chapter 4. problems of boundary values in static electric fields
DESCRIPTION
1) Separation of variables in system of rectangular coordinate. 2) Separation of variables in system of cylindrical coordinate. 3) Separation of variables in system of spherical coordinate. 4) Image method ( 镜像法不作要求 ). Chapter 4. Problems of Boundary Values - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/1.jpg)
Chapter 4.Problems of Boundary Values
in Static Electric Fields1) Separation of variables in system of rectangular coordinate
2) Separation of variables in system of cylindrical coordinate
3) Separation of variables in system of spherical coordinate
4) Image method ( 镜像法不作要求 )
◇ The boundary value problem of electrostatic and steady fields can be solved by Laplace or Poission equation under given boundary conditions
![Page 2: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/2.jpg)
回忆两无源介质间 ( 无电荷无电流 ) 的边界条件
![Page 3: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/3.jpg)
1 Separation of variables in system of rectangular coordinate
Separation of variablesSeparation of variables is a classical differential equation method, which is suitable for a sort of boundary value with ideal boundary conditions. ◇ General steps
Select the coordinate according to the geometric shape and field distribution of the boundary, and list the differential equations and boundary conditions. Separate the variables. Solve the equation (general solution: the linear combination of all special solutions) Determine the integral constant (based on the given boundary conditions) to obtain the final solution
2
0
+ 0k
2
2 2
2
2
电磁学(电动力学)中有三个重要方程:
Lapl ace方程(无源): =
Poi sson方程(有源): =- ,如对于点电荷, =-e r
Hel mhol tz方程(无源+波动): =
它们都带有Lapl ace算符 。这些方程与它们的解具有典型性(普遍适用性),在电动力学、量子力学、流体力学、声学、天体物理中均会出现。
![Page 4: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/4.jpg)
An example:
Determine the potential function in 2D rectangle space.
xa
y
b0U
0 0
0
?
2
0
0 0
0 0 ,0
0
0,
xx a
y y b
x a y b
U
According to the problem, the potential function and its boundary conditions are given as follows:
![Page 5: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/5.jpg)
In rectangular coordinate, can be expressed as:2 0
2 2
2 20
x y
(1)
Separate the variables as follows:
,x y f x g y (2)
Substituting equation (2) into (1), one can arrive at
2 2
2 2
d d0
d d
f x g yg y f x
x y (3)
![Page 6: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/6.jpg)
The two sides of the above equation is divided by
2 2
2 2
d d1 10
d d
f x g y
f x x g y y (4)
Only depends on variable x Only depends on variable y
So the two terms on the left side in Eq. (4) can be equal to two constant numbers, respectively, i.e.,
22
2
22
2
d1
d
d1
d
x
y
f xk
f x x
g yk
g y y
(5)
g y f x
![Page 7: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/7.jpg)
2 2 0x yk k
Where
(6)
If k x is a real number and k y an imaginary number
1 2sin cosx xf x A k x A k x (7)
Thus the solution of the equation (5) is:
1 2
' '1 2
sinh cosh
x x
x x
x x
f x B x B x
f x B e B e
Or (8)
If k x is an imaginary number
1f x C x C
If k x=0
(9)
![Page 8: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/8.jpg)
Analyzing the boundary conditions, we can obtain
(7)
Since =0 when x=a and 0
1 2sin cosx xf x A k x A k x
2
0
0 0
0 0 ,0
0
0,
xx a
y y b
x a y b
U
x
nk
a
where is called the eigenvalue of the present boundary problem:
(8)
Substituting the boundary conditions into Eq. (7), we can get
1 sinn
f x A xa
![Page 9: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/9.jpg)
(9)
(10)
1 2sinh coshy yg y B y B y
Thus, g(y) is
Note:
2 2
22
0
0
x y
x y
y x
k k
k j
k
1 sinhn
g y B ya
Hence,
1
, sin sinhnn
n nx y C x y
a a
(11)
The general solution is given by
![Page 10: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/10.jpg)
00
0
0
42sinh sin d
2 1
4
sinh 2 1
a
n
n
Un nC b U x x
a a a n
UC
nb n
a
Employing the theory of Fourier series expansion
(12)
0
1
4, sin sinh
sinh 2 1n
U n nx y x y
n a ab na
So the final solution is:
(13)
![Page 11: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/11.jpg)
2 Separation of variables in system of cylindrical coordinate
2 2
2 2 2
1 10r
r r r r z
The Laplace equation in cylindrical coordinate system can be written as:
(14)
,r f r g (16)
If the field is two-dimensional field and is independent of z, the Laplace equation can be rewritten (simplified) as
2
2 2
1 10r
r r r r
Assume the solution (r, ) can take the following form
(15)
![Page 12: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/12.jpg)
Substitution of (16) into (15) yields
(17) 2
2 20
g f r f r gr
r r r r
(18)
Function of Function of r
2r
f r g Simplify Eq. (17) by multiplying
2
2
10
f r grr
f r r r g
(19)
So,
2
22
1f r grr
f r r r g
![Page 13: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/13.jpg)
(20)
(21)
2
22
d0
d
gg
Then,
and we obtain
sin cosg A B
(22)
As the potential function should satisfy..., this means = n (integer)
2
sin cosg A n B n
Therefore,
(23)
![Page 14: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/14.jpg)
2
2 22
d d0
d d
f r f rr r n f r
r r
(24)
(25)
The left side of Eq. (19) can be rewritten as
i.e.,
2dd0
d d
f rr r n f r
r r
( Euler equation )
(26)
The solution to Eq. (25) is
0 0
0
ln 0
n nf r Cr Dr n
f r C D r n
(27)
The general solution of in the two-dimensional cylindrical coordinate is
1
sin cos sin cosn nn n n n
n
r A n B n r C n D n
![Page 15: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/15.jpg)
Example: an infinitely long dielectric cylinder with radius a and permittivity is placed in an external uniform electrical field E0
vertical to cylinder. If the field is directed along x-axis and the axis of cylinder is z-axis, determine the potential functions inside and outside the cylinder.
a
x0E
, ,r z
0
o
If the potentials inside and outside the cylinder are 1 and 2, respectively, the equations and boundary conditions are given as follows
In cylindrical coordinate, x=r cos, The corresponding potential of E0 is:
0 0 cd osx
x x Ex r
e e (28)
![Page 16: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/16.jpg)
1
1
sin cos
sin cos
nn n
nn n n
r A n B n
r C n D n
So,
(30)
11 1 cos cos
DB r
r
and1 0B E
From condition 1 0 cosr rE
(31)
(32)
(29)有限
21
22
2
1 0
1 2
1 20
0
0
0
cos
r a r a
r a
r a
r
r rE
a a
r r
Finite
![Page 17: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/17.jpg)
2
1
' sin 'cos
'sin 'cos
nn n
nn n n
r A n B n
r C n D n
(33)
Since 1 2a a , we have
1
1
1 0
0
1
' sin 'cos
cos
' cos cos c
o
o
c
s
s
nn n
n
DB a E a
a A n B n
DE a
a
a
(35)
21
' sin 'cosnn n
n
r A n B n
(34)
2 0r Because is finite
![Page 18: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/18.jpg)
11 0 0 2' cos cos cos
DB E
a
1 20 r a r ar r
Because
Thus,
20 01 0 1
0 0
2, ' cosD a E B
(36)
(37)
The potentials are given by
201 0 0
0
1cos cosE r a E
r
02 0
0
2cosE r
(38)
(39)
![Page 19: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/19.jpg)
3 Separation of variables in spherical coordinate system
The Laplace equation in spherical coordinate can be written as
(40)
22
2 2 2 2 2
1 1 1sin 0
sin sinr
r r r r r
The potential function can be expressed as:
(41) ,r f r g Without considering
![Page 20: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/20.jpg)
Separate variables (in the similar fashion), we can get the general solution to equation (14)
(42) 1
0
, P cosmmm m m
m
r f r g A r B r
2
0
12 2
0 1
1 dP 1
2 ! d
P cos P cos sin d 0
2P cos sin d P d
2 1
mm
m m m
m n
m m
x xm x
x xm
Pm x Legendre polynomial:◇
where
![Page 21: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/21.jpg)
Example: a spherical dielectric with radius a and permittivity is placed in a uniform electrical field E0. If the field is along z-axis, determine the potentials inside and outside the sphere.
a
z
0E , ,r
0
o
0 0 cd osz
z zE z E r
e e
In spherical coordinate system, z=r cos . The corresponding potential of E0 can be expressed as
Assume the potentials inside and outside the sphere are 1 and 2, respectively. The equations and boundary conditions are as follows
![Page 22: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/22.jpg)
Because 1 0 cosr rE
21 1 1 cosA r B r
and 1 0A E
The general solution is
11
0
, P cosmmm m m
m
r A r B r
有限
21
22
2
1 0
1 2
1 20
0
0
0
cos
r a r a
r a
r a
r
r rE
a a
r r
finite
![Page 23: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/23.jpg)
12
0
, ' ' P cosmmm m m
m
r A r B r
is finite 2 0r Because
20
, ' P cosmm m
m
r A r
2
20
20 1
2
, ' P c
'c
os
c
o
o
s
s
mm m
m
r A r
E a
A
B a
Since 1 2a a
![Page 24: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/24.jpg)
4 Method of Image ( 镜像法不作要求 )
According to the uniqueness theorem, in some appropriate position outside of the field under consideration, we can utilize some dummy ( 虚拟的 ) charges (image charge) to replace the induced charges of the conductor interface or the polarization charges on the media interface. Transform the original problem into the equivalent problem with the same boundary conditions.
The principles to select image charges:
1. Image charge should be outside of the region to be investigated (The image charge must be external to the volume of interest)
2. The boundary conditions should not be changed after introducing the image charges.3. Image method can only be suitable for those special boundaries (such as plane, circular and spherical boundaries).
![Page 25: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/25.jpg)
● 镜像法基本思路:在所研究的场域外的某些适当位置,用一些虚拟电荷等效替代导体分界面上的感应电荷或媒质分界面上的极化电荷的影响。
● 也即:在研究区域之外,用一些假想的电荷分布代替场问题的边界。——将原问题转为与原问题边界条件相同的等效问题。
● 镜像法求解电位问题的理论依据:唯一性定理。待求区域的电位由其电荷分布与边界条件共同决定。
● 等效电荷一般位于原电荷关于边界面的镜像点处,故称为镜像电荷。大多是一些点电荷或线电荷。
● 镜像电荷位置选择原则:1 、镜像电荷必须位于求解区域以外的空间。
2 、镜像电荷的引入不能改变原问题的边界条件。
![Page 26: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/26.jpg)
x
z
hq
Conductor
1. plane image
Infinite grounding conductor plane (z=0), where a point charge is placed at z=h. Determine the potential distribution of the upper half space.
q
h
z
0
h
q
'R
R
, ,P x y z
x
Because the original charge is placed at z=h, its mirror image should be at z=-h with the quantity being -q. Thus the potential at boundary z=0 keeps zero, and the conductor plane at z=0 can be removed in the equivalent system.
![Page 27: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/27.jpg)
1 1( , , ) ( )
4 '
qx y z
R R
2 2 2 2 2 2
1 1( )( 0)
4 ( ) ( )
qz
x y z h x y z h
So the potential in z>0 is written as
(43)
0s n n
z
D En z
2 2 2 3/ 22 ( )
qh
x y h
The total induced charges in infinite conductor is
(44)
2 2 2 3/ 22 ( )in ss
qhq ds dxdy q
x y h
(45)
The quantity of original and induced charges are equal to each other.
![Page 28: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/28.jpg)
2. Spherical face image
A grounding conductor sphere has the radius a. A point charge is placed at P1 point. The distance between charge and spherical center O is d1. Determine the potential distribution outside the sphere.
2S
2d
1d
1R
1S
2Ra
o2P
P
r
2q 1P
1q
Because the potential outside the sphere is decided by point charge and induced charge on the conductor surface, the latter of which can be replaced by the image charge q2 if the potential keep zero on spherical surface.
![Page 29: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/29.jpg)
1 2
0 1 2
1
4
q q
R R
So, the potential outside the sphere can be written as
(46)
1 22
0 1 2
1 21
0 1 2
1: 0
4
1: 0
4
q qS
a d a d
q qS
d a a d
In order to determine the position and quantity of the image charge, we can consider two special points, say, S1 and S2, inside the sphere, where the potentials are zero, namely,
(47)
![Page 30: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/30.jpg)
The solutions to the above two equations are given by
(48)2 1 2 1
1
2
2 2 11
(
aq q q q
d
ad d d
d
或 舍去)or
given up
(49)
(50)
1 2 1
0 1 2 0 1 1 2
1 1
4 4
q q q a
R R R d R
Thus, the potential outside the sphere can be rewritten as
12 2 2
1 1 1
12 2 2
2 2 2
2 cos
2 cos
R r d rd
R r d rd
where
![Page 31: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/31.jpg)
( 以下内容根据课时情况决定增删 )
Laplace Equations in Spherical and Cylindrical Coordinate Systems
球坐标和柱坐标系中的拉普拉斯方程
2
0
+ 0k
2
2 2
2
2
电磁学(电动力学)中有三个重要方程:
Lapl ace方程(无源): =
Poi sson方程(有源): =- ,如对于点电荷, =-e r
Hel mhol tz方程(无源+波动): =
它们都带有Lapl ace算符 。这些方程与它们的解具有典型性(普遍适用性),在电动力学、量子力学、流体力学、声学、天体物理中均会出现。
![Page 32: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/32.jpg)
Laplace 方程的解(勒让德函数、贝塞尔函数等)称为“特殊函数”。这些解都是确定的、固定的,已经在 100多年前被英美数学家求出,主要用了幂级数展开法( power series expansion ) . 求解过程是很复杂的,但今天的我们的任务就是利用这些解来表示任何一个边值问题的解,因为任何一个边值问题的解均可以用这些特殊函数来展开。我们的目的就是利用边界条件来定出这些展开系数。
下面的内容来自 Jackson 经典电动力学“ Classical Electrodynamics”(John Wiley & Sons, 1998, pp.95-119) ( 可见复印的资料 ).
建议自学。
![Page 33: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/33.jpg)
33
Boundary-Value Problems
1 Laplace’s Equation in Spherical Coordinates
2 0V
In spherical coordinates ( , , )r 1 1 1
02
2 2 2 2
2
2r rrV
r
V
r
V
( )sin
(sin )sin
We can solve this eq. by separation of variables, let VU r
rP Q
( )( ) ( )
2 2
2 2 2 2sin 0
sin sin2
d U UQ d dP UP d QPQ + + =
dr r d d r d
![Page 34: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/34.jpg)
34
r UPQ2 2sin /Multiplying 2 2
2 22
1 1 1sin sin
sin2 2
d U d dP d Qr θ + ( θ ) =
U dr Pr θ dθ dθ Q d
1 2
22
Q
d Q
dm
We have
0sin
)(sinsin
112
2
2
22
m
d
dP
d
d
Pdr
Ud
Ur
Similarly, we separate andP( ) U r( )
imeQ Clearly the solution is
In order that Q be single valued , m must be an integer 2
![Page 35: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/35.jpg)
35
2
2
2
2
1sin 1 0
sin sin
10
2
d dP mθ + l(l + ) P =
θ dθ dθ θ
d U l(l + )U =
dr r
Introducing another real constant )1( ll
( , , )l 0 1 2
U Ar Brl l 1For r we have,
mlP
For m=0, - ----- Legndre Polynomials Pl (cos )
Let x=cos 01
)1()1(2
22
x
mll
dx
dPx
dx
d
Its solutions are the associated Legendre functions
0)1()1( 2
ll
dx
dPx
dx
d
![Page 36: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/36.jpg)
36
P x P x dxll l ll' '( ) ( )
2
2 11
1
P xl
d
dxxl l
l
ll( )
![ ( ) ]
1
212
)13(2
1)(,)(1)(,1)1( 2
210 xxPxxPxPPl
Orthogonal:
ml
imml eP
r
rUrV
,
)(),,(
So the general solutions of Lapace eq. in Spherical Coordinates
llm
llm rBrArU 1)(
(Rodrigues formula)
![Page 37: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/37.jpg)
37
2 Boundary-value problems with azimuthal Symmetry
V r A r B r Pl
ll
ll
l( , ) [ ] (cos )( )
0
1
A Bl l,
A problem possessing azimuthal symmetry m=0, then
As usual, the coefficients can be determined by the BCs.Suppose Vis the potential on S of sphere of radius a. We want to know the potential V inside the sphere.If no charge at the origin,
V r( )0 -- finite, leading to 0lB for all l.
0
)(cos)(l
ll
l PaAV
From the relation'
1
1
12
2cos)(cos)(cos llll l
dPP
we have Al
aV P dl l l
2 1
2 0( ) (cos ) sin
![Page 38: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/38.jpg)
38
If, for example, in hemisphere,
VV
V( )
/
/
0 2
2
V r Vr
aP
r
aP( , ) [ (cos ) ( ) (cos ) ]
3
2
7
813
3
To find V outside the sphere
( / ) ( / )r a a rl l 1
)0,,( ll Arr
The series is a unique expansion with BCs. So from a knowledge of V in a limited domain, namely on the symmetry axis, we may derive the solution
e.g. z=r V z r A r B rll
ll( ) [ ]( ) 1
Valid for positive z.
If this potential function can be expanded in a power series in z=r , with known coefficient, the solution is derived by multiplying by )(coslP
![Page 39: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/39.jpg)
39
Let us see hemisphere again. We have obtained
V z r Vr a
r r a( ) [ ]
12 2
2 2
This can be expanded in powers of a r2 2/
V z rV j j
ja rj
j
j( ) ( )( / ) ( / )
!( / )
1
2 1 2 1 21
1
2
By comparison withV z r A r B rl
ll
l( ) [ ]( ) 1
only l j 1 2 , i.e., terms with odd l are non-zero.
V rV j j
j
a
rPj
j
jj( , ) ( )
( / ) ( / )
!( ) (cos )
1
2 1 2 1 21
1
22 1
This is the same as the previous result.
![Page 40: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/40.jpg)
40
An important expansion is that the potential at due to a unit point charge at
r
11
0| ' |(cos ) r r
r
rP
l
l ll
)( rrWhere is the smaller (larger) of | '||, | rr
,
-- the angle between
.
| '||, | rr
This can be proved by rotating axis so that r '
lies alone the z axis. Then the potential satisfies Lapace’s eq., possesses azimuthal symmetry, and can be expanded to
except at point r r '
)(cos]['
1 )1(
0
ll
ll
ll
PrBrArr
![Page 41: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/41.jpg)
41
rIf is on the z axis, the RHS reduces to
[ ]( )
ll
ll
lA r B r
0
1
While the LHS becomes
rrrrrrrrrr
1
| '|
1
)cos 2 (
1
| |
12/122
1 1
1
10r r
rr
r
rl
l( )( )
For points off the axis , it is only necessary to multiply each term in the above eq. by
Pl (cos )
1
)1(l
ll
ll
l r
rrBrA
The general result is proved.
11
0| ' |(cos ) r r
r
rP
l
l ll
![Page 42: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/42.jpg)
42
Another example is the potential due to a total charge q uniformlydistributed around a circular ring of radius a, located as shown in theFigure, with its axis the z axis and its center at z=b.
V z rq
r c cr( )
cos
2 2 2
The inverse distance AP, 1/AP can be expanded by using
)()(cos)(0
1crP
r
cqrzV
lll
l
)()(cos)(0
1crP
c
rqrzV
lll
l
The potential at a point P on the axis
11
0| ' |(cos ) r r
r
rP
l
l ll
![Page 43: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/43.jpg)
43
The potential at any point in space is then derived by multiplyingeach member of the series by the Legendre polynomials
V r qr
rP P
l
l l ll
( , ) (cos ) (cos )
1
0
Where is the smaller (larger) of r and c)( rr
![Page 44: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/44.jpg)
44
01
)1()1(2
22
x
mll
dx
dPx
dx
d
3 Associated Legendre Polynomials and the Spherical Harmonics
0)1()1( 2
ll
dx
dPx
dx
d
)(xPlLegendre’s polynomials
So far, we have considered the azimathal symmetry m=0, these involve only ordinary Ledgendze polynomials.
ml
imml eP
r
rUrV
,
)(),,(
![Page 45: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/45.jpg)
45
The general potential problem can, however, have variations so that inm 0 Q e im
In this case, we need the generalization of mll PP )(cos
It can be shown that in order to have finite solution on the interval 1 1x
m l l l l , ( ) ( ),1 1
)()1()1()( 2/2 xPdx
dxxP lm
mmmm
l
lml
mlm
l
mm
l xdx
dx
lxP )1()1(
!2
)1()( 22/2
the parameter l must be zero or a positive integer and the integer mcan take on only
(+|m|,-|m|; Rodrigues’formula)
positive m
![Page 46: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/46.jpg)
46
Since differential eq. for depends only on and m is an integer, Plm
m2
Pl
m and mlP are proportional
P xl m
l mP xl
m mlm
( ) ( )( )!
( )!( )1
For fixed m, Plm
form an orthogonal set in the index l on the interval
1 1x
1
1
'' )!(
)!(
12
2)()( ll
ml
ml ml
ml
ldxxPxP
The sol. of Laplace eq. was decomposed into a product of factors for
r, ,
Q emim( ) form a complete set of orthogonal in the index m
on the interval 0 2
![Page 47: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/47.jpg)
47
Plm
form a similar set in the index l for each m for 1 1cosTherefore,
)()(cos mm
l QP will form a complete orthogonal set on the surface of the unit sphere in the two indices l, m.
From the normalization condition, it is clear that suitable normalized function, denoted by Ylm ( , ) called spherical harmonics
Yl l m
l mP elm l
m im
2 1
4 ( )!
( )!(cos )
It can be seen
*, )1()(
)!(
)!()1(
)!(
)!(
4
12lm
mimml
mml YexP
ml
ml
ml
mllY
g A Ym l
m l
lm lml
( , ) ( , )
0
![Page 48: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/48.jpg)
48
The normalization and orthogonality conditions are
d d Y Yl m lm l l m m sin ( , ) ( , )' '
*' '
00
2
The complete relation is
)'cos(cos)'(),()','(*
0
lmlm
l
lm
lm
YY
l Y 01
400,
lY e
Y
i
1
3
83
4
11
10
sin
cos
![Page 49: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/49.jpg)
49
Note that, for m=0,
)(cos4
12),(0
ll P
lY
Since the spherical harmonics form a complete orthogonal set of function on the surface of the unit sphere, then an arbitrary function can be expanded in spherical harmonics
g( , )
g A Ym l
m l
lm lml
( , ) ( , )
0
ddgYA lmlm sin),(),(*
Where the coefficients
which recovers the ordinary Legendre functions.
![Page 50: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/50.jpg)
50
A point of interest to us is the form for 0
[ ( , )]gl
All
0 0
0
2 1
4with
Al
d P gl l0
2 1
4
(cos ) ( , )
The general sol. for a Boundary-value problem in m 0
spherical coordinates can be written
V r A r B r Ylml
lml
lmm l
m l
l( , , ) [ ] ( , )( )
1
0
If V is specified on a spherical surface, the coefficient can be determined by this BC.
![Page 51: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/51.jpg)
51
6 Laplace’s Equation in Cylindrical Coordinates; Bessel Function
In cylindrical coordinates ( , , ) z
2 0V
2
2 2
2
2
2
2
1 10
V V V V
z
The separation of variables V z R Q Z z( , , ) ( ) ( ) ( )
d Z
dzk Z
2
22 0
d Q
dv Q
2
22 0
0)(1
2
22
2
2
Rv
kd
dR
d
Rd
We get three ordinary differential equations:
![Page 52: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/52.jpg)
52
The solutions of the first two eqs. are elementary
Z z e kz( ) Q e iv( )
For the potential o be single valued, v--integer. But except (barring) some boundary-condition requirement in the z direction, the parameter k is arbitrary. For the present we will assume k is real. The radial eq. can be presented in a standard form by the change of variable x=k
d R
dx x
dR
dx
v
xR
2
2
2
2
11 0 ( )
This is the Bessel’s Eq., and the solutions are called Bessel functions of order
bygiven solution seriespower a isit ,)(xJ v
jj
j
vv x
vjjxxJ 2
0)2/(
)1(!
)1()2/()(
jj
j
vv x
vjjxxJ 2
0
)2/()1(!
)1()2/()(
![Page 53: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/53.jpg)
53
N xJ x v J x
vvv v( )( ) cos ( )
sin
solution.t indenendenlinearly
another find toneed We).()1()( have we(integer), If
function)(Neumann kind second theoffunction Bessel (2)
solutions. independ
linearly ofpair a form )(),( integer,an not is If x.of valuesfinite allfor
converge series The .order first theof functions Bessel called are solutions These
kindfirst theof functions Bessel (1)
)( xJxJm
xJxJ
mm
m
vv
integeran not is ifeven t independenlinearly are )( and )(
kind). second theoffunction (Besselfunction Neumann is This
xJxN vv
eq. Bessel the tosolutions ofset lfundamenta a form functions Hankel The
)()()(
)()()(
:as defined are functions, Hankel called kind, third theoffunction Bessel The )3(
2
1
xiNxJxH
xiNxJxH
vvv
vvv
![Page 54: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/54.jpg)
54
Fourier-Bessel series
Bessel function has an infinite number of roots,
3,2,1,0)( nxJ vnv
. ofroot nth theis (x)Jx vn
:integralion normalizat The
.0 inteval on theset orthogonalan form ,....,2,1,0 fixedfor
),/( Functions kind.first theof functions Besselonly consider We
anv
axJ vnv
nnvnvvnvvnv
a
xJa
axJ
axJd
21
2
0
)]([2
)()(
f A J xavn v vn
n( ) ( )
1
Aa J x
f J xa
dvnv vn
v vn
a
2
21
20( )
( ) ( )
:series Bessel-Fourier ain 0 interval on the )(function
arbitraryan expandcan wecomplete, is functions Bessel ofset that theAssuming
af
Where the coefficients:
![Page 55: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/55.jpg)
55
Modified Bessel functions
)(2
)(),()( )1(1 ixHixKixJixI vv
vvv
v
011
becomes )R(for eq. the,k-by replaced is kconstant sparation theif :Note
2
2
2
2
22
R
x
v
dx
dR
xdx
Rd
The solutions of this equation are called modified Bessel functions; They arejust Bessel functions of pure imaginary argument. The usual choices of linearlyindependent solutions are denoted by
![Page 56: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/56.jpg)
56
7 Boundary Value Problems in Cylindrical Coordinates
Consider the specific Boundary - value problem in Fig.
0)()0(),,()( aVzVVLzV
The cylinder has a radius a and a height L, the top and bottom surfaces being at z=L and z=0. The potential on the side and the bottom of the cylinder is zero, while the top has a potential V=VWe want to find the potential at any point inside the cylinder.
V z R Q Z z( , , ) ( ) ( ) ( )
In order that V be single valued and vanish at z=0
Q A m B m( ) sin( ) cos( )
kzececzZ kzkz sinh}{)( 21
where v=m is an integer and k is a constant to be determined.
![Page 57: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/57.jpg)
57
The radial factor is
)()()( kDNkCJR mm
If V is finite at D should be equal to 0.
0)/2()(
05772.0)2/ln(2
)(,1vx
v
vxxNx
vv
.0)( of roots theare where
,........2,1 ,0)(
: valuesspecial only thoseon can takek
thatmeans as vanish potential t that therequiremen The
mnmmn
mnmnmnm
xJx
na
xkakJ
a
![Page 58: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/58.jpg)
58
Combining all these conditions, we find that the general form of the solution is
0 1
)]cos()sin()[sinh()(),,(m
mnmnmnmnmn
mBmAzkkJzV
At z=L, we are given the potential as V There we have
0 1
)]cos()sin()[sinh()(),(m
mnmnmnmnmn
mBmALkkJV
a),(0 )( and
)2(0 )cos(,)(sin of properties orthogonal theFrom
.in Fourier -Bessel a and in seriesFourier a is This
mnm kJ
mm
nnmnmmnmmnm
a
mm
akJa
dkJkJ
dmm
'2
1
2
'
0
2
0
'
)(2
)()(
)'sin()sin(
![Page 59: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/59.jpg)
59
2
0 02
12
sin)(),()(
)(cos2 a
mnmmnm
mnmn mkJVdd
akJa
LkechA
2
0 02
'2
cos)(),()(
)(cos2 a
mnmmnmn
mnmn mkJVdd
akJa
LkechB
integral.an intoover goes series the,a If a.0 ,in interval finite afor
eappropriat isresult The series. in the 2/ use should we0,mFor 0
nB
be shouldsolution general Then the ).,V( be to0z plane whole theand
0)V(z when 0zfor potential theknow want to weexample,For
V z dke J k A k m B k mkzm m m
m( , , ) ( )[ ( ) sin( ) ( ) cos( )]
00
V dkJ k A m B mm m mm
( , ) ( )[ sin( ) cos( )]
00
The coefficients are determined by
![Page 60: Chapter 4. Problems of Boundary Values in Static Electric Fields](https://reader038.vdocuments.mx/reader038/viewer/2022102818/56812b0b550346895d8ef1fa/html5/thumbnails/60.jpg)
60
2
0 0
')'(
)'()'(
cos
sin),(
1dk
kB
kAkJd
m
mV
m
mm
The variation in is just a Fourier series. Consequently the coefficients are separately specified by the integral relations:
Since)'(
1)'()(
0
kkk
dxxkJkxJx mm
A
Bk
d d V J km
mm
mm
0
2
0( , ) ( )
sin
cos
series. in the 2/)( usemust we0,mfor usual, As 0 kB
:0plane theof area wholeover the integralsby determined are tscoefficien
that thefind we,over gintegratin and )(by sidesboth gMultiplyin
z
kJm