```Chapter 1 Electromagnetic Field Theory
1-1 Electric Fields and Electric Dipoles

 Q 1

Gauss’s law of E :  E  dS     dv'
s
 

divergence
theorm
    Edv' 
v
1
 
v'

1
 10 9 (F/m) in the
and ε0=

36

 dv'    E 
free space.



qaˆ RR'

For q at R ' , field point at R  E 




E due to a system of discrete charges: E 

Volume source ρ  E 

Surface source ρs  E 
1
4 v '

aR

R2


and aˆ RR'
2
4 | R  R' |
1
n
q k ( R  R' k )
k 1
| R  R' k |3
4 | R  R ' |


q ( R  R')

4


3
 
R  R'
   .
R  R'



dv' 
 
1
s
a
dS '
R

s
'
4
R2
1
R

4 v ' | R
|3
dv'

Line source ρl  E 
1
4

 aR
l'
l
R2
dl '



q1 q 2
q1 q 2 R
R

Eg. Show that Coulomb’s law F  a R
, where aˆ R  , R  R .
2
3
R
4R
4R







q1
q1 R
q1
2

(Proof) ∵ F  q 2 E and  E  d S 
 4r E , E  a R
4R 2 4R 3




∴ F  aR

q1q2
4R 2
Eg. Determine the electric field intensity of an infinitely long line charge of a
uniform density ρl in air.
(Sol.)


L
 E d S   
s
2rLE 
0
2
0
Erd dz  2rLE
l L   l
, E  ar
0
2 0 r
Eg. Determine the electric field intensity of an infinite planar charge with a
uniform surface charge density ρs.


 E d S  2 ES  2 EA , 2EA 
(Sol.)
s
  s
  z 2 ,

E  0

  z s ,

2 0
s A
0
z0
z0
Eg. A line charge of uniform density ρl in free space forms a semicircle of radius
b. Determine the magnitude and direction of the electric field intensity at the
center of the semicircle. [高考]





 (bd )
l 
l
(Sol.) d E y   l
,

sin

E

y
E


y
sin

d



y
y
2

0
40 b
20 b
40 b
Eg. Determine the electric field caused by spherical cloud of electrons with a
volume charge density ρ=-ρ0 for 0  R  b (both ρ0 and b are positive) and ρ=0
for R>b. [交大電子物理所]
(Sol.)
(a) R  b
Q  0
  b3
4 3 
Q
0
b , E  aˆ R


a
R
3
40 R 2
3 0 R 2
(b) 0  R  b




E  a R E , d S  a R dS ,

si

si
Q   dv    0  dv    0
v

E d S  E  dS  E 4 R 2
v
  R
4 3 
R , E   aR 0
3
3 0
Eg. A total charge Q is put on a thin spherical shell of radius b. Determine the
electrical field intensity at an arbitrary point inside the shell. [台大電研]
(Sol.)
s 

Q
, dE  s
2
40
4b
d 
dS1
r1
2
cos  
dS 2
r2
2
 dS1 dS 2
 2  2
r
r2
 1
cos  , dE 




 s  d
d 


0
40  cos  cos  
Electric dipole: A pair of equal but opposite charges with separation.





d
d  d
d2
| R  | 3  [( R  )  ( R  )] 3 / 2  [ R 2  R  d  ] 3 / 2
2
2
2
4
 
 
R d
3 R d
 R [1  2 ] 3 / 2  R 3 [1 
]
2 R2
R
3

 
d
3 R d
| R  | 3  R 3 [1 
]
2
2 R2



 
  

d
 R d
R 

q 
2 
2  q
E


3
 3
4   
4R 3


d
d
 R

R
2
2 






    
3 R d R  d   1
 R2
 4R 3




 

p  z p  p a R cos   a sin  




E
p
4R 3
 
R p  p cos 
    
3 R P R  P 
 R2







 

 a R 2 cos   a sin  


Eg. At what value of θ does the electric field intensity of a z-directed dipole have
no z-component.

(Sol.) E 

p
40 R





(a r 2 cos   a sin  ) , z  a r cos   a sin 
3
No z-component  2 cos  cos  sin   sin   0  tan 2   2  θ=54.7° or 125.3°
1-2 Static Electric Potentials

p2 

 V2  V1    Ed l and  2V    
E  V
p1
Electric potential due to a point charge:
R 

q
q

a
R dR 
2
4R
4R
q 1
1
 V p1 
(  )
4 R2 R1
V   aR

V21  V p2
Electric potential due to discrete charges:
qk
1 n
V

4 k 1 R  R' k
Electric potential due to an electric dipole:
q 1
1
V 
(
 )
4 R R
If d<<R, we have
1 
d

  R  cos 
R 
2

and
1 
d

  R  cos 
R 
2

qd cos 
V 
,
4R 2
1
1
d


 R 1 1 
cos 
 2R

d


 R 1 1 
cos 
 2R



p a R
p  qd V 
4R 2




 E  V   a R
(V )
 V


V
p
 a

(
a
2
cos


a
R
 sin  )
R
 4R 3
Scalar electric potential due to various charge distributions:
Volume source ρ  V 
Surface source ρs  V 
Line source ρl  V 
1

1
4 
R
v'
1
4 
s'
s
R
dv ' .
dS '
l
dl '
4  R

Note: 1. V is a scalar, but E is a vector.

2. E  V is valid only in the static EM field.
Eg. Obtain a formula for electrical field intensity along the axis of a uniform line
charge of length L. The uniform line-charge density is ρl. [高考]
(Sol.) R  z  z ' , z 
L
2

 z  L / 2  
L
dz '
 l ln 
, z

2
z  z ' 40  z  L / 2 

 dV

l L
L
E  z
z
, z
2
2
dz
2
40 z  L / 2
V
l
40

L/2
L / 2


Eg. A finite line charge of length L carrying uniform line charge density ρl is

coincident with the x-axis. Determine V and E in the plane bisecting the line
charge.
(Sol.) V  
L / 2

 l dx
L/2
40

and E  V  y

 l
x 2  y 2 20
l 

2 0 y 
2
 
L
  L
2
  ln
ln

y




2
 2


 


y




2
2 
 L / 2   y 
L/2
Eg. A charge is distributed uniformly over an L  L square plate. Determine V

and E at a point on the axis perpendicular to the plate and through its center.
(Sol.)  s 
2

 s L / 2    L 
Q
L
2
2
2
2
2
2
2


, y y z , V
ln    y  z   ln y  z dy
20  L / 2    2 
2
L2


 
2

  L 2

L
L

 2   z 2  

 
2
Q L   2 
2
1 

ln 
 z tan 
2 

2
2
 0 L  2   L 
L
L

2
2

 2 2   z  2 
 z 2 2   z




2

L

 


Q
2
1 
E  V  z
tan 
2
2
 0 L
 z 2 L   z 2
 

2















Eg. A positive point charge Q is at the center of a spherical conducting shell of an

inner radius Ri and an outer radius Ro. Determine E and V as functions of the
 
R
Q
Q
Q
V


EdR 
(Sol.) R>Ro,  EdS  E 4R 2 
, E
,
2

s

0
4 0 R
40 R
Ri<R<Ro, E  0 , V  V
R<Ri, E 
C
Q
40 R
2
R  R0

Q
40 Ro
, V    EdR  C 
Q  1
1
Q

   V 
40  Ro Ri 
40
Q
4 0 R
C
1
1
1
 
 
 R R o Ri 
Eg. A charge Q is distributed uniformly over the wall of a circular tube of radius

b and height h. Determine E and V on its axis (a) at a point outside the tube, (b)
at a point inside the tube.
(Sol.) dV  
2
0
(a) Vo  

2 0 b 2  z  z '
2

Eo   z
(b) Vi  

2 0 b 2  z  z '
2



 s bdz '
2 0 b 2  z  z '
2
b s
b  b2  z 2
Q
ln
, s 
2
2bh
2 0 z  h   b 2  z  h 
dV  b s 
1
1

z

2
2
2
dz
2 0  b   z  h 
b  z2

 s bdz '
z
0
40 b 2  z  z '
2
 s bdz '
h
0
 s bd ' dz '


h
z
 s bdz '
2 0 b 2  z ' z 
2
 sb  s
2
ln 2 z  b 2  z 2  h  z   b 2  h  z  



2 0 b



Eg. Consider two spherical conductors with radii b1 and b2 (b2>b1) that are
connected wire. The distance between the conductors is very large in comparison
to b2 so that charges on spherical conductors may be considered uniformly
distributed. A total charge Q is deposited on the spheres. Find (a) the charges on
the two spheres, and (b) the electric field intensities at the sphere surfaces?
(Sol.)
Q1
Q2
Q
b
(a)
, 1  1 , Q1  Q2  Q

40 b1 40 b2 Q2 b2
b1
b2
Q and Q2 
Q
b1  b2
b1  b2
Q1 
(b) E1n 
Q1
and
40 b12
E2n
b
E
Q2
, 1   2

2
40 b2 E 2  b1
2
 Q1 b2


 Q2 b1
Eg. Obtain a formula for the electric field intensity on the axis of a circular disk
of radius b that carries uniform surface charge density ρs. [高考]
(Sol.) ds'  r ' dr ' d ' , R  z 2  r ' 2
V 
s
40
2
0
r'
b
  z
0
2
V
E  V   z
z



As z>>1  z z  b
2
 r'

2 1/ 2
dr ' d ' 


s 2
1/ 2

z  b2   z
2 0

  s
2
2 1 / 2
,
 z 2 1  z z  b
0
 

1 / 2
 z s 1  z z 2  b 2
,
 2 0



2 1 / 2





z0

z0


 b 2 
b2
s
 1 2 , E  z
2z
40 z 2
  Q
 z 4 z 2
  0
Q
 z
 4 0 z 2
z0
z0
Eg. Make a two-dimensional sketch of the equipotential lines and the electric
field lines for an electric dipole.
(Sol.)
qd cos 
For an electric dipole, V 
=constant  R  cv cos
4 0 R 2


dl  kE , where k is a constant.





 

a R dR  a Rd   a R sin  d  k  a R E R  a E  a E 


dR
Rd 
dR Rd  R sin  d

,
, R  c E sin 2 


2 cos  sin 
ER
E
E
1-3 Magnetic Fields


 B

B
Magnetic field: H  
 M , where μ0=4π×10-7 (A/m) in the free space.

0
  I d  aˆ R
Magnetic flux density: B  0 
4 C R 2

 
H

d
l

I



H
J


Ampere’s law of H :
1

  




B
  B  J  J m  J    M or   (  M )  J ,
0
o



 





B
H
 M   H  dl  I , and B   0 ( H  M )   0 (1   m ) H   0  r H ,
0

where  r  1   m 
0



Gauss’s law of B :   B  0   B  d S  0
S
∵   B  0 , A fulfills B    A
  B   J      A  (  A)   2 A
Choose   A  0   2 A    J (Note:  2V  

 
A

∵ V
,
∴
dv
'
4
4 
R
V'
1

J
dv'

R
V'

is scalar Poisson’s equation)

(Wb/m)


 
 
Magnetic flux:    B  dS   (  A)  dS   A  dl '
S
 I
Biot-Savart’s law: B 
4

A
4
J
I
dv' 

R
4
V'

∴ B    A     I
C

dl 'aˆ R I
 R 2  4

dl ' R
 3
C' R

dl
 R , ∵   ( f G)  f  G  (f )  G ),
C'
d l '  I

R  4
C'

 4

S
(Wb)
d l'
I  1
1

I
   ( R )  4   R   dl '( R )  dl '  4 
C'
C'
d l '  aˆ R
R2




  xˆ x  yˆ y  zˆ z
,



dl '  xˆdx' yˆdy ' zˆdz '    dl '  0

Note:


 I dl 'aˆ R
I dl 'R
and then dB 
(
)
4
4 R 3
R2
Eg. A direct current I flows in a straight wire of length 2L. Find the magnetic
flux density B at a point located at a distance r from the wire in the bisecting
plane.

1
(Sol.) dl 'R  zˆdz '(aˆ r r  zˆz ' )  aˆ rdz ' , R  ( z 2  r 2 ) 2

oI
B  aˆ
4

L
L
rdz '
oIL
 aˆ
2 32
( z' r )
2r L2  r 2
2
Eg. Find the magnetic flux density at the center of a square loop, with side w
carrying a direct current I.
w
w
, r
in this case,
2
2
(Sol.) L 
oI

B  4  zˆ
2
w
2
 zˆ
w w 2 w 2
( ) ( )
2 2
2
2 2oI
w
Eg. Find the magnetic flux density at a point on the axis of a circular loop of
radius b that a direct current I.
(Sol.) d l '  aˆ bd ' , R  zˆz  aˆ r b , R  ( z 2  b 2 )1
2
d l '  R  aˆ r bzd ' ẑ b 2 d '
 oI
B
4

2
0
oIb 2
aˆ r bz  zˆb 2
(z 2  b2 )
d '  zˆ
32
2( z 2  b 2 ) 3 2

oI
. In case of z=0, B  zˆ
2b
Eg. Determine the magnetic flux density at a point on the axis of a solenoid with
radius b and length L, and with a current in its N turns of closely wound coil.

(Sol.) dB 
zˆ 0 Ib 2
3
2 2
2[( z  z ' ) 2  b ]
B
L
0

 NI 
dB  o

(
N
)dz '
L
Lz
2L  ( L  z) 2  b 2


12



z
zL
   o NI 

2
2 
2
2
2L  z  b
z b
( z  L) 2  b 2


z




 

Ampere’s law of B :   B   J   B  dl  I , where μ=μ0 in the free space.
c
Eg. An infinitely long, straight conductor with a circular cross section of radius b
carries a steady current I. Determine the magnetic flux density both inside and
outside the conductor. [交大光電所]
(Sol.)
(a) Inside the conductor, r  b :
 B dl  
2
0
Brd   2rB  o(
C1
r 2
orI
r
) I  o( ) 2 I => B  aˆ 
b
b
2b 2
2


oI
ˆ
B

d
l

2

rB


oI  B  a
C 2
2r
(b) Outside the conductor:
Eg. A long line carrying a current I folds back with semicircular bend of radius b.
Determine magnetic flux density at the center point P of the bend. [高考]
(Sol.)

 I 
 I
B  B1  B2 , where B1  2  zˆ o , B2  zˆ o
4b
4b
Eg. A current I flows in the inner conductor of an infinitely long coaxial line and
returns via the outer conductor. The radius of the inner conductor is a, and the
inner and outer radii of the outer conductor are b and c, respectively. Find the
magnetic flux density B for all regions and plot B versus r. [高考電機技師]
(Sol.) 0  r  a , B  aˆ 
b  r  c , B  aˆ (
I
rI
, a  r  b , B  aˆ 
2
2r
2a
c2  r 2
c b
2
2
)
I
2r
Eg. Determine the magnetic flux density inside an infinitely long solenoid with
air core having n closely wound turns per unit length and carrying a current I.
(Sol.) BL  onLI => B  onI
Eg. The figure shows an infinitely long solenoid with air core having n closely
wound turns per unit length. The windings are slanted at an angle α and carry a
current I. Determine the magnetic flux density both inside and outside the
solenoid.
0, 0  r  b

(Sol.) B1   20 bnI sin 
,
 bnI sin 
 aˆ 0
, r b
aˆ
2r
r

 zˆ o nI cos  , 0  r  b
, B  B1  B2
B2  
0, r  b
Eg. Determine the magnetic flux density inside a closely wound toroidal coil with
an air core having N turns and carrying a current I. The toroid has a mean
(Sol.)
 B  d l  2rB   NI
o


oNI
B  aˆ B  aˆ
, (b-a)<r<(b+a), B  0 for r<(b-a) and r>(b+a)
2r
Eg. In certain experiments it is desirable to have a region of constant magnetic
flux density. This can be created in an off-center cylindrical cavity. The uniform
axial current density is J  zˆJ . Find the magnitude and direction of B in the
cylindrical cavity whose axis is displaced from that of the conducting part by a
distance d. [台大電研、清大電研、中原電機]
(Sol.) J  zˆJ ,
 Bdl  
o
I
If no hole exists,
2r1 B 1   o r12 J  B 1 
 o r1
2

 o J
B x1 
y1


2
J 
B   o J x
y1
1

2

For - J in the hole potion, B 2  
 o r2
2

o J
y2
 B x 2 
2
J 
B    o J x
2
 y 2
2
At y1  y 2 and x1  x 2  d  B x  B x1  B x 2  0 , and B y  B y1  B y 2 
o J
2
d
1-4 Electromagnetic Forces

Lorentz force equation: F  q( E  v  B)



Electric force: Fe  qE . Magnetic force: Fm  qv  B
Eg. An electron is injected with an initial velocity vo  yˆvo into a region where
both an electric field and a magnetic field B exist. Describe the motion of the
electron if E  zˆEo and B  xˆBo . Discuss the effect of the relative magnitude of
E0 and B0 on the electron paths in parts.
(Sol.) m
v
 e( E  v  B) ,
t
 v x

0


t
v x  0


 E  zˆEo
e
Eo
Eo
 v y


  Bov z
 v y  (vo  ) cos  0 t 

m
Bo
Bo
 B  xˆBo  t

Eo
e
 v z

e
  ( Eo  Bov y ) v z  (  vo ) sin  0 t ;  0  Bo

Ro
m

m
 t


x  0
Eo

Eo
E
c2
c
c
(If v o 
)
 ( y  o t)2  (z  2 )2  ( 2 )2
Bo   y   sin  o t  B t
Bo
o
o
o
o


E
c
 z   2 (1  cos  o t ), c 2  vo  o
o
Bo


Magnetic force due to B and I:

 
 
d   dq  
 
Fm  qv  B  dFm  dq  B 
d   B  Id   B , ∴ Fm  I  d   B
dt
dt
C
Eg. Determine the force per unit length between two infinitely long parallel
conducting wires carrying currents I1 and I2 in the same direction. The wires are
separated by a distance d. [清大電研]




 I
 II
B12   xˆ 0 1  F12 '   yˆ 0 1 2
(Sol.) F12 '  I 2 ( zˆ  B12 ) ,
2d
2d
Eg. Calculate the force per unit length on each of three equidistant, infinitely
long, parallel wires d apart, each carrying a current of I in the same direction.
Specify the direction of the force.

(Sol.) I 1  I 2  I 3  I , B2  xˆ 2 B12 cos 30   xˆ

3 0 I 
3 0 I 2
, f 2   zˆI  B2   yˆ IB 2   yˆ
2d
2d
Eg. The bar AA’, serves as a conducting path for the current I in two very long
parallel lines. The lines have a radius b and are spaced at a distance d apart. Find
the direction and the magnitude of the magnetic force on the bar. [中山物理所]
(Sol.)

 I 1
1
B   zˆ 0 ( 
),
4 y d  y

d   yˆ dy

 

d b 
 I2 1
 I2 d
1
 dF  Id   B   xˆ 0 ( 
)dy  F   dF   xˆ 0 ln(  1)
b
4 y d  y
2
b
Application of the electric forces: The e-paper
Scalar electric potential function: V 
1

dv'

4 V ' R
 
Vector magnetic potential function: A 
4

J
dv '

R
V'
Retarded potentials:


 (t  R v)
1
A
(
R
,
t
)

,
V ( R, t ) 
dv
'
4
4 
R
V'

J (t  R / v)
dv'

R
V'
1-5 Faraday’s Law and Magnetic Dipoles


 

B 
B
 dS
Faraday’s law:   E  
or  E  d    

t
t
C
S




 A
B

∵  E  
  (  A)    ( E  )  0
t
t
t


 A

A
∴  V fulfills E 
 V  E  V 
t
t



A
Note: In static field: E  V , but in time-varying field: E  V 
t

 
 
 
B 
d
Emf: V   E  dl . Magnetic flux:    B  dS ,  E  d      dS  V  
dt
t
C
S
S

 
Motional emf: V '   (v  B)  dl (Volt)
C






Fm  
 
 
Fm  qv  B 
 v  B   E m  V '    E m  dl   (v  B)  dl
q
C
Eg. A circular loop of N turns of conducting wire line in the xy-plane with its

center at the origin of a magnetic field specified by B  zˆB0 cos(r 2b) sin t ,
where b is the radius of the loop and ω is the angular frequency. Find the emf
induced in the loop.
(Sol.)
b
   [ zˆB0 cos(
r
) sin( t )]  [ zˆ 2rdr ] 
8b 2 
(  1) B0 sin( t )
 2
2b
Nd
8N 2 
V 

b (  1) B0 cos(t )
dt

2
0
Eg. A metal bar slides over a pair of conducting rails in a uniform magnetic field

B  zˆB0 with a constant velocity v. (a) Determine the open-circuit voltage V0 that
appears across terminals 1 and 2. (b) Assuming that a resistance R is connected
between the two terminals, find the electric power dissipated in R. Neglect the
electric resistance of the metal bar and of the conducting rails. [交大電子物理所]
1'
(Sol). (a) V0  V1  V2   ( xˆv  zˆB0 )  ( yˆdl )  vB0 h
2'
(b) Pe  I 2 R  (
vB0 h 2
(vB0 h) 2
) R
R
R
(W)
(V)

2
Eg. The circuit in Fig. is situated in a magnetic field B  zˆ3 cos(5 10 7 t  x)
3
μT. Assuming R=15Ω, find the current i. [中山物理所]
0.6
2
(Sol.)    3 cos(5  10 7 t  x)10 6  (0.2dx)
0
3
V 
i
d
2
 45[cos(5  10 7 t   0.6)  cos(5 10 9 t )]
dt
3
,
V
 1.76 sin( 5 10 7 t  0.2 )
2R
Eg. A conducting sliding bar oscillates over two parallel conducting rails in a

sinusoidally varying magnetic field B  zˆ5 cos(t ) T. The position of the sliding
bar is given by x=0.35(1-cosωt), and the rails are terminated in a resistance R=0.2
Ω. Find i.
(Sol.)   5 cos t  0.2(0.7  x) , x=0.35(1-cosωt), i  
1 d
R dt
 i  1.75 sin t  (1  2 cos t )
Eg. The Faraday disk generator consists of a circular metal disk rotating with a
constant angular velocity ω in a uniform and constant magnetic field of flux
density B  zˆB0 that is parallel to the axis of rotation. Brush contacts are the
open-circuit voltage of the generator if the radius of the disk is b.
4
0
B0 b 2
  
(Sol.) V0   (v  B)  dl   [( aˆ r )  zˆB0 ]  (aˆ r dr )  B0  rdr 
(V)
3
b
2

Magnetic dipole moment: m = zˆIS  zˆm , where S is the area of the loop that carries
I and m=IS.

 0 m
 aˆ R
Vector potential of a magnetic dipole:
, where
A
4R 2
  m

B    A  o 3 aˆ R 2 cos   aˆ sin 
4R


Eg. For the small rectangular loop with sides a and b that carries a current I.

Find the vector magnetic potential A at a distant point P(x,y,z). And determine


the magnetic flux density B and A . [交大光電所]
(Sol.)

  m
 0 m

 aˆ R
, where m=Iab, B    A  o 3 aˆ R 2 cos   aˆ sin 
A
2
4R
4R


V
mk



k 1
Magnetization vector: M  lim
(A/m), where m k is the magnetic dipole
V 0 V
moment of an atom.
dA
 M 
 o M  aˆ R
o
o  1
1
 
dv
'

M


'
(
)
dv
'



'

M


'

 R 
4
R
4  R
4R 2
 

 A

o
4
o
4

V'

V'
'M

dv' o
R
4

'M
d v ' o
R
4
M 
    R dv'

V'


(  'F d v'    F  dS ' )
V'
s
M  aˆ n '
d S'
R

∴ Magnetization volume current density: J m    M

Magnetization surface current density: J ms  M  aˆ n
(A/m2)
(A/m)
Equivalent Magnetization Charge Densities:
Vm 
1
4

S'
M  aˆ n '
1
d S '
R
4
 
V'
('M )
d v'
R
(Note: V  1
4 o

P  aˆ n '
1
dS '
R
4 o
Define the magnetization surface charge density as
magnetization volume charge density as  m    M

V'
 
 'P
dv ' )
R
 ms  M  aˆ n
and the
Eg. A circular rod of magnetic material with permeability μ is inserted coaxially
in the long solenoid. The radius a of the rod is less than the inner radius b of the
solenoid. The solenoid’s winding has n turns per unit length and carries a
 

current I. (a) Find the values of B , H , and M inside the solenoid for r<a and
for a<r<b. (b) What are the equivalent magnetization current densities Jm and
Jms for the magnetized rod? [清大電研]
(Sol.) (a) r < a: H  zˆnI , B  zˆnI , M 
B
0
 H  zˆ(

 1)nI ,
0
a < r < b: H  zˆnI , B  zˆ o nI , M  0



(b) J m  'M  0 , J ms  M  aˆ n  ( zˆ  aˆ r )(  1)nI  aˆ (  1)nI
0
0
Eg. A ferromagnetic sphere of radius b is uniformly magnetized with a

magnetization M  zˆM 0 . (a) Determine the equivalent magnetization current

densities J m and J ms . (b) Determine the magnetic flux density at the center of
the sphere. [台大電研]

(Sol.) (a) J m  'M  0 , J ms  ( aˆ R cos   aˆ  sin  ) M o  aˆ R  aˆ  M o sin 
(b) d B  zˆ
 o ( J ms b d  )(b sin  ) 2
2(b 2 )
3
 zˆ
2
M
o M o 3
sin  , B  zˆ o o
2
2


0
2
sin 3  d   zˆ  0 M 0 .
3
Eg. Determine the magnetic flux density on the axis of a uniformly magnetized
circular cylinder of a magnetic material. The cylinder has a radius b, length L

and axial magnetization M  zˆM 0 . [台大物研]

(Sol.) J m  'M  0 , J ms  M  aˆ n  ( zˆM 0 )  aˆ r  aˆ M 0 , dI  J ms dz '
B  zˆ 
L
0

 0 M 0 b 2 dz '
2 ( z  z' ) 2  b 2

3
 zˆ
2
 0 M 0 
2



2
2 
 z 2  b2
( z  L)  b 

z
zL
Eg. A cylindrical bar magnet of radius b and length L has a uniform
magnetization M  zˆM 0 along its axis. Use the equivalent magnetization charge
density concept to determine the magnetic flux density at an arbitrary distant
point. [交大電子所]
(Sol.)  ms
M 0 , top

 M  aˆ n   M 0 , bottom , ρm=0 in the interior region

0, sidewall
qm  b 2  ms  b 2 M 0  Vm ( x, y, z ) 

qm 1
1
(

) (A)
4 R  R 
2
q m L c o s (b M o ) L c o s
M c o s

 T 2 , where M T  b 2 LM 0
2
2
4R
4R
4R
B    o Vm 
o M T
(aˆ R 2 cos   a sin  ) (T)
4R 3
Consider an infinitely long solenoid with n turns per unit length around to create a
magnetic field; a voltage V1=  n d
dt
is induced unit length, which opposes the
current change. Power P1=-V1I per unit length must be supplied to overcome this
induced voltage in order to increase the current to I. The work per unit volume
Bf
required to produce a final magnetic flux density Bf is W1=  HdB .
0
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