution between a spherical and planar electrode constitutes an academic problem, which was already described by physicists almost 150 years ago. Nevertheless, it is not easy to find a reference with a solution that is immediately applicable. Although many authors, dealing with electrostatics, refer to the classical textbook by Smythe, unfortunately, this book does not present the potential distribution in a simple way [1].

The idea of image charge for field problems is due to Lord Kelvin, but Maxwell [2], Lodge [3], and Searle [?] extended the scope of the method. An excellent discussion on the distribution of the potential as a function of the system's coordinates was given by A. Foster in his PhD thesis [4] for a point charge between a sphere-plane capacitor. Foster uses the method of images to calculate the contribution of a point charge to the electric potential of the system.

Theoretical models for the electric potential distribution in the space between electrodes are useful for the calculation of the trajectories of charged particles and the prediction of flashover voltages over a given range of field conditions. Experiments based on high-voltage breakdown tests play an important rule in electrical engineering education. Real laboratory experiments designed to determine the voltage at breakdown have been combined with computer-based simulations and have produced stimulating teaching experiences. The experience gained with such a combination of teaching procedures is presented in the work of Lowther and Freeman [5]. The work of J.H. Cloete and J. van der Merwe [6] also describes an experiment for the determination of the voltage at breakdown by slowly decreasing the spacing between two conducting spherical electrodes. Their paper gives a detailed explanation on how to apply the method of images to model this practical problem. For many electrostatic systems, the method of images provides a simple solution, when solving Laplace equation would be very complicated, as is the case of the sphere-plane system of electrodes.

In this paper we also use the method of images to obtain an analytical solution for the potential distribution in the space between a conducting sphere and a plane electrode. Our goals are to familiarize students with this method by solving the sphere-plane electrostatic problem and to present easy to use equations for the potential and the electric field. Once the potential distribution is obtained, the electric field distribution and system parameters like capacitance, stored energy and force between the electrodes can be deduced straightforwardly.

2. Method of image charge

Almost without exception, old and new textbooks on applied electromagnetism discuss the method of images, and this clearly shows the importance of the method for science and engineering. According to Binns, Lawrenson and Trowbridge [7], the essence of the method of images consists in replacing the effects of a boundary related to an applied field by distributions of charges "behind" the boundary line as illustrated in Fig. 1 for two configurations that will be combined to deduce the potential for sphere-plane electrodes. The first configuration consists of a point charge and a grounded conducting infinite plane and this is shown in Fig. 1(a). The second configuration consists of a point charge and a grounded conducting sphere shown in Fig. 1(b). The solid and dashed lines represent the real and virtual fields respectively. The point from where the dashed lines diverge is the image charge. The field pattern, and consequently, the potential distribution for the real electrodes system is equivalent as the one generated by two point charges. The advantage of the image method is that the evaluation of the potential with point charges is simple and straightforward.

A schematic representation of the sphere-plane electrode system is presented in Fig. 2. The two electrodes are subject to a potential difference V and the derivation is presented in cylindrical coordinates. According to the chosen reference frame, the conducting plane is placed at the coordinate z= 0 and its electric potential _p is set to zero. The electric potential _s at the spherical electrode is V , being the potential difference between the spherical and the planar electrode. The sphere has radius a and its center is placed at z = z₀. The minimum distance d between the sphere and the grounded plane is d = z₀ – a.

To find the potential for this system using image charges, we start considering an isolated sphere with charge q₀, which generates the potential V at the surface. Then, q₀ can be expressed in terms of its resulting potential V as

where k =9 × 10⁹ Vm/C is the electrostatic constant. In the presence of the plane at z = 0, charge q₀ generates an image of same magnitude and opposite sign –q₀ at position -z₀. The image charge –q₀ also generates an image in the sphere with position and magnitude given by

This situation is depicted in Fig. 3(a). The image charge q₁, in turn, generates -q₁ at the plane, which generates q₂ in the sphere and so on. Fig. 3(b) represents the final charge distribution. The position and magnitude of the i^th image charge are given by the recurrent relations

The derivation of Eqs. (2) to (5), which can be found in many textbooks, will not be repeated here.

It is convenient to define a normalized charge ξ_i = q_i/q₀, that will be used later. Dividing Eq. (5) by q₀ we get

with ξ₀ = 1.

Charge distributions for three configurations of the problem are shown in Table 1. The results show the effect of the minimum distance d between the electrodes on the charge distribution in the spherical domain. The first six values of z_i and ξ_i for each d are also presented. It can be noticed from this table that ξ_i → 0 and z_i → z∞ (constant) as i → ∞. In these pictures the relative magnitude of the charges is represented in a gray scale from black, for ξ₀ (unity), to white, for ξ_∞ (zero). In fact, these charges have images beneath the plane boundary, but they are not shown for the sake of simplicity. The position and magnitude of these image points are -z_i and -q_i. The potential for the sphere-plane is equivalent of the potential of these two groups of charges {q_i, –q_i}.

3. Solution for the potential, the electric field and other physical quantities

The potential due to a charge of index i in the sphere and its image in the plane is given by

The potential due to all charges is completely determined by summing

_i and using Eqs. (1) and (5) in Eq. (7), results in

It is worth to point out that Eq. (8) is a solution for point charges, which is not the actual system of a sphere-plane. The solution in Eq. (8) is equivalent to the solution of the sphere-plane system only for (z – z₀)² + r²> a² (outside the sphere) and for z > 0 (above the plane). For the region inside the sphere = V at any point and beneath the plane = 0 at any point. To account for these regions Eq. (8) has to be redefined as

Once is given, electric field, capacitance, electrostatic energy and force on the electrodes can be obtained promptly. The electric field can be obtained from E(r, z)=-∇ and can be written as

where E_r and E_z are the components of the field in r and z directions respectively, so that E(r, z)= E_r + E_z . Fig. 4 shows the potential distribution as a density plot and the electric field lines. Details on the procedure to obtain these figures are shown in ^{Appendix B}.

The sphere-plane electrode system is a capacitor of which the total charge q is the sum of all charges in the sphere, given by

From the sequences of ξ_i in Table 1 it can be seen that the larger d is the faster ξ_i converges. So the number of terms to be summed for q in Eq. (12), to reach a specified accuracy, depends on d. A convergence analysis of q and of is presented in the next section.

The capacitance is obtained from C = q/V . Combining Eqs. (1) and (12) results in

The electrostatic energy U = ¹/₂CV² is given by

and the force between sphere and plane F = – ∂U/∂z₀ is

The prime mark indicates a differentiation with respect to z₀. In this way, differentiation of (6) leads to the following recurrent relation to compute

where is given by

The starting conditions for the recurrent relations above are = 0 and = 1. All formulas are summarized in ^{Appendix A}. The treatment given above can also be applied to two spheres. If the spheres have the same size the formulas for the potential distribution are identical to Eq. (8). In this case the symmetry plane of the sphere-sphere problem becomes the plane of the sphere-plane system.

4. Convergence analysis of charge and potential

The number of terms needed for the total charge q in Eq. (12) to converge depends only on the ratio z₀/a. Table 2 shows the number of terms needed for q to converge to 99.9 % of its asymptotic value as a function of z₀/a.

An analysis of the convergence of may be important as, for example, in fitting processes, where fitting parameters must be varied, then evaluated, and compared to experimental results repeatedly. A fitting procedure may take a lot of time and one must avoid computing unnecessary terms of . This analysis aims to give a good insight about the convergence behavior, i.e. where the convergence is faster/lower, how the convergence varies with z₀/a. In Fig. 5 we present the convergence behavior for two cases of z₀/a. In this figure the numbers represent the number of terms needed for in Eq. (8) to converge to 99.9 % of the asymptotic value at that point of the space. It can be seen that the region under the sphere needs more terms to converge. The number of terms needed is, at most, up to 50 even for z₀/a as small as 1.01. For much different cases than the examples used here, like: much higher accuracy, much wider range of the coordinates, ratio z₀/a much closer to unity, etc, requires an analysis for these specific cases. The asymptotic values of q and are taken with 100 terms in their sum.

5. Conclusion

We developed the solution for the potential, electric field, capacitance, energy and force for electrodes in a sphere-plane configuration using the method of image charges. Our equation, represented in a cylindrical coordinate system, can easily and conveniently be applied to systems with plane-sphere geometry. The methodology for solving this problem may be used in several other problems of electrostatics. We are confident that our treatment can conveniently be applied by students and engineers working on electrostatics.

Acknowledgments

We greatly acknowledge the help of Dr. Pablo Paredez, Ms. Aline Marque and Dr. Daniel den Engelsen for the help in preparing this manuscript. Authors are grateful to the Brazilian funding agencies CNPq and Fapesp for financial support.

Recebido em 26/9/2008

Revisado em 11/3/2009

Aceito em 24/4/2009

Publicado em 23/9/2009

Once the equations for the potential and the electric fields are established their corresponding graphics can be rendered for a visual presentation of the solution. This is not a trivial procedure. Many high level program languages are available for algebraic manipulation and graphics. In this section we will line out a procedure in Mathematica® [8]- a software with a very high level programming language- that we used to render the figures shown in this article. Next sections will provide an overall procedure to do the programs in Mathematica and will present the programs used.

B1. Image charges in the sphere

The figures shown in Fig. 3 and Table 1 was obtained as follows:

1. Numerical values are attributed to z₀, a.

2. Normalized charges ξ_i and their positions z_i are calculated for i from 0 to 100, which is more than sufficient.

3. The representation of the sphere, the plane and the points (0,z_i) are drawn using built-in commands Circle[...], Line[...] and Point[...] respectively. The points are rendered in gray level using command GrayLevel[...].

4. Graphics are merged and rendered.

The program used is shown below. An animation can be found in the internet with URL given in Ref. [9].

Clear[i,a,z,zi,r,z0]; "Clear parameters attributions";

a=10^-3; "radius of the sphere";

d=10^-5; "distance sphere-plane";

z0=a+d; "z coordinate of the sphere center";

nter=100; "number of image charges to compute";

zi[0]=z0; "position of first charge";

xsi[0]=1; "magnitude of first charge";

Do[

zi[i]=z0-a^2/(z0+zi[i-1]); "z positions of image charges";

xsi[i]=a xsi[i-1]/(z0+zi[i]); "normalized magnitude of image charges";

,{i,nter}];

ImageCharges= Table[{PointSize[0.02], GrayLevel[1-xsi[i]], Point[{0,zi[i]}]}, {i,nter,0,-1}];

"Image charges characteristics to be drawn as points";

G=Graphics[{Circle[{0,z0},a], Line[{{-a,0},{a,0}}], ImageCharges}, AspectRatio-> 1, PlotRange->All]; "Draw of the circle and the line and the points";

Show[G, AspectRatio->(z0+a)/(2a)];

"Render the circle, the line and the points in the same graphic";

B2. Density plot of the potential

To render Fig. 4(a):

1. We attributed numbers to parameters a and z₀.

2. We computed z_i and ξ_i for i from 0 to 100.

3. We wrote the potential as in Eq. (9) like = If[condition, then, else], where condition is (z – z₀)² + r²> a², block then must be the expression for outside the sphere and block else is the expression for inside the sphere (V).

4. We rendered the potential with built-in command DensityPlot[(r,z)].

The program used is shown below. See also URL in Ref. [10].

Clear[i,a,d,z0,nter,zi,xsi,Fi]; "Clear all variables that will be used";

a=0.001; "radius of the sphere";

d=0.0001; "distance sphere-plane";

z0=a+d; "z coordinate of the center of the sphere";

nter=100; "number of terms to be computed in the potential";

V=1; "potential at the sphere";

zi[0]=z0; "position of the i(th) image charge";

xsi[0]=1; "relative magnitude of the i(th) image charge";

Do[

zi[i]=z0-a^2/(z0+zi[i-1]); "z position of the image charges";

xsi[i]=a xsi[i-1]/(z0+zi[i-1]); "normalized magnitude of the image charges";

,{i,nter}];

Fi=If[(z-z0)^2+r^2>=a^2,

a V Sum[xsi[i]/Sqrt[(z-zi[i])^2+r^2]-xsi[i]/Sqrt[(z+zi[i])^2+r^2], {i,0,nter}], V]; "Attribution of a function to the potential outside and inside the sphere.";

DensityPlot[Fi, {r,-2a,2a}, {z,0,4a}, Mesh-> False, Frame-> False, PlotPoints-> 200, AspectRatio-> 1];

"Render the potential as a Density plot.";

B3. Field lines between the electrodes

To render Fig. 4(b):

1. We attributed numerical values to a ,z₀ and few other parameters used.

2. We computed z_i and ξ_i for i from 0 to 100.

3. We attributed expressions for the electric field functions E_r and E_z as in Eqs. (10) and (11).

4. Points on the circle that represents the sphere where selected as starting points of the electric field line. The starting point of each electric field line is not equally spaced from each other. Their spacing is proportional to the strength of the electric field, relative to the maximum field at the bottom of the sphere. This is done so the reader can visualize the field strength by the density of the field lines.

5. The field line is built, point by point, using the coordinate of the previous point plus a constant step toward the field direction to calculate the next point of the field line. New points are calculated until the field line reaches the border of the region to be shown in the graphic. The points are stored in a list and plotted.

The program is shown below. See also Ref. [11]. Fig. 1(a) and (b) are particular cases of this program and will not be shown here. Animations of Fig. 1(a) can be seen in Ref. [12].

"Clear parameters";

Clear[a,d,z0,V,nter,zi,xsi,nlines,del,i,ii,j,k,r,z,rfl,zfl,fieldline,Er,Ez,Emod,tet,pretet];

a=0.001;"Sphere radius";

d=0.0001;"distance sphere-plane";

z0=a+d;"z coordinate of the center of the sphere";

V=1;"Voltage at the sphere";

zi[0]=z0;"z positions of the initial charge";

xsi[0]=1;"normalized magnitude of the initial charge";

nter=100;"number of terms to compute the electric field";

nlines=50;"number of field lines to draw";

del=a/100.;"step distance between two points in a field line";

Do[fieldline[j]={},{j,nlines}];"list to store the points of the k(th) field line ";

Do[

zi[i]=z0-a^2/(z0+zi[i-1]);"z positions of the image charges";

xsi[i]=a xsi[i-1]/ (z0+zi[i-1]);

"normalized magnitude of the image charges";

,{i,nter}];

Er=a V r Sum[xsi[i]/((z-zi[i])^2+r^2)^(3/2)-xsi[i]/((z+zi[i])^2+r^2)^(3/2),{i,0,nter}];

"r component of the electric field";

Ez=a V Sum[xsi[i](z-zi[i])/((z-zi[i])^2+r^2)^(3/2)-xsi[i](z+zi[i])/

((z+zi[i])^2+r^2)^(3/2),{i,0,nter}];

"z component of the electric field";

Emod=Sqrt[Er^2+Ez^2]; "module of the electric field";

Do[

pretet=Abs[1/Emod]/.{r->a Sin[2 Pi k/nlines],z->z0-a Cos[2 Pi k/nlines]};

"pretet is auxiliary to determine the points on the sphere at which a field line will depart";

tet=2Pi Sum[pretet,{k,j}]/Sum[pretet,{k,nlines}];

"tet is the angle between z-axes and the point at the sphere at which a field line will depart";

rfl[0]=a Sin[tet];"initial r coordinate of the j(th) field line";

zfl[0]=z0-a Cos[tet];"initial z coordinate of the j(th) field line";

"-------------Block to build the j(th) field line------------------";

ii=0;"couter";

While[rfl[ii]\[GreaterEqual]-2a && rfl[ii]<=2a && zfl[ii]>=0 && zfl[ii]<=4a,

ii++;"Increment ii";

rfl[ii]=(r+Er/Emod del)/.{r->rfl[ii-1],z->zfl[ii-1]};

"r coordinate of the ii(th) point of the j(th) field line";

zfl[ii]=(z+Ez/Emod del)/.{r->rfl[ii-1],z->zfl[ii-1]};

"z coordinate of the ii(th) point of the j(th) field line";

AppendTo[fieldline[j],{rfl[ii],zfl[ii]}];

"Append last calculated point (rfl,zfl) to the j(th) field line";

];

"-------------------------------------------------------------------";

P[j]=ListPlot[fieldline[j],PlotJoined->True,PlotRange->{{-2a,2a},{0,4a}},

DisplayFunction->Identity];

,{j,nlines}];"Store the graphics of the j(th) field line in P[j]";

Plist=Table[P[j],{j,nlines}];"Does a list with all field lines";

G=Graphics[{RGBColor[1,0,0], Circle[{0,z0},a]}];

"Store in G the graphics of the field lines and the sphere";

Show[{Plist,G},Ticks->None,Axes->{True,False},AspectRatio->1, DisplayFunction-> $DisplayFunction];

"Render the field lines and the sphere together";

"--------------------------------------------------";

• [1] W.R. Smythe, Static and Dynamic Electricity (McGraw-Hill, New York, 1950), 2nd ed.
• [2] J.C. Maxwell, A Treatise in Electricity and Magnetism (Dover, New York, 1958), v. 1 3rd ed.
• [3] O.J. Lodge, Proc. Phys. Soc. London. 2, 24 (1875).
• [4] A. Foster, Theoretical Modeling of Non-contact Atomic Force Microscopy on Insulators PhD Thesis, University College London, 2000. Available at http://www.fyslab.hut.fi/~asf/physics/thesis1/thesis1.html, in 01/09/2009.
• [5] D.A. Lowther and E.M. Freeman, IEEE Trans. Educ. 36, 219 (1993).
• [6] J.H. Cloete and J. van der Merwe, IEEE Trans. Educ. 41, 141 (1998).
• [7] K.J. Binns, P.J. Lawrenson and C.W. Trowbridge, The Analytical and Numerical Solution of Electric and Magnetic Fields (John Wiley & Sons, Chichester, 1992), p. 93.
• [8] Available at http://en.wikipedia.org/wiki/Mathematica_(programming_language)
  » link
• [9] Available at http://www.youtube.com/watch?v=mnTXosm9VRY See also http://www.youtube.com/watch?v=DBSjvGSE8tk, in 02/10/2009.
  » link
• [10] Available at http://www.youtube.com/watch?v=rpBDLtzt_VI, in 02/10/2009.
  » link
• [11] Available at http://www.youtube.com/watch?v=ZHVpaqPVuBU, in 02/10/2009.
  » link
• [12] Available at http://www.youtube.com/watch?v=HmZjHGG_4cQ, in 02/10/2009.
  » link

Appendix A:  Formulary

^{Click to enlarge}

Appendix B:  Algorithms and programs for rendering graphics

Publication Dates

• Publication in this collection
  02 Oct 2009
• Date of issue
  Sept 2009

History

• Accepted
  24 Apr 2009
• Reviewed
  11 Mar 2009
• Received
  26 Sept 2008