Chapter. 4
Solution of Electrostatic
Problems
ChinWook Chung
Dept. of Electrical Engineering
Hanyang University
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Summary
• Electrostatic cases (, V, E)
Charge density, 
V
1
4 0

V

R
dv
E
 2V  

0
1
4 0

V
 Rˆ
R2
dV
 E   / 0,
E  0
E  V
Electric Field, E
Potential, V
V    E dl
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Electrostatic Boundary Value Problems
• Poisson's & Laplace’s eqn.
 2V  2V  2V
V 2  2  2
x
y
z
In generalized coordinate,
2
 2V   V 
1
h1h2 h3
(Laplacian)
   h2 h3 V    h1h3 V    h1h2 V  








u
h

u

u
h

u

u
h

u
1 
2 
2
2 
3 
3
3 
 1 1
E  V , D= Ε
 D  
   E     V      2V  

(Poisson Eqn.)

  0   2V  0  Laplace's equation 
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
The solutions of Laplace's equation
• One dimensional case
– V(x) is a function of only one of the three coordinates.
 2V  2V  2V
 2  2 0
2
x
y
z
1   V  1   2V   2V
r
  2  2   2
r r  r  r    z
1   2 V 
1
 
V 
1
 2V
R
 2
 sin 
 2 2
2
R R  R  R sin   
  R sin   2
2
1 d  dV  3. 1  d V 
d 2V
1.
 0 2.
2 
2  4.
r

2
r
d

r dr  dr 


dx
1 d  2 dV 
1
d 
dV 
R
5.
sin





R 2 dR 
dR 
R 2 sin  d 
d 
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Electrostatic Boundary Value Problems
• Example 4.1
a) The potential at any point between the plates
b) the surface charge densities on the plates
d 2V  y 
dy
2
0
dV
 C1
dy
  y   0
V  y   C1 y  C2
E1n 
at y  0, V  0
at y  d , V  V0
Surface charge density
a n  a y ,  sl   0 E
V
V 0 y
d
E  y   a y
y 0
a n  a y ,  su    0 E
V
V 
 a y  0 
y
d 
s
0

yd
 0V0

d
 0V0
d
 sl    su in this case
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Electrostatic Boundary Value Problems
• Example 4.2
– By solving Poisson’s and Laplace’s equations for V,
– Determine the E field both inside and outside a spherical cloud of
electrons with a uniform volume charge density, = -0 (where
=0 is positive) for 0<R<b and =0 for R>b
– Cf. example 3-23
1 d  2 dV 

R




R 2 dR  dR 
0
E  V  aR
dV
dR
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Uniqueness Theorem
• A solution of Poisson’s equation that satisfies the given boundary
conditions is a unique solution.
V1 , V2 : two solutions
 2

 V1   ,  V2  


2
Assume that both V1 and V2 satisfy
the same boundary condition on S1 , S 2 , S3 ,...S n
Vd  V1  V2
 2Vd  0  Vd  2Vd  0
  (Vd Vd )  Vd  2Vd  Vd
2
    fA   f   A  A f
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Uniqueness Theorem
 V V   a ds  
S
d
d
n
Vd d
2
1
1
 S Vd Vd   an ds  0  Vd  R , Vd  R 2
Vd  0  Vd  const or 0
Vd  0 on the surfaces  Vd  0 everywhere
• A unique solution
 The solution obtained even by intelligent guessing !!
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Method of image
• Point charge and ground plate
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Method of image
V
Q
1
Q 1

4 0 R 4 0 R
R  x 2   y  d   z 2 , R  x 2   y  d   z 2
2
2
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Method of image
V
Ez  y  0   
y
 S   0 Ez  y  0   

Qd
2  x 2  z 2  d 2 
Qtotal    S  2 rdr  Qd 
0

0
rdr
2
r
 d
2
2
2
,
x

z

r
3/ 2

2 3/ 2
 Q
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Method of image
Negatively charged conduction plane
s 
Q
Q
Q
d
2  r 2  d 2 3/ 2
Q
• What if the conduction plane is not grounded ?
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Method of image
• Example 4-3
F1  a y
F2  a x
F3  
Q2
4 0  2d2 
2
Q2
4 0  2d1 
2
Q2
2 3/ 2
4 0  2d1    2d2  


2
a
x
2d1  a y 2d2 
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Line charge & line images
• Conducting cylinder and line charges
E
•
l 1
2 0 r
Potential from a line charge (r0 = reference point)
l
r0
V    Er dr 
ln
r
2 0 r
r
r
l
V?
0
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Line charge & line images
Two line charges
V
l
r
ln 0
2 0 r
Finding equipotential surfaces
VM 
l
r

r

r
ln 0  l ln 0  l ln i
20 r 20 ri 20 r
Equipotential surface  ri /r =const.
PM
OPi OM
i


PM OM OP
ri di a
a2
   const.  di 
r a d
d
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Two wire transmission line
• Example 4-4
– Capacitance ? C = Q / V
V= 220 Volts
V=0
D
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Two wire transmission line
V2 
C
l

a
a
ln , V1   l ln
2 0 d
2 0 d
l
 0
V1  V2

ln  d / a 

a2
1
d  D  di  D 
 d  D  D 2  4a 2
d
2
C
 0
ln  D / 2a  

 D / 2a 
 0
D
 1, C 
2a
ln  D / a 
2
 1



 0
 ln  x  x 2  1   cosh 1 x


cosh  D / 2a 
1
 F/m 
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Point charge & conducting sphere
• Example 4-5
P
– OMQ, OMP : similar triangles
a di

d a
1  Q Qi 
  0
4 0  r ri 
r
Q
a
 i   i  Qi   Q
r
Q
d
VM 
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Point charge & conducting sphere
• Example 4-5
P
VM  r  a,  ,   
1  Q Qi
 
4 0  r ri

,

r  a 2  d 2  2ad cos  , ri  a 2  d i2  2ad i cos 
For grounded sphere
Qi
1 
Q
V  a,  ,   


4 0  a 2  d 2  2ad cos 
a 2  d i2  2ad i cos 
V  a,   0   0, V  a,      0

0


a2
a
 di  , Qi   Q
d
d
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Point charge & conducting sphere
• Example 4-5
Q  1 a 1 
V  R,   


,
4 0  RQ d RQi 
RQ  R 2  d 2  2 Rd cos  ,
2
 a2 
a2
2
RQi  R     2 R cos 
d
d 
Electric field
V
R
Charge density
ER  R,    
 s   0 ER  a,    
Q  d 2  a2 
4 a  a 2  d 2  2ad cos  
3/ 2
Total induced charge on the sphere
Qind    s ds  
2
0
S


0
a
d
 s a 2 sin  d d   Q  Qi
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Point charge & conducting sphere
When V  a,  ,    V0 ,
additional image charge is required
Qi
V0 
 Qi  4 0 aV0
4 0 a
Qi
V=V0
 additional image charge 
Electric field profile
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Charged Sphere and conducting plane
• 4 - 4.4


2
Q  Q0  Q1  Q2  ...  Q0 1   

....

2
1




a

2c
V0 
Q0
4 0 a
, C


Q
2
 4 0 a 1   
 ... 
2
V0
1


Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Electrostatic Boundary Value Problems
•
Laplace’s equation in cylindrical coordinates
1   V  1   2V   2V
r
  2  2   2  0
r r  r  r    z
•
Cylindrical symmetry and the lengthwise dimension is very large
 2V
 2V
 0, 2  0
 2
z
1   dV
r
r dr  dr
•

  0  V  r   C1 ln r  C2

If the problem is such that electric potential changes only in the
circumferential direction and not in r- and z-directions,
1
r2
 d 2V
 2
 d

  0  V    K1  K 2

Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Electrostatic Boundary Value Problems
• Example (Problem 4-23)
Determine the potential distribution fo the regions :
a) 0    
b)     2
d 2V
For 0     ,
0
2
d
 V  0   0

V    V0
 V   
 V  A  B
V0

, 0    
For     2 ,
V    V0   K1 +K 2
V  2   0  2 K1 +K 2
V0
2 V0
, K2 
2  
2  
Finally,
V0
V   
 2    ,     2
2  
K1  
E?
E
1 dV
1V
a   0 a
r d
r
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Electrostatic Boundary Value Problems
• Example – spherical capacitor
– Determine the potential distribution
Spherical symetry, V is independent of  and 
C1
d  2 dV 
dV C1
 2  V    C2
R
0
dR 
dR 
dR R
R
C1
at R  Ri , V  Ri   V1    C2
Ri
C1
at R  Ro , V  Ro   V2    C2
Ro
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Electrostatic Boundary Value Problems
C1  
R0 Ri V1  V2 
R0  Ri
V  R 
R0V2  RiV1
, C2 
R0  Ri
1  Ri R0

V

V

R
V

R
V
 1 2  0 2 i 1  , Ri  R  R0

R0  Ri  R

V is independent of the dielectric constant of the insulating material
C ?
4 0 Ri Ro
4 0
C

 R0  Ri   1  1 


 Ri R0 
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Electrostatic Boundary Value Problems
• Example (problem 4-26)
1
d 
dV 
sin


0
2
R sin  d 
d 
excluding R  0 &   0 or 
dV
A
d
d


 V  A
 B  A ln tan   B
sin 
2

sin 

 
ln
 tan 
V    / 2  0
2
 V  V0 

 
V      / 2  V0
ln  tan 

2

E
1 dV
a  
R d
V0
a


R sin  ln tan 
2

S  
V0


R sin  ln tan 
2

Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Electrostatic Boundary Value Problems
V0
 2

2V0
R sin ddR
Q

dR





R




sin  ln tan  0 0
ln tan  0
2
2


infinity !
Q
Q
2V0
2R1
R1  C 



V0


ln tan 
ln tan 
2
2


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Hanyang University, Seoul, Korea
BVP in Cartesian coordinates
• Problems governed by partial differential equations with prescribed
boundary conditions are called boundary value problems (BVPs)
• BVPs for potential can be classified into three types:
– Dirichlet problems
• The value of the potential is specified everywhere on the boundaries
– Neumann problems
• The normal derivative of the potential (electric field) is specified everywhere
on the boundaries
– Mixed boundary-value problems
• The potential is specified over some boundaries and normal derivative of the
potential is specified over the remaining ones.
•
The solutions of Laplace’s equation are often called harmonic functions.
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
BVP in Cartesian coordinates
• Laplace’s equations
– Separation of variable
 2V  2V  2V
 2  2 0
2
x
y
z
V  x , y, z   X  x  Y  y  Z  z 
d2 X
d 2Y
d 2Z
1 d 2 X 1 d 2Y 1 d 2Z
YZ
 XZ
 XY
0


0
dx 2
dy 2
dz 2
X dx 2 Y dy 2 Z dz 2
– A function of only one coordinate variable, each of the three terms must
be a constant
1 d2 X
d2 X
2
2


k



k
X,
x
x
2
2
X dx
dx
d 2Y
d 2Z
2
2
Similarly,


k
Y
,


k
Z
y
z
2
2
dy
dz
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
BVP in Cartesian coordinates
• The condition for the separation constants
kx2  ky2  kz2  0
• Possible solutions for X(x)
1 d2 X
2


k
x
X dx 2
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
BVP in Cartesian coordinates
• Example 4-6
V  0, y   V0 , V  , y   0
V  x , 0  V  x ,b  0
kx2  ky2  0  ky2  kx2  k 2
1 d2 X
2
2


k

k
,
x
2
X dx
1 d 2Y
2
2


k


k
y
Y dy 2
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
BVP in Cartesian coordinates
• Example 4-6
V  0, y   V0 , V  , y   0
V  x , 0  V  x ,b  0
ky2  kx2  k 2
X  x   D2e  kx ,Y  y   A1 sin ky  Vn  x , y   Cne  kx sin ky
V  x , b   0  Vn  x , b   Cn e  kx sin kb  0  k 
Vn  x , y   Cn e
 n x / b

n
n
sin
y  V  x , y    Cn e n x / b sin
y
b
b
n 1
n
V  0, y   V0   Cn sin
y 0 yb
b
n 1
4V0  1  n x / b
n
V  x, y 
e
sin
y

 n odd n
b

n
b
 4V0

if n is odd 

Cn   n

0
if n is even 
1
sin A sin B   cos  A  B   cos  A  B  
2
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
BVP in Cylindrical coordinates
• Laplace’s equation for V in cylindrical coordinate
1   V
r
r r  r
2
2
 1 V V
 2  0  Bessel functions
 2
2
r


z

• In such cases, 𝜕 2 𝑉/𝜕𝑧 2 = 0. After separation of variables
V  r ,    R  r    
r d  dR  1 d
r d  dR 
1 d
2
2
r


0

r

k
,


k
R dr  dr   d 2
R dr  dr 
 d 2
k  integer=n
d2R
dR
r

r
 n 2 R  r   0  R  r   Ar r n  Br r  n
2
dr
dr
2
•
    A sin n  B cos n
General solutions
Vn  r,    r n  An sin n  Bn cos n   r  n  A 'n sin n  B 'n cos n 
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
BVP in Cylindrical coordinates
• The simplest form when k=0
1 d
2


k
     A0  B0      B0  no circumference variation 
 d 2
d  dR 
r
 0  R  r   C0 ln r  D0 , for k  0
dr  dr 
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
BVP in Cylindrical coordinates
• Example 4-8 – coaxial cable
V b   0, V  a   V0
– No z dependence and by symmetry, no  dependence (k=0)
C1 ln b  C2  0
C1 ln a  C2  V0
 C1  
V r  
V0
V ln b
, C2  0
ln b / a 
ln b / a 
V0
b
ln
ln b / a  r
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
BVP in Cylindrical coordinates
• Example 4-9 – infinite thin tube
 V0 for 0< <
V b,    
V0 for  < <2
– General solution
Vn  r,    r n  An sin n  Bn cos n   r  n  A 'n sin n  B 'n cos n 
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
BVP in Cylindrical coordinates
• Example 4-9 – infinite thin tube
 V for 0< <
V b,     0
V0 for  < <2
– Inside the tube ( r < b)

V  r,     r n An sin n  V  finite @ r  0 & odd function of 
n 1
 4V0
if n is odd

V b,    V0 for 0< <  An   n bn
 0
if n is even
– Outside the
tube (r > b)

V  r,     r  n Bn sin n  V  finite @ r   & odd function of 
n 1
 V for 0< <
At r  b, V b,     b n Bn sin n   0
n 1
V0 for  < <2

 4V0bn
if n is odd

Bn   n
 0
if n is even

Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
BVP in Cylindrical coordinates
• Example 4-9
– Inside the tube
4V0
n
1r
V  r,   

  sin n , r  b
 n odd n  b 

– Outside the tube
V  r,   
4V0
n
1 b

  sin n , r  b
 n odd n  r 

Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
BVP in spherical coordinates
• Laplace’s equation for V in spherical coordinate with azimuthal
symmetry
1   2 V 
1
 
V
R

sin




R 2 R 
R  R 2 sin   

•

0

After separation of variables, V  R,     R    
1 d  2 d 
1
d 
d 
R

sin

0
 dR  dR   sin  d 
d 
k2
k 2
d 2
d
 n 1
2
n
2
R

2
R

k


0


R

A
R

B
R
,
n
n

1

k




n
n
n
dR 2
dR
2
d
d
d 

sin

 n  n  1 sin   0      Pn  cos   , Legendre polynomials


d 
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
BVP in spherical coordinates
• General solution with no azimuthal variation


V  R,     An R n  Bn R
n 0
 n 1
 P  cos 
n
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
BVP in spherical coordinates
• Example 4-10
– Uncharged conducting sphere of radius b is placed in an initially
uniform electric field E0=z E0. Determine V(R,) and E(R,)
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
BVP in spherical coordinates
• Example 4-10
– V(R,) for R>b


V  R,     An R n  Bn R
n 0
V b,    0
 n 1
 P  cos 
n
V  R,    E0 z  E0 R cos  for R  b
V  finite @ R   & V  E0 R cos 

V  R,    E0 RP1  cos     Bn R
 n 1
n 0
Pn  cos  

V  R,    B0 R   B1R  E0 R  cos    Bn R
1
2
n2
0 no charging
–
B1  E0b3
 n 1
Pn  cos  
0  V b,    0
  b 3 
V  R,    E0 1     R cos 
  R  
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
BVP in spherical coordinates
• Example 4-10
– Electric field intensity
E  a R ER  a E
3

V
b 
ER  
 E0 1  2    cos 
R
 R  

  b 3 
1 V
E  
  E0 1     sin 
R 
  R  
 ps  P0 cos 
R
b 
V  R  ?
z
– Surface charge density
s     0 ER b,   3 0 E0 cos    s  cos 
p  qd
– Dipole
p  a zmoment
4 0b3 E0
V  E0 R cos  
E0b cos 
R2
3
V
qd cos 
4 0 R 2
dipole potential term
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Numerical solution of Laplace’s equation
• Finite difference method
V V1  V2

x
d
 2V V2  2V1  V3
 2V V4  2V1  V5
 2 
, Analogously,

x
d2
y 2
d2
 2V  2V V2  V3  V4  V5  4V1
V  2  2 
0
2
1
x
y
d
1
 V1  V2  V3  V4  V5 
4
2
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
Numerical solution of Laplace’s equation
• Finite difference method
– Potential profile ( +100V to 0V ) in a parallel plate
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea
H.W.
• 4-1,2,3,4,7,5,9,15,17,22,23,24,27,29
Plasma Electronics Laboratory
Hanyang University, Seoul, Korea