ISSN 1063-7842, Technical Physics, 2008, Vol. 53, No. 8, pp. 1086–1090. © Pleiades Publishing, Ltd., 2008.Original Russian Text © E.M. Vinogradova, N.V. Egorov, K.A. Krimskaya, 2008, published in Zhurnal Tekhnichesko

œ

Fiziki, 2008, Vol. 78, No. 8, pp. 128–131.

1086

FORMULATION OF THE PROBLEM

Let us consider the axisymmetric problem of deter-mining the electrostatic potential of a system consistingof an arbitrary number of nonconcentric spherical seg-ments located between two spheres.

This axisymmetric problem will be solved in abispherical system of coordinates. The relation betweenbispherical coordinates (

α

,

β

) with cylindrical coordi-nates (

r

,

z

) has the form [1]

Coordinate surfaces

β

= const define a family ofnonintersecting nonconcentric spheres, which aredescribed by the following equation:

Coordination surfaces

α

= const define a family ofspindle-shaped surfaces of revolution, which areorthogonal to surfaces

β

= const and are defined by theequation

The parameters of the problem are as follows:

N

isthe number of segments;

α

i

≤

α

≤

π

,

β

=

β

i

are the sur-

faces of segments (

i

= );

f

i

(

α

) are the first-kindboundary conditions on corresponding segments (

i

=

); 0

≤

α

≤

π

,

β

=

β

0

,

β

=

β

N

+ 1

are the surfaces ofthe spheres; and

f

0

(

α

),

f

N

+ 1

(

α

) are the first-kind bound-ary conditions on the spheres.

z ir+ icα iβ+

2---------------,cot=

0 α π, ∞ β ∞< <– .≤ ≤

r2 z c βcoth–( )2+c

βsinh--------------⎝ ⎠

⎛ ⎞2

.=

r c αcot–( )2 z2+c

αsin-----------⎝ ⎠

⎛ ⎞2

.=

1 N,

1 N,

Without loss of generality, we can set

β

i

<

β

k

for

i

<

k

.If product

β

0

β

N

+ 1

is zero, one of the spheres opens toform a plane

β

= 0. When

β

0

β

N

+ 1

> 0, the system of seg-

Calculation of the Electrostatic Field of a System of Spherical Segments

E. M. Vinogradova, N. V. Egorov, and K. A. Krimskaya

St. Petersburg State University, St. Petersburg, 198904 Russiae-mail: vincat@rednet.ru

Received November 26, 2007

Abstract

—The electrostatic potential distribution is determined for a system of axisymmetric electrodes in theform of nonconcentric spherical segments.

PACS numbers: 41.20.Cv, 41.85.Ne

DOI:

10.1134/S1063784208080185

SHORTCOMMUNICATIONS

r

c

z

α

=

α

k

β

=

β

k

β

=

β

N

β

=

β

N

+ 1

β

=

β

0

> 0

… …

Fig. 1.

r

c

z

c

……

α

=

α

k

β

=

β

k

> 0

β

=

β

N

> 0

β

=

β

N + 1

> 0

β

=

β

1

< 0

β

=

β

0

< 0

Fig. 2.

TECHNICAL PHYSICS

Vol. 53

No. 8

2008

CALCULATION OF THE ELECTROSTATIC FIELD 1087

ments is within one of the closed spheres, but outsidethe other sphere (see Fig. 1). If

β

0

β

N

+ 1

< 0, the seg-ments lie outside the spheres (Fig. 2).

The distribution of electrostatic potential

U

(

α, β)satisfies the Laplace equation and the following bound-ary conditions:

(1)

SOLUTION OF THE PROBLEM

To solve boundary-value problem (1), we divide theentire region occupied by the electron-optical systeminto N + 1 subregions: βi ≤ β ≤ βi + 1. For each subregion,we can present the potential distribution U(α, β) =

Ui(α, β) (i = ) in the form of the following expan-sion in Legendre polynomials [1, 2]:

(2)

Coefficients A0, n, AN + 1, n can be calculated fromboundary conditions (1) on spheres β0, βN + 1:

(3)

The representation of potential in form (2) satisfiesthe Laplace equation and the potential continuity con-ditions at the interfaces between the regions. Theboundary conditions and the continuity conditions forthe derivative of the potential along the normal to theinterfaces between subregions lead to the system ofpaired equations

∆U α β,( ) 0=

U α βi,( ) αi α π≤ ≤ f i α( ), i 1 N,= =

U α β0,( ) f 0 α( )=

U α βN 1+,( ) f N 1+ α( ).=⎩⎪⎪⎨⎪⎪⎧

0 N,

Ui α β,( ) βcosh αcos–=

×n

12---+⎝ ⎠

⎛ ⎞ β βi–( )sinh

n12---+⎝ ⎠

⎛ ⎞ βi 1+ βi–( )sinh

-----------------------------------------------------Ai n,

n 0=

∞

∑

+

n12---+⎝ ⎠

⎛ ⎞ βi 1+ β–( )sinh

n12---+⎝ ⎠

⎛ ⎞sinh βi 1+ βi–( )-------------------------------------------------------Ai 1+ n, Pn cos α,( ),

βi β βi 1+ , i< < 1 N, .=

Ak n,2n 1+

2---------------

f k α( )βk αcos–cosh

----------------------------------------Pn αcos( ) αsin α,d

0

π

∫=

k 0, N 1.+=

(4)

Let us introduce the following notation:

(5)

Equations (5) form a system of linear algebraic equa-

tions in coefficients Ai, n (i = ). A similar system was

n12---+⎝ ⎠

⎛ ⎞ 1

n12---+⎝ ⎠

⎛ ⎞ βi βi 1––( )sinh

-----------------------------------------------------Ai 1– n,–n 0=

∞

∑

+ n12---+⎝ ⎠

⎛ ⎞ βi βi 1––( ) n12---+⎝ ⎠

⎛ ⎞ βi 1+ βi–( )coth+coth⎝ ⎠⎛ ⎞

× Ai n,1

n12---+⎝ ⎠

⎛ ⎞ βi 1+ βi–( )sinh

-----------------------------------------------------Ai 1+ n, Pn αcos( )– = 0,

0 α αi,≤ ≤

Ai n, Pn αcos( )n 0=

∞

∑

= f i α( )

βicosh αcos–---------------------------------------Pn αcos( ) αdαsin ,

αi α π, i≤ ≤ 1 N, .=⎩⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎧

C1 n, = n12---+⎝ ⎠

⎛ ⎞ β1 β0–( )coth n12---+⎝ ⎠

⎛ ⎞ β2 β1–( )coth+⎝ ⎠⎛ ⎞

× A1 n,1

n12---+⎝ ⎠

⎛ ⎞ β2 β1–( )sinh

--------------------------------------------------A2 n, ,–

Ci n,1

n12---+⎝ ⎠

⎛ ⎞ βi βi 1––( )sinh

-----------------------------------------------------Ai 1– n,–=

+ n12---+⎝ ⎠

⎛ ⎞ βi βi 1––( )coth n12---+⎝ ⎠

⎛ ⎞ βi 1+ βi–( )coth+⎝ ⎠⎛ ⎞

× Ai n,1

n12---+⎝ ⎠

⎛ ⎞ βi 1+ βi–( )sinh

-----------------------------------------------------Ai 1+ n, , i– 2 N 1–, ,=

CN n,1

n12---+⎝ ⎠

⎛ ⎞ βN βN 1––( )sinh

---------------------------------------------------------AN 1– n,–=

+ n12---+⎝ ⎠

⎛ ⎞ βN βN 1––( )coth⎝⎛

+ n12---+⎝ ⎠

⎛ ⎞ βN 1+ βN–( )coth ⎠⎞ AN n, .

1 N,

1088

TECHNICAL PHYSICS Vol. 53 No. 8 2008

VINOGRADOVA et al.

solved in [3] for a particular case β0 = –∞, βN + 1 = ∞. In thegeneral case [4], having solved system (5), we obtain

(6)

For convenience of subsequent calculations, weintroduce new coefficients Bk, n:

(7)

Then system (4) of paired summatory equations canbe reduced to the form

(8)

where

Ai n,

n12---+⎝ ⎠

⎛ ⎞ βN 1+ βi–( )sinh

n12---+⎝ ⎠

⎛ ⎞ βN 1+ β0–( )sinh

--------------------------------------------------------=

× n12---+⎝ ⎠

⎛ ⎞ βk β0–( )Ck n,sinhk 1=

i

∑

+

n12---+⎝ ⎠

⎛ ⎞ βi β0–( )sinh

n12---+⎝ ⎠

⎛ ⎞ βN 1+ β0–( )sinh

---------------------------------------------------------

× n12---+⎝ ⎠

⎛ ⎞ βk β0–( )Ck n, ,sinhk i 1+=

N

∑i 1 N, .=

B1 n,1

n12---+⎝ ⎠

⎛ ⎞ β1 β0–( )sinh

--------------------------------------------------– C1 n, ,+=

Bi n, Ci n, , i 2 N 1–, ,= =

BN n,1

n12---+⎝ ⎠

⎛ ⎞ βN 1+ βN–( )sinh

---------------------------------------------------------- CN n, .+–=

n12---+⎝ ⎠

⎛ ⎞ Bi n, Pn αcos( )n 0=

∞

∑ 0, 0 α αi,≤ ≤=

gikn Bk n,

k 1=

i 1–

∑ 12--- gii

n+⎝ ⎠⎛ ⎞ Bi n,+

n 0=

∞

∑

+ gkin Bk n,

k i 1+=

N

∑ Pn αcos( ) Fi α( ),=

αi α π,≤ ≤⎩⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎧

giin 1

2--- e

– n12---+⎝ ⎠

⎛ ⎞ βN 1+ βi–( )

n12---+⎝ ⎠

⎛ ⎞ βi β0–( )sinh–=

(9)

(10)

(11)

(12)

+ e– n

12---+⎝ ⎠

⎛ ⎞ βi β0–( )

n12---+⎝ ⎠

⎛ ⎞ βN 1+ βi–( )sinh

× 1

n12---+⎝ ⎠

⎛ ⎞ βN 1+ β0–( )sinh

---------------------------------------------------------,

gikn

n12---+⎝ ⎠

⎛ ⎞ βN 1+ βi–( ) n12---+⎝ ⎠

⎛ ⎞ βk β0–( )sinhsinh

n12---+⎝ ⎠

⎛ ⎞sinh βN 1+ β0–( )---------------------------------------------------------------------------------------------------------,–=

k i,<

gkin

n12---+⎝ ⎠

⎛ ⎞ βN 1+ βk–( ) n12---+⎝ ⎠

⎛ ⎞ βi β0–( )sinhsinh

n12---+⎝ ⎠

⎛ ⎞sinh βN 1+ β0–( )---------------------------------------------------------------------------------------------------------,–=

i k,<

F1 α( )f 1 α( )β1cosh αcos–

----------------------------------------=

–12---

n12---+⎝ ⎠

⎛ ⎞ β1 β0–( )en

12---+⎝ ⎠

⎛ ⎞ βN 1+ β1–( )–

sinh

n12---+⎝ ⎠

⎛ ⎞ βN 1+ β0–( )sinh

------------------------------------------------------------------------------------

⎝⎜⎜⎜⎜⎛

–n 0=

∞

∑

+

n12---+⎝ ⎠

⎛ ⎞ βN 1+ β1–( )en

12---+⎝ ⎠

⎛ ⎞ β1 β0–( )–

sinh

n12---+⎝ ⎠

⎛ ⎞ βN 1+ β0–( )sinh

--------------------------------------------------------------------------------------

⎠⎟⎟⎟⎟⎞

A0 n,

+

n12---+⎝ ⎠

⎛ ⎞ β1 β0–( )sinh

n12---+⎝ ⎠

⎛ ⎞ βN 1+ β0–( )sinh

--------------------------------------------------------AN 1+ n, Pn αcos( ),

Fi α( )f i α( )βicosh αcos–

---------------------------------------=

–

n12---+⎝ ⎠

⎛ ⎞ βN 1+ βi–( )sinh

n12---+⎝ ⎠

⎛ ⎞ βN 1+ β0–( )sinh

--------------------------------------------------------A0 n,

⎝⎜⎜⎜⎛

n 0=

∞

∑

+

n12---+⎝ ⎠

⎛ ⎞ βi β0–( )sinh

n12---+⎝ ⎠

⎛ ⎞ βN 1+ β0–( )sinh

--------------------------------------------------------AN 1+ n,

⎠⎟⎟⎟⎞

Pn αcos( ),

i 1 N, ,=

TECHNICAL PHYSICS Vol. 53 No. 8 2008

CALCULATION OF THE ELECTROSTATIC FIELD 1089

To solve system (8) of paired equations, we intro-duce new unknown functions φi(t) instead of coeffi-cients Bi, n with the help of the substitution

(13)

With the help of this substitution, the second equa-tions from system (8) of paired equations are trans-formed into identity [1]. Carrying out substitution (13)in the first equations in system (8), we obtain a systemof Fredholm coupled integral equations of the secondkind in functions φi(t):

(14)

(15)

Kernels Kik(x, t) of Eqs. (14) are symmetric and canbe written in the explicit form,

(16)

Thus, to solve boundary-value problem (1), we mustsolve system (14) of second-kind Fredholm equationswith right-hand sides defined by formulas (12) and (15)and with symmetric kernels (16). Consequently, formu-

FN α( )f N α( )βNcosh αcos–

-----------------------------------------=

–

n12---+⎝ ⎠

⎛ ⎞ βN 1+ βN–( )sinh

n12---+⎝ ⎠

⎛ ⎞ βN 1+ β0–( )sinh

---------------------------------------------------------A0 n,

n 0=

∞

∑

–12---

n12---+⎝ ⎠

⎛ ⎞ βN β0–( )en

12---+⎝ ⎠

⎛ ⎞ βN 1+ βN–( )–

sinh

n12---+⎝ ⎠

⎛ ⎞ βN 1+ β0–( )sinh

--------------------------------------------------------------------------------------

⎝⎜⎜⎜⎜⎛

+

n12---+⎝ ⎠

⎛ ⎞ βN 1+ βN–( )en

12---+⎝ ⎠

⎛ ⎞ βN β0–( )–

sinh

n12---+⎝ ⎠

⎛ ⎞ βN 1+ β0–( )sinh

----------------------------------------------------------------------------------------

⎠⎟⎟⎟⎟⎞

AN 1+ n,

× Pn αcos( ).

Bi n, φi t( ) n12---+⎝ ⎠

⎛ ⎞ tsin t.d

αi

π

∫=

12---φi x( ) 2

π--- φk T( )Kik x t,( ) td

αk

π

∫k 1=

N

∑+ Φi x( ),=

αi x π,< <

Φi x( ) 2π--- d

dx------

Fi α( ) αsin

2 xcosh αcos–( )--------------------------------------------- α,d

x

π

∫=

i 1 N, .=

Kik x t,( ) gi k,n n

12---+⎝ ⎠

⎛ ⎞ x n12---+⎝ ⎠

⎛ ⎞ t.sinsinn 0=

∞

∑=

las (2), (3), (6), (7), and (13) define an analytic solutionin the entire domain of the initial boundary-value prob-lem on electrostatic potential distribution.

THE CASE OF CONSTANT VALUESOF POTENTIAL AT THE ELECTRODES

If constant values of potentials fi(α) = fi = const (i =

) are preset at all electrodes of the system,coefficients A0, n and AN + 1, n defined by formulas (3)have the form

In this case, the right-hand sides of system (14) canbe calculated explicitly:

If one of the spheres in the initial system of electrodesis missing (this corresponds to β0 = –∞ or βN + 1 = ∞), for-mulas (15) and (16) for calculating the kernels andright-hand sides of the system of coupled second-kindFredholm integral equations (14) are simplified consid-erably.

Let us suppose that βN + 1 = ∞; in this case, fromEqs. (9)–(11), we obtain

We introduce the following notation:

0 N 1+,

Ak n, 2 f ken

12---+⎝ ⎠

⎛ ⎞ βk–

, k 0 N 1.+,= =

Φi x( ) 2π--- 1

2--- f i

x2---

βi

2----coshsin

βi

2----cosh x

2---cos–

----------------------------------–

⎝⎜⎜⎜⎛

=

+ 2

n12---+⎝ ⎠

⎛ ⎞ βN 1+ βi–( )sinh

n12---+⎝ ⎠

⎛ ⎞ βN 1+ β0–( )sinh

--------------------------------------------------------- f 0en

12---+⎝ ⎠

⎛ ⎞ β0–

⎝⎜⎜⎜⎛

n 0=

∞

∑

+

n12---+⎝ ⎠

⎛ ⎞ βi β0–( )sinh

n12---+⎝ ⎠

⎛ ⎞ βN 1+ β0–( )sinh

-------------------------------------------------------- f N 1+ en

12---+⎝ ⎠

⎛ ⎞ βN 1+–

⎠⎟⎟⎟⎞

---

------× n

12---x+⎝ ⎠

⎛ ⎞sin

⎠⎟⎟⎟⎞

, i 1 N, .=

giin 1

2---e

2n 1+( ) βi β0–( )–,–=

gikn 1

2--- e

n12---+⎝ ⎠

⎛ ⎞ βi βk––

en

12---+⎝ ⎠

⎛ ⎞ βi β0–( ) βk β0–( )+( )–

–⎝ ⎠⎜ ⎟⎛ ⎞

, k i.≠–=

µik βi βk– , νik βi β0–( ) βk β0–( ).+= =

1090

TECHNICAL PHYSICS Vol. 53 No. 8 2008

VINOGRADOVA et al.

Using summation of trigonometric series [5], wecan present kernels Kik(x, t) and right-hand sides Φi(x)of system (14), which are defined by formulas (15)and (16), in the form

If both spheres are absent in the system of elec-trodes, then β0 = –∞ and βN + 1 = ∞. In this case, formu-

las (15) and (16) for calculating the kernels and right-hand sides of system (14) have the form

CONCLUSIONSWe have found the electrostatic potential distribu-

tion in a system containing an arbitrary number of axi-symmetric electrodes in the form of nonconcentricspherical segments located between two closed spheri-cal surfaces. The problem of determining the unknowncoefficients in the expansion of the potential is reducedto the solution of the system of coupled second-kindFredholm integral equations. All geometrical sizes andpotentials of the segments and spheres are parametersof the problem. Such systems can be used in modelingan electron gun with a field tip. The determination ofthe potential distribution in such systems is the mostcomplicated part of the problem since the geometricalsizes of the electrodes differ by several orders of mag-nitude, which complicates the application of numericalmethods of calculation.

REFERENCES1. Ya. S. Uflyand, Method of Paired Integral Equations in

Problems of Mathematical Physics (Nauka, Leningrad,1977) [in Russian].

2. N. V. Egorov and E. M. Vinogradova, Vacuum 72, 103(2004).

3. Yu. N. Kuz’kin, Zh. Tekh. Fiz. 40, 2276 (1970) [Sov.Phys. Tech. Phys. 15, 1777 (1970)].

4. E. M. Vinogradova, N. V. Egorov, and R. Yu. Baranov,Radiotekh. Elektron. (Moscow) 45, 638 (2007).

5. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals,Series, and Products (Nauka, Moscow, 1971; Academic,New York, 1980).

Translated by N. Wadhwa

Kik x t,( )14---

12---µiksinh

µiksinh x t–( )cos–-----------------------------------------------

⎝⎜⎜⎛

=

–

12---νiksinh

νikcosh x t–( )cos–------------------------------------------------

⎠⎟⎟⎞

12--- x t–( )cos

–

12---µiksinh

µiksinh x t+( )cos–------------------------------------------------

⎝⎜⎜⎛

–

12---νiksinh

νikcosh x t+( )cos–-------------------------------------------------

⎠⎟⎟⎞

12--- x t+( )cos , i k≠ ,

Kii x t,( )12--- βi β0–( )sinh–=

×

12--- x t–( )cos

2 βi β0–( )cosh x t–( )cos–-----------------------------------------------------------------

–

12--- x t+( )cos

2 βi β0–( )cosh x t+( )cos–----------------------------------------------------------------- ,

Φi x( ) 2π--- 1

2--- f i

x2---

βi

2----coshsin

βi

2----cosh x

2---cos–

----------------------------------– 2 f 0x2---sin+

⎝⎜⎜⎜⎛

=

--

----×

12--- βi β0–( ) β0+( )cosh

βi β0–( ) β0+( )cosh xcos–---------------------------------------------------------------------

⎠⎟⎟⎟⎞

.

Kik x t,( ) 14--- 1

2---µiksinh=

×

12--- x t–( )cos

µikcosh x t–( )cos–------------------------------------------------

12--- x t+( )cos

µikcosh x t+( )cos–-------------------------------------------------–

⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

,

Φi x( ) 1π--- f i

x2---

βi

2----coshsin

βi

2---- x

2---cos–cosh

----------------------------------.–=