Chapter 4 Selected Topics for Circuits and Systems
4-1 Poisson’s and Laplace’s Equations


Poission’s equation:   E     2V  

Laplace’s equation: If no charge exists, ρ=0,  2V  0
Eg. The two plates of a parallel-plate capacitor are separated by a distance d and
maintained at potentials 0 and V0. Assume negligible fringing effect at the edges,
determine (a) the potential at any point between the plates, (b) the surface charge
densities on the plates. [清大電研]
(Sol.)
d 2V
V
 0  V  c1 y  c2 , V  y  0  0, V  y  d   V0  V  0 y
dy 2
d

 
V
dV
E   yˆ
  yˆ 0 , En  aˆ n  E 
dy
d

ˆn  y
ˆ ,  sl  
At the lower plate: a
At the upper plate: aˆ n   yˆ ,  ul 
V0
d
.
V0
d
Eg. The upper and lower conducting plates of a large parallel-plate capacitor are
separated by a distance d and maintained at potentials V0 and 0, respectively. A
dielectric slab of dielectric constant εr and uniform thickness 0.8d is placed over
the lower plate. Assuming negligible fringing effect, determine (a) the potential
and electric field distribution in the dielectric slab, (b) the potential and electric
field distribution in the air space between the dielectric slab and the upper plate.
[台大電研]

(Sol.) Set Vd  y   c1 y  c 2 , E d   yˆc1 , Dd   yˆ 0 r c1

Va  y   c3 y  c4 , Ea   yˆc3 , Da   yˆ 0 c3
Vd 0  0,Va d   V0 , Vd 0.8d   Va 0.8d  , Dd 0.8d   Da 0.8d 
1   r V0
V0
 rV0
, c2  0, c3 
d , c4 
0.8  0.2 r d
0.8  0.2 r 
1  0.25 r


5 yV0
5V0
5 V


a  Vd 
b  Va  5 r y  4  r  1 d V0 , E a   yˆ r 0
, E d   yˆ
4   r d
4   r d
4   r d
4   r d
 c1 
Eg. Show that uniqueness of electrostatic solutions.
(Proof) Let V1 and V2 satisfy  2V1  


and  2V2   . Define Vd=V1-V2,


▽2Vd=0
1. On the conducting boundaries, Vd =0  V1= V2

2. Let f= Vd, A =▽Vd


  Vd Vd      fA  f  A  A  f  Vd  2Vd  Vd
R  , Vd  V1  V2 
1
1
, Vd  2 , ds  R 2
R
R
2
  Vd Vd   aˆ n ds   Vd dv
2
v
s
  Vd Vd   aˆ n ds  0,   Vd dv  0 
2
s
v
Vd =0  V1= V2
Image Theorem P(x,y,z) in the y>0 region is
V  x, y , z  
Q  1
1 

 , where R+ and R- are


4 0  R R 
the distances from Q and –Q to the point P,
respectively.
Eg. A point charge Q exists at a distance d above a large grounded conducting
plane. Determine (a) the surface charge density ρs, (b) the total charge induced
on the conducting plane. [交大光電所]

(Sol.) E/ y 0   yˆ
a 

Q
40 R
 s  yˆ  E / y 0  
2

 2 sin    yˆ
Qd
2 d 2  r

2 32
, b 

Qd
20 d 2  r 2


0

32
 s 2rdr  Q
Eg. Two dielectric media with dielectric constants ε1 and ε2 are separated by a
plane boundary at x=0. A point charge Q exists in medium 1 at distance d from
the boundary. Determine Q1 & Q2. (Q & -Q1 in medium 1, or Q & Q2 in medium
2)
(Sol.) V1 ( x  0) 
V1=V2,  1
Q
41 s 2  d 2

Q1
41 s 2  d 2
, V2 ( x  0) 
V1
V
Q  Q1 Q  Q2
, and
  2 2 at x  0 

x
x
1
2
Q  Q1  Q  Q2  Q1  Q2 
 2  1
Q
 2  1
Q  Q2
42 s 2  d 2
Eg. A line charge density ρl located at a distance d from the axis of a parallel
conducting circular cylinder of radius a. Both are infinitely long. Find the image
position of line charge.
r
(Sol.) Assume  i    l ,V    Edr  
r0
 VM 
l r 1

r
dr  l n o

20 r r
20
r
0
l
r

r

r
n 0  l n 0  l n i
20
r 20
ri 20
r
ri d i a
a2
   C  di 
r a d
d
Eg. A point charge Q is placed at a distance d to a conducting sphere. Find its
image.
(Sol.)
VM 
1  Q Qi 
 
0
4 0  r
ri 

a  di a
ri
Q
a
a
a2
  i   Qi   Q ,
  di 
r
Q d
d
d a d
d
4-2 Boundary-Value Problems in Rectangular Coordinates
 2V 
 2V  2V  2V

 2  0 . Let V(x,y,z)=X(x)Y(y)Z(z), kx2+ky2+kz2=0
x 2 y 2
z
d 2 X x 
d 2Y  y 
d 2 Z z 
2
2





k
X
x

0
,

k
Y
y

0
,
 k z2 Z z   0
x
y
2
2
2
dx
dy
dz
→
For X(x), 1. kx2=0, X(x)=A0x+B0 is linear.
2. kx2>0, X(x)=A1sinkxx+B1coskxx, X(x=a) is finite, X(x=b) is finite
3. kx2<0, X(x)=A2sinhkxx+B2coshkxx, X(∞) is finite, X(-∞) is finite
Similar cases exist in Y(y) and Z(z).
Eg. Two grounded, semi-infinite, parallel-plane electrodes are separated by a
distance b. A third electrode perpendicular to and insulated from both is
maintained at a constant potential V0. Determine the potential distribution in the
region enclosed by the electrodes. [高考]
(Sol.)
V x, y, z   V x, y   X x Y  y 
V 0, y   V0 , V x,0   0
B.C.: V , y   0,V x, b   0
d 2 X ( x)
 k x2 X ( x)  X ( x)  D1e k x x  D2 e k x x  D2 e k x x
2
d x
d 2Y  y 
 k y2Y  y   Y  y   A1 sin k y y
2
dy
 Vn  x, y   Cne  k x x sin k y y  k x  k y 
 Vn  x, y   Cne


V 0, y   V0  
n 1
nx
b
n
, n  1,2,3
b
ny
b
ny
Cn sin
b
sin

b
my
ny
my
V
sin
dy

Cn  sin
sin
dy

0
0
0
b
b
b
n 1
b
 2bV0
C
, m : odd  n , m  n

  m
 2


0, m  n
0, m : even
 4V0
, n : odd

 C n   n

0, n : even
V  x, y  
1 
e

 n:odd n
4V0
nx
b
sin
ny
, n  1,3,5, for x>0, 0<y<b.
b
4-3 Boundary-Value Problems in Cylindrical Coordinates
 2V 
1   V
r
r r  r
2
2
 1 V V


0

2
2
z 2
 r 
 2V
1. Assume
 0 , then V(r,φ)=R(r)Φ(φ),
z 2
d 2 Rr 
dRr 
d 2  
2


→ r2

r

n
R
r

0
,
 n 2    0
2
2
dr
dr
d
n
-n
→R(r)= Arr +Brr , Φ(φ)=Aφsinnφ+ Bφcosnφ
→Vn(r,φ)=rn(Asinnφ+Bcosnφ)+ r-n(A’sinnφ+B’cosnφ), n≠0

→V(r,φ)=  Vn r ,  
n 1
2. Assume n=0,
d 2  
 0  Φ(φ)=A0φ+B0, R(r)=C0lnr+D0,
d 2
In the φ-independent case, V(r)=C1lnr+D1
In the φ-dependent case, Φ(φ)=Aφ+B, V(r,φ)=(Clnr+D)(Aφ+B)
Eg. Consider a very long coaxial cable. The inner conductor has a radius a and is
maintained at a potential V0. The outer conductor has an inner radius b and is
grounded. Determine the potential distribution in the space between the
conductors. [電信特考]
(Sol.) V b  0, V a   V0  C1n(b)  C2  0, C1n(a)  C2  V0
C1 
V0
V ln b
V0
r
, C2   0
, ∴ V r  
ln  
ln b a 
ln b a 
ln b a   b 
Eg. Two infinite insulated conducting planes maintained at potentials 0 and V0
form a wedge-shaped configuration. Determine the potential distributions for the
a 0    
regions:
. [中央光電所、成大電研]
b    2
(Sol.)
(a) V(φ)=Aφ+B,
V 0  0  B0  0
V0


V0  V     ,0    

V    V0  A0  A0  

V    V0  A1  B1
V0
2V0
V0
(b) V 2   0  2A  B  A1  
2    ,
, B1 
 V   
1
1

2  
2  
2  
    2
Eg. An infinitely long, thin, conducting circular tube of radius b is split in two
halves. The upper half is kept at a potential V=Vo and the lower half at V=-Vo.
Determine the potential distributions both inside and outside the tube.
(Sol.)
V0 ,0    
V b,   
 V0 ,    2
(a) Inside the tube:
r  b  Vn r ,   An r n sin n  V r ,  


An r n sin n
n 1
r b


n 1
V0 ,0    
Anb n sin n  
 V0 ,    2
 4V0
, n : odd

 An   nb n
0, n : even

 V r ,  
4V0


1r
 n  b 
n
sin n , r  b
n  odd
(b) Outside the tube:
r  b  Vn r ,   Bn r  n sin n  V r ,  


n 1
r b

 Bb
n
n 1
n
V0 ,0    
sin n  
 V0 ,    2
 4V0b n
, n : odd

 Bn   n
0, n : even

 V r ,  
4V0


1b
 n  r 
nodd
n
sin n , r  b
Bn r  n sin n
Eg. A long, grounded conducting cylinder of radius b is placed along the z-axis in
 
an initially uniform electric field E  x E 0 . Determine potential distribution

V(r,φ) and electric field intensity E r ,   outside the cylinder. Show that the
electric field intensity at the surface of the cylinder may be twice as high as that
in the distance, which may cause a local breakdown or corona (St. Elmo’s fire.)
[中央光電所、高考]

(Sol.) V r ,     E 0 r cos    Bn r  n cos n
At r  b,
n 1

E  xˆE 0 , V   E 0 r cos  

At r  b : V r,     E0 b cos    Bn b n cos n  0  B1  E0 b 2 , Bn  0 for n  1.
n 1
 b2 




r

b
:
V
r
,



E
r
0 1  2  cos 
Outside the cylinder,
 r 

 b2

 b2

E r ,   V  aˆr E0  2  1 cos   aˆr E0  2  1 sin 
r

r

Eg. A long dielectric cylinder of radius b and dielectric constant εr, is placed

along the z-axis in an initially uniform electric field E  xˆE 0 . Determine V(r,φ)

and E r ,   both inside and outside the dielectric cylinder.

n
(Sol.) For r  b, V0 r ,    E0 r cos    Bn r cos n
n 1

n
r  b, Vi r ,    An r cos n
n 1
V0 b,   Vi b, 
V0
Vi 

  r
r r  b
r r  b

 E0b  B1b 1  A1b, Bn b  n  An b n , n  1
E0  B1b 2   r A1 , nBn b n1   r nAn b n1 , n  1
 2 E0
 1 2

, B1  r
b E0
 A1 
 r 1
r 1

 A  B  0 for..n  1
n
 n

   1 b2 
 2  E0 r cos 
V0 r ,   1  r

 r 1 r 
 
V r ,    2 E r cos 
0
 i
r 1


  r  1 b2 
  r  1 b2 


1 

ˆ
ˆ
E

a
E
1


cos


a
E

0
r
0

0
2

  1 r 
   1  r 2  sin 
V
V

r
r




E  V  aˆr
 aˆ
 

r
r
E  2 E aˆ cos   aˆ sin 

 i  1 0 r
r



4-4 Boundary-Value Problems in Spherical Coordinates
1   2 V 
1
 
V 
1
 2V
R

sin


0




R  R 2 sin   
  R 2 sin 2   2
R 2 R 
2
Assume φ-independent: 2  0 , V(r,θ)=R(r)Θ(θ)
 
 2V 
d 2 Rr 
dRR 
d 
d 
 2r
 k 2 Rr   0,
sin 
 nn  1 sin    0 , and
2

dr
d 
d 
dr
k2=n(n+1)→ R(r)=Anrn+Bnr-n-1, Θ(θ)=Pn(cosθ)→V(r,θ)=[Anrn+Bnr-n-1]Pn(cosθ)
r2
Table of Legendre’s Polynomials
Pn cos  
n
0
1
1
cos
2
1
3 cos 2   1
2
3
1
5 cos 2   3 cos 
2




Eg. An infinite conducting cone of half-angle α is maintained at potential V0 and
insulated from a grounded conducting plane. Determine (a) the potential
distribution V(θ) in the region α<θ<π/2, (b) the electric field intensity in the
region α<θ<π/2, (c) the charge densities on the cone surface and on the grounded
plane.
(Sol.)
d 
dV 
dV
C


 1  V    C1n tan   C2
 sin 
  0,
d 
d 
d
sin 
2

(a)


V    C1n tan   C 2  V0
2



 

V    C1n tan   C 2  0
4
2


dV
 aˆ
(b) E  aˆ
Rd
V0
V0
   
n  tan  
  2 
C2  0
C1 

   
V0 n  tan  
  2 
V   
   
n  tan  
  2 
   :  s   0 E   
 0V0
   
Rn  tan   sin 
, (c)
   
  2 
Rn  tan  sin 
 0V0

 
  2 
  :  s   0 E    
2
2

 
  
Rn  tan  
  2 
Eg. An uncharged conducting sphere of radius b is placed in an initially uniform

electric field E  zˆE 0 . Determine the potential distribution V(R,θ) and the

electric field intensity E R,  after the introduction of the sphere. [中山電研]
(Sol.) V b,   0
If R>>b, V R,    E0 z   E0 R cos 



V R,     An R n  Bn R n 1 Pn cos   ,
Rb
n 0
An  0, n  1
(
A1   E0
)
↓

  E 0 RP1 cos     Bn R n 1 Pn cos  , R  b
n 0
(sphere is uncharged,
B0  0 )
↓

B

  12  E0 R  c o s 
Bn R  n 1 Pn c o s , R  b
R

n2


B

R  b, 0   21  E0 b  cos    Bn b n 1 Pn cos    B1  E0 b 3 , Bn  0, n  2 ,
b

n2
  b 3 
∴ V R,    E0 1     R cos  , R  b
  R  

 V 
 V 
E R,   aˆ R ER  aˆ E  V R,   aˆ R  
  aˆ  

 R 
 R 
3

  b 3 
b 
ˆ
ˆ
 aR E0 1  2   cos  a E0 1     sin  , R  b
 R  

  R  

2
A dipole moment P  zˆ 4 0b E0 is at the center of the sphere. Surface charge
density is  s     0 ER R b  3 0 E0 cos
4-5 Capacitors and Capacitances
Q=CV  C=Q/V
Eg. A parallel-plane capacitor consists of two parallel conducting plates of area S
separate by uniform distance d, the space between the plates is filled with a
dielectric of a constant permittivity. Determine the capacitance.
 
 Q
Q 
(Sol.)  s  , E   y s   y
S

S
V  
y 0
C

 Q 

d
Q
   y dv  
E d l     y
d
0
S  
 S

y d 

 V
Q
S
  . In this problem, E   y
V
d
d
 Surface charge densities on the upper and conducting planes are ρs and -ρs,
ρs=εEy=εV/d.
Eg. The space between a parallel-plate capacitor of area S is filled with dielectric
whose permittivity varies linearly from ε1 at one plate (y=0) to ε2 at the other
plate (y=d). Neglecting all the edge effect, find the capacitance. [台科大電子所]
(Sol.) Assume Q on plate at y=d,  


E y
s

, s 
 2  1
d
y  1

d 
Qd ln  2  1 
Q S ( 2   1 )
Q
C  
 V    E d l 
0

V
S  2   1 
S
d ln( 2 )
1
Eg. A cylindrical capacitor consists of an inner conductor of radius a and an
outer conductor of radius b is filled with a dielectric of permittivity ε, and the
length of the capacitor is L. Determine the capacitance of this capacitor.


(Sol.) E  a r E r 
Vab   

a 
Q   
Q
Q
2L
b

E d l     a r
ln   , C 

   a r dr  
b
Vab
b
 2L  a 
 2Lr  
ln  
a
r a 
r b
Q
,
2Lr
Eg. A spherical capacitor consists of an inner conducting sphere of radius Ri and
an outer conductor with a spherical wall of radius Ro. The space in between them
is filled with dielectric of permittivity ε. Determine the capacitance.



(Sol.) E  a r E r  a r
Q
4R 2
Ri   
Ri
Q
Q 1
1 



V=   E  ar dR   
dR


Ro
Ro 4R 2
4  Ri Ro 


C
Q
4
.

1
1
V

Ri Ro
For an isolating conductor sphere of a radius: Ri , Ro   , C  4Ri
Eg. Assuming the earth to be a large conducting sphere (radius=6.37×103km)
surrounded by air, find (a) the capacitance of the earth, (b) the maximum charge
that can exist on the earth before the air breaks down.
(Sol.) (a) C  4 0 R  4 
(b) Eb  3  10 6 


1
 10 9  6.37  10 3  10 3  7.08  10  4
36
QMax
40 R 2
 QMax  1.35  1010
(F )
(C )
Eg. Determine the capacitance of an isolated conducting sphere of radius b that
is coated with a dielectric layer of uniform thickness d, the dielectric has an
electric susceptibility χe.



Q
Q
b

d

R
,
E

(Sol.) b  R  b  d , E  a r
,
2
4 0 1   e R
4 0 R 2
b

V    E d l 

Q 40 1   e 
Q
1
 e
 , C  

e
1
40 1   e   b  d b 
V

bd b
Eg. A cylindrical capacitor of length L consists of coaxial conducting surface of
radii ri and ro. Two dielectric media of different dielectric constants εr1 and εr2,
and fill the space between the conducting surface. Determine the capacitance. [台
大物理所、高考電機技師]
(Sol.)
rL 0 r1   0 r 2 E   l L  E 
ri
V    Edr 
ro
C
l L
V

l
r
ln  o
0  r1   r 2   ri
0  r1   r 2 L
ln ro ri 



l
r 0  r1   r 2 
Eg. The radius of the core and the inner radius of the outer conductor of a very
long coaxial transmission line are ri and ro respectively. The space between two
conductors is filled with two coaxial layers of dielectrics. The dielectric constants
of the dielectrics are εr1 for ri<r<b and εr2 for b<r<ro. Determine its capacitance
per unit length.


ri 

(Sol.) E1  a r


l
l
, ri  r  b , E2  ar
, b  r  ro
20 r1r
20 r 2 r
V    E d r 
ro
C
l
V

l
2 0
 1  b  1  ro 
ln   ,
 ln   

r

 b 
r2
 r1  i 
2 0
1  b  1  ro 
ln   
ln  
 r1  ri   r 2  b 
( F / m)
Eg. Determine the capacitance per unit length between two long, parallel,
circular conducting wires of radius a. The axes of the wires are separated by a
distance D. [台大電研]
(Sol.) V2 
C

V1  V2


a
a
n , V1    n
2 d
2 d


nd a 
C=
, d  D  di  D 

n D 2a  

 D 2a 
2

a2
1
, d  D  D 2  4a 2
d
2
 1



cosh
1
D
2a 

(F/m)
Eg. A straight conducting wire of radius a is parallel to and at height h from the
surface of the earth. Assume that the earth is perfectly conducting; determine the
capacitance and the force per unit length between the wire and the earth.
(Sol.) D=2h, C 
0
cosh
1
D 2a 

F 

h a  m
0
cosh
1
Eg. A capacitor consists of two coaxial metallic cylindrical surface a length 30mm
and radii 5mm and 7mm. The dielectric between the surfaces has a relative
permittivity εr2=2+(4/r), where r is measured in mm. Determine the capacitance
of the capacitor.


(Sol.) E  a r
ri 

l
 ar
2r

V    E d r 
ro
l
4

2 0  2  r
r

l

 ar
4 0 r  2
r7
 L 40 L
l

9
 1500 0
ln r  2
 l ln   , C  l 
r  5 40  7 
V
40
9
ln  
7
Series or parallel connection of capacitance:
V 
Q
Q Q
Q
Q
1
1
1
1
1



  




  
C sr C1 C 2 C 3
Cn
C sr C1 C 2 C3
Cn
Q  Q1  Q2    Qn
 C // V  C1V C 2V    C nV
 C //  C1  C2    Cn
4-6 Electrostatic Energy
To remove Q1 from infinite to a distance R12 from Q2, the amount of work required is
Q1
Q2
1
W2  Q2V2  Q2
 Q1
 Q1V1  Q1V1  Q2V2 
40 R12
40 R12
2
induction
method

We 
1 N
1
Qk Vk , where Vk 

2 k 1
40
N
Qj
j 1( j  k )
R jk

Eg. Determine the work done in carrying a -2μC charge from P1(2,1,-1) to



P2(8,2,-1) in the field E  x y  y x along (a) the parabola x=2y2, (b) the straight
line joining P1 and P2.



(Sol.) dl  xˆdx  yˆdy , W  q  E d l  q  (  ydx  xdy)
2
(a) Along x  2 y 2 , dx  4 ydy  W  q  6 y 2 dy  14q  28 ( J ) .
1
(b) Along x  6 y  4, dx  6dy  W  q  12 y  4dy  14q  28 ( J )
2
1
Eg. Find the energy required to assemble a uniform charge of radius b and
volume charge density ρ. [清大電研]
QR
4
Q R   R 3
(Sol.) VR 
3
4 0 R
dQR   4R 2 dR , dW  VR dQR 
W   dW 
Q
4 2 4
 R dR
3 0
4 2 b 4
4 2 b 5
  R dR 
0
3 0
15 0
4 3
3Q 2
b , W
3
200 b
(J )
Eg. According to We=
energy is We 
1
1
 

Vdv

   D Vdv , show that the stored electric
2 v '
2 v ' 

 
1
D
 E dv
2 v '



 
  
(Proof) ∵   V D   V  D  D V , ∴ V  D    V D   D  V




∴ We 

 
 
1
1
1
1
 


V
D
dv

D


Vdv

V
D

a
ds

D
 E dv


n
2 v ' 
2 v '
2 s '
2 v '

When R  , S  R 2 , V 
 
 
1
1
1 
1
, D  2   V D a n ds  0  We   D E dv
2 v'
2 s'
R
R
 2

D
2

If D   E , then We 

1
1
 E dv  
dv   we dv

v'
2 v'
2 v' 
Note: 1. SI unit for energy: Joule(J) and 1 eV =1.6×10-19J.
2. Work (or energy) is a scalar, not a vector.
 2
D
2
1  1 
Electrostatic energy density: we= D E   E 
2
2
2
Eg. A parallel-plate capacitor of area S and separation d is charged by a d-c
voltage source V. The permittivity of the dielectric is ε. Find the stored
electrostatic energy.
2
2
1
1 V 
1 S 
V
V 
(Sol.) E  , We      dv     Sd     V 2
v
'
d
2
2 d 
2 d 
d 
Eg. Use energy formulas to find the capacitance of a cylindrical capacitance
having a length L, an inner conductor of radius a, an outer conductor of inner
radius b, and dielectric of permittivity  .
2
1 b  Q 
Q
Q 2 b dr
Q2  b 

L 2rdr  
(Sol.) E  a r
, We    

ln   ,

2Lr
2 a  2Lr 
4L a r
4L  a 


Q2
Q2
2L
b

ln    C 
2C 4L  a 
b
ln  
a
Eg. Find the electrostatic energy stored in the region of space R>b around
an electric dipole of moment p.

(Sol.) E 

p
40 R
aˆ R 2 cos   aˆ sin   ,
2
W 
1
p2
2
 0  E dv  
2
120 b 3
v
4-7 Electrostatic Forces and Torques
Electrostatic force and torque due to the fixed charge:


dW  FQ  d l is mechanic work done by the system, it costs the stored energy.




∵ dWe  dW   FQ  d l  We   d l , ∴ FQ  We
(N )
We
We
  Q 2  Q 2 C

 
, dW  TQ z d  TQ z  
  
 FQ   
2

l
l  2C  2C l
 l
Electrostatic force and torque due to the fixed potential:


1
1
dWs  Vk dQk , dW  Fv  d l , dWe  Vk dQk  dWs
2 k
2
k
dW  dWe  dWs
 dW 



1
dWs  dWe  Fv  d l  We   d l
2
2
2
We   
We   1
 V C Q C
,  Fv  
  CV 2  

2

l
l  2
 l
 2 l 2C l
Eg. Determine the force on the conducting plates of a charged parallel-plate

∴ Fv  We ,
Tv z

capacitor. The plates have an area S and separate in air by a distance x.
(Sol.) (a) Assuming fixed charge, We 
F 
Q x

1
1
QV  QE x x ,
2
2
 1
Q2

 QE x x   
x  2
2 0 S

(b) Assuming the fixed potential,
2
2
Fv x  We    1 CV 2   V    0 S     0 SV2
x
x  2
2x
 2 x  x 
 SV
∵ Q  CV  0
, ∴ FQ x  Fv x
x
Eg. A parallel-plate capacitor of width w, length L, and separation d is partially
filled with a dielectric medium of dielectric constants εr. A battery of V0 volts is
 
connected between the plates. (a) Find D , E , ρs in each region. (b) Find
distance x such that the electrostatic energy stored in each region is the same. [台
大電研]

V 
V
V
E1   yˆ 0 , D1   yˆ 0 r 0 ,  s1   0 r 0
(Sol.)(a)
d
d
d

V 
V
V
E2   yˆ 0 , D2   yˆ 0 0 ,  s 2   0 0
d
d
d
(b)
We1
 x
L
 r 1 x 
We 2 L  x
 r 1
Eg. A parallel-plate capacitor of width w, length L, and separation d has a solid
dielectric slab of permittivityεin the space between the plates. The capacitor is
charged to a voltage V0 by a battery. Assuming that the dielectric slab is
withdrawn to the position shown, determine the force action on the slab. (a) with
the switch closed, (b) after the switch is first opened. [台大電研、清大電研]
(Sol.) (a)
2
2

w
1
w
C
We  CV02 , C  x   0 L  x   Fx  We  xˆ V 0
 xˆ V 0    0 
2
d
2 x
2d
(b)
Eg. The conductors of an isolated two-wire transmission line, each of radius b,
are spaced at a distance D apart. Assuming D>>b and a voltage V0 between the
lines, find the force per unit length on the lines.
(Sol.)


 

l


Db
D
D b
E  xˆ  l 
,
V

V

V

E
 dxˆ  l ln
 x l ln
1
2
b
 0

0
b
0 b
 20 x 20 D  x 
2

l
0
V0 C 
0V02


ˆ
ˆ
F m , F  We  x
C 

 x
2
V0 ln D
2 D
2 D ln D
b
b
 
  
4-8 Resistors and Resistances
Ohm’s law: V=RI
 
V
I
V
  
V  E  E  , I   J  dS  JS  J     V    I  RI

S

 S 
s
∴ R

1
S
, G  
S
R

 
   
Power dissipation: P   E  Jdv   E  d   J  dS  VI   I 2 R
v
ss
Eg. A long round wire of radius a and conductivity σ is coated with a material of
conductivity 0.1σ. (a) What must be the thickness of the coating so that the
resistance per unit length of the uncoated wire is reduced by 50%? (b) Assuming
a total current I in the coated wire, find J and E in both the core and the coating
material. [台科大電子所]
1
1
, R2 
(Sol.) R1 
2
a
 [( a  b) 2  a 2 ]
(a) R1  R2  b 
(b) I1  I 2 
 11  1a ,
I
I
I
, J1 
 E1 , J 2 
 0.1E2
2
2
2
2a
2 a  b   b 2


 J 1  10 J 2 , E1  E2
Eg. A d-c voltage of 6V applied to the ends of 1km of a conducting wire of 0.5mm
radius results in a current of 1/6A. Find (a) the conductivity of the wire, (b) the
electric field intensity of the wire, (c) the power dissipation in the wire, (d) the
electron drift velocity, assuming electron mobility in the wire to be

1.4  10 3 m 2

V s .
(Sol.) (a) R 
 V
I
V
  
 3.54  10 7 S m  , (b) E   6  10 3 V m  , (c)

S I
SV
P  VI  1Watt, (d) ve  E  8.4  10 6 (m / sec)
Calculation of resistance:





 2V  0  V  E  V  J  E  I   Jds  R  V / I
Eg. A conducting material of uniform thickness h and conductivity σ, has the
shape of a quarter of a flat circular washer, with inner radius a and outer radius
b. Determine the resistance between the end faces. [清大電研]
(Sol.)  2V  0 , V=0 at   0 , V=V0 at  

2


d 2V
2V0
V
 0 , V=c1φ+c2, V  2V0  , J  E  V  aˆ 
 aˆ 
2

r
r
d
I   J d s 
S
2V0

h
b
a
V
dr 2hV0 b ,
R 0 

ln
I
r

a

b
2h ln  
a
Eg. A ground connection is made by burying a hemispherical conductor of radius
25mm in the earth with its base up. Assuming the earth conductivity to σ=10-6
S/m, find the resistance of the conductor to far-away points in the ground. [交大
電信所]
b
I
I
I
(Sol.) J  aˆ R
, E  aˆ R
=> V0    EdR 
2
2

2b
2R
2R
R
V0
1
1


 6.36  10 6 .
6
3
I
2b 2 (10 )( 25  10 )
Eg. The space between two parallel conducting plates each having an area S is
filled with an inhomogeneous ohmic medium whose conductivity varies linearly
from σ1 at one plate (y=0) to σ2 at the other plate (y=d). And d-c voltage V0 is
applied across the plates. Determine the total resistance between the plates.
(Sol.) J   yˆ Jo => E 
d
V0   E  yˆ dy 
0
J

  yˆ
y
Jo
,  ( y )   1  ( 2   1) ,
d
 ( y)
V
V
J0d
d
2
2
ln( )
ln , R  0  0 
I
JoS ( 2   1) S  1
 2 1 1
Relation between R and C: C 
Q s D  d s   E d s


,
V  E  d l  E  dl


L
R
  E d l
V  LE  d l
L
, ∴


I
 J  d s   E  d s
S
RC=
L
C 

G 
S
Eg. Find the leakage resistance per unit length (a) between the inner and outer
conductors of a coaxial cable that has an inner conductor of radius a, an outer
conductor of inner radius b, and a medium with conductivity σ, and (b) of a
parallel-wire transmission line consisting of wires of radius a separated by a
distance D in a medium with conductivity σ. [台科大電研]
(Sol.) (a) C 
2
 1
1
b
, R    
ln  
b
  C  2  a 
ln( )
a

(b) C 
cosh 1 (
D
)
2a
, R    1   1 cosh 1 ( D )
  C  
2a
Eg. Find the resistance between two concentric spherical surfaces of radii R1 and
R2 (R1<R2) if the space between the surfaces is filled with a homogeneous and
isotropic material having a conductivity σ.
(Sol.) C 
4
1
1

R1 R 2
, RC 

1 
1 1
1
(  )
=> R  
=> R 
4 R1 R 2

C 
4-9 Inductors and Inductances


Mutual flux:  12   B1  d S 2  L12 I 1 (Wb)
S2
General mutual inductance: L12 
Self-Inductance: L11 
N 2  12  12

(H)
I1
I1
 11
I1
Eg. Assume that N turns of wire are tightly wound on a toroidal frame of a
rectangular cross section. Then, assuming the permeability of the medium to be
μ0, find the self-inductance of the toroidal coil. [台大電研]
(Sol.)

d l  aˆ rd ,
 
 o NI
B
C  d l   Br d   2rB   o NI  B  2r
   NIh b d r  o NIh b
N  o N 2 h b
    B  d s  o

ln(
)
L


ln( )
,

a r
2

2

a
I
2

a
S
Eg. Find the inductance per unit length of a very long solenoid with air core
having n turns per unit length. And S is the cross-sectional area.
(Sol.) B   o nI ,   BS   o nSI '  n   o n 2 SI , L'   o n 2 S
Eg. Two coils of N1 and N2 turns are would concentrically on a straight
cylindrical core of radius a and permeability μ. The windings have lengths l1 and
l2, respectively. Find the mutual inductance between the coils.
N

(Sol.) 12   ( 1 )(a 2 ) I 1 ,  12  N 2  12  N 1 N 2a 2 I 1
1
1
=> L12 
12 
 N1 N 2a 2
I1
1
Eg. Determine the mutual inductance between a very long, straight wire and a
conducting circular loop. [台大電研、清大物理所]
(Sol.)
B at p is

0 I
2
0 I
2 ( d  r cos  )
b
2
0
0

I
rddr
 0
d  r cos 
2
L  0 (d  d 2  b 2 )
b
2rdr
0
d r

2
2
 0 I (d  d 2  b 2 )
Eg. Determine the mutual inductance between a conducting triangular loop and
a very long straight wire.
(Sol.)


 
 I
B2  aˆ 0 2 ,      B  ds , where dS  aˆ zdr
S
2r
z= 3 ( d  b  r )
3 0 I d  b 1
3 0 I
(d  b  r )dr 
[( d  b) ln( 1  b / d )  b]

2 d r
2
3 0

L 
[( d  b) ln( 1  b / d )  b]
I
2

Eg. Determine the mutual inductance between a very long, straight wire and a
conducting equilateral triangular loop. [高考]
(Sol.)

 I
B  aˆ  0  aˆ  B
2r

d
d
L
3
b
2
B 
2
3
(r  d )dr 
0 I 3
3b
[ b  d ln( 1 
)]
2d
3 2


3
3b
 0 [ b  d ln( 1 
)]
I
2d
3 2
Eg. Find the mutual inductance between two coplanar rectangular loops with
parallel sides. Assume that h1 >> h2 (h2 > w2 > d). (台大電研)
(Sol.)
12 
0 h2 I
2
L12 
12 0 h2
( w  d )( w2  d )

ln[ 1
]
I
2
d ( w1  w2  d )

w2
0
(
 h I w d
1
1
w1  d

)dx  0 2 ln( 2

)
d  x w1  d  x
2
d
w1  w2  d
 NN
Neumaun formula: L12  0 1 2
4
L12 
N 2  12 N 2

I1
I1

 B
1
S

N
 dS 2  2
I1
 
d 1d  2
 R
C1 C2


 (  A )  dS
1
S2


 NN
 0 N1 I 1 d 1
∵ A1 
, ∴ L12  0 1 2

4
4 C1 R1
2

N2
I1


 A  d
1
C2
 
d 1d  2
 R
C1 C2
2
Eg. A rectangular loop of width w and height h is situated near a very long wire
carrying a current i1. Assume i1 to be a rectangular pulse. Find the induced
current i2 in the rectangular loop whose self-inductance is L.
(Sol.)
L12
di1
di
 L 2  Ri 2 ,
dt
dt
where L12 
h
12 h d  w  0 i1
w
 
dr  0 ln( 1  )
d
i1
i1
2r
2
d
R
t=0, L
( ) t
di2
L
 Ri 2  L12 I 1 (t )  i2  12 I1e L
dt
L

L12
I 1e
t=T, i2 
L
t>T,
RT
L
, when –I1 is applied
R
 ( )( t  T )
L12
i2  
I1e L
L
4-10 Magnetic Energy
 
1 N N
1 N
1
L
I
I

I k  k   A  Jdv '
Wm=  jk j k

2 j 1 k 1
2 k 1
2 V'
Let V1  L1
I1
di1
1
1
 W1   V1i1dt  L1  i1di1  L1 I12  I11 : Magnetic energy
0
dt
2
2
I2
di2
 W21   V21 I1dt  L21 I1  di2  L21 I1 I 2
0
dt
1
1
1 2 2
 Wm  L1 I12  L21 I1 I 2  L2 I 22   L jk I j I k
2
2
2 j 1 k 1
V21  L21
Similarly,
And W2 
1
L2 I 22
2
Wm 
Generally,
 
∵  k   B  dS n ' 
Sk
∴ Wm 
1 N N
1 N
L
I
I

Ikk
 jk j k 2 
2 j 1 k 1
k 1
N
when
 k   L jk I j
j 1
 
A
  dk
Ck
  1
 
1 N
I k  A  dl k   A  Jdv'

2 k 1
2 V'
Ck
( I k dlk '  J ( aˆ k ' )dlk '  J  vk ' )
 




 




∵   ( A  H )  H  (  A)  A  (  H )  A  (  H )  H  (  A)    ( A  H )
   



And J    H  A  J  H  B    ( A  H )
  

 

1
1
1
1
H  2 ,
 Wm   H  Bdv'  ( A  H )  an dS ' as R    A  ,
R
R
2 V'
2 S'

 
1
dS  R 2    ( A  H )  aˆ n dS '  0
2 S'
 
1
∴ Wm   H  Bdv'   wm dv'
2 V'
V'
1  
H B
2
2W
| B |2
Magnetic energy density: wm 
and L= 2m .
2
I
1
wm   | H |2
2
wm 
Eg. Determine the inductance per unit length of an air coaxial transmission line
that has a solid inner conductor of radius a and a very thin outer conductor of
radius b. [台科大電機所]
(Sol.)
0 I 2 a 3
0 I 2
Wm1=
B 2rdr 
r dr 
2 0 0
16
4a 4 0
2
0 I b 1
0 I 2 b


1 b 2
2
b
Wm2=
B2 2rdr 
dr 
ln , L'  2 (Wm1  Wm 2 )  0  0 ln


a
a
2 0
4
r
4
a
8 2 a
I
1
a
2
1
Eg. Consider two coupled circuits having self-inductance L1 and L2, which carry
currents I1 and I2, respectively. The mutual inductance between the circuits is M.
a) Find the ratio I1/I2 that makes the stored magnetic energy Wm a minimum.
b) Show that M  L1 L2 . [清大核工所]
(Sol.) Wm 
1
1
L1 I 12  MI 1 I 2  L2 I 22
2
2
I 22
I1 2
I1
I 22
I
(a) Wm  [ L1 ( )  2M ( )  L2 ]  [ L1 x 2  2Mx  L2 ], x  1
2
I2
I2
2
I2
dWm
I2
I
M
for minimum Wm
 0  2 (2 L1 x  2 M )  x  1  
dx
2
I2
L1
(b) (Wm)min=
I 22
M2
(
 L2 )  0  M  L1 L2
2
L1
4-11 Magnetic Forces and Torques
Force due to constant flux linkage:

 

Wm
F  d   dWm  (Wm )  d   F  Wm and (T ) z  

Force due to constant current:
dWs   I k 'd k  dW  dWm
k
dWm 

 

1
1
I k  k  dWs  dW  FI  dl  dWm  (Wm )  dl  FI  Wm

2 k
2
Torque in terms of mutual inductance:
1
1
L
Wm  L1 I 12  L12 I 1 I 2  L2 I 22  FI  I1 I 2 (L12 ) , TI  I 1 I 2 12
2
2

Eg. One end of a long air-core coaxial transmission line having an inner an
conductor of radius a and an outer conductor of inner radius b is short-circuited
by a thin, tight-fitting conducting washer. Find the magnitude and the direction
of the magnetic force on the washer when a current I flows in the line.
(Sol.) Wm 
FI  xˆ
 x b
1 2
 x b
x b I
LI , L    B dr   0 dr  0 ln
2
I
I a
I a 2r
2
a
Wm
 I2 b
I 2 L
 xˆ ( )
 xˆ 0 ln( )
x
2 x
4
a
Eg. A current I flows in a long solenoid with n closely wound coil-turns per unit
length. The cross-sectional area of its iron core, which has permeability μ, is S.
Determine the force acting on the core if it is withdrawn to the position. [高考電
機技師]
(Sol.)
1
1
1
1
H 2 dv, Wm ( x  x)  Wm ( x)  (  0  r n 2 I 2   0 n 2 I 2 ) Sx   0 (  r  1)n 2 I 2 Sx

2
2
2
2
Wm  0
 ( FI ) x 
 (  r  1)n 2 I 2 S
x
2
Wm 
  
Magnetic torque: T  m  B
(B=B⊥+B∥, m∥B⊥=>m×B⊥=0)

dT  xˆdF 2b sin   xˆ( IdlB|| sin  )2b sin   xˆ 2Ib 2 B|| sin 2 d



T   dT  xˆ 2 Ib 2 B//  sin 2 d  xˆI (b 2 ) B//  xˆmB//
0
  
 T  m B
Eg. A rectangular loop in the xy-plane with sides b1 and b2 carrying a current I

has in a uniform magnetic field B  xˆBx  yˆBy  zˆBz . Determine the force and
torque on the loop.
  
(Sol.) T  m  B  Ib1b2 ( xˆBy  yˆBx )
4-12 Magnetic Circuits
Define Vm  NI : mmf,   BS : magnetic flux,  
(1)
N I
j
j
  k  k . (2) ∵   B  0 , ∴
k

j
l
: reluctance
S
0
j
Eg. (a) Steady current I1 and I2 flow in windings of N1 and N2 turns, respectively,
on the outside legs of the ferromagnetic core. The core has a cross-sectional area
Sc and permeability μ. Determine the magnetic flux in the center leg.
(Sol.)
1 
l1
l
l
,  2  2 , 3  3
S C
S C
S C
Loop 1: N 1 I 1  (1   3 ) 1  1 2
Loop 2: N1 I1  N 2 I 2  1 1  (1  2 ) 2
1 
2 N1 I1  1 N 2 I 2
1 2  1 3  23
Eg. A toroidal iron core of relative permeability 3000μ0 has a mean radius R=
80mm and a circular cross section with radius b=25mm. An air gap lg=3mm exists,
and a current I flows in a 500-turn winding to produce a magnetic flux of 10-5Wb.
Neglecting flux leakage and using mean path length, find (a) the reluctances of
the air gap and of the iron core, (b) Bg and Hg in the air gap, and Bc and Hc in the
iron core, (c) the required current I.
lg
3  10 3

 1.21  106 ( H 1 )
(Sol.) (a)  g 
7
2
o S 4  10  (  0.025 )
2 0.08  0.003
 6.75  104 ( H 1 )
7
2
3000  (4  10 )  (  0.025 )
Bg
Bc
10 5
3
ˆ

a
5
.
09

10
H

(b) B g  B c  aˆ 
(T)
, Hc 
g

2
0
3000 0
  0.025
c 


(c) NI   C   g  I  0.0256 (A)
Eg. Consider the electromagnet in Figure. In which a current I in an N-turn coil
produce a flux Φ in the magnetic circuit. The cross-sectional area of the core is
S. Determine the lifting force on the armature.
(Sol.)
dWm  d (Wm ) airgap  2(
N
B2
2
Sdy ) 
dy
20
0 S
NI
Rc  2 y 0S
N

I
I
 dWm
2
and
F  yˆ
  yˆ
dy
0 S
L
FI  yˆ
d 1 2
1
NI
2
( LI )   yˆ
(
) 2   yˆ
dy 2
 0 S Rc  2 y /  0 S
0 S
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4-6 Electrostatic Energy

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