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Field and Wave Electromagnetics
Chapter 3 Static Electric FieldsChapter 3. Static Electric Fields
3 1 Introduction3-1 Introduction
Coulomb’s law Experimental lawCoulomb’s law Experimental law
1 2ˆ q qF k12
1 212 2
12
ˆRq qF a kR
=
Radio Technology Laboratory 2
3 2 Fundamental Postulates of Electrostatics in Free Space3.2 Fundamental Postulates of Electrostatics in Free Space
Force : (N)FForce :
El i fi ld i i (전계 강도)
(N)F
Electric field intensity (전계 강도)
F0
lim [ V m ] or [ N C]q
FEq→
=
why? - Not to disturb the charge distributionwhy? Not to disturb the charge distribution of the source
Radio Technology Laboratory 3
3 2 Fundamental Postulates of Electrostatics in Free Space
Two fundamental postulates of electrostatics
3.2 Fundamental Postulates of Electrostatics in Free Space
Two fundamental postulates of electrostatics
E ρε
⎧ ∇ =⎪⎨
(MKS)i ―① 4E πρ∇ =note (cgs)i0
0 E
ε⎨⎪ ∇× = →⎩ irrotational ―②
0
1 v v
Edv dv
Q
ρε
→ ∇ =∫ ∫
∫
i①
0
S
QE d sε
= =∫ Gauss's lawi
0E dl→ =∫ i② 0
( ) 0c
S
E dl
E ds E dl
→ =
∇× = =
∫∫ ∫
i
∵ i i
②
Radio Technology Laboratory 4
Kirchhoff’s voltage law
3 2 Fundamental Postulates of Electrostatics in Free Space
Conservative field
3.2 Fundamental Postulates of Electrostatics in Free Space
Conservative field
1 2
0C C
E dl E dl+ =∫ ∫i i1 2
2 1
1 2
1 2Along Along
P P
P PC C
E dl E dl= −∫ ∫i i1 2
2
1
2
g g
Along
P
PC
E dl= ∫ i2g
Why?
E dl
E∫
Why?
(Usually depends on integral path)
cf) is irrotational!!
i
∵
Radio Technology Laboratory 5
Ecf) is irrotational!!∵
3 3 Coulomb’s Law3-3 Coulomb s Law
Coulomb’s lawCoulomb’s law
ˆ ˆ( ) RS S
qE d s R E R ds= =∫ ∫i i0
20
( )
4
RS S
RqE
R
ε
πε=∴
∫ ∫
0
20
4
ˆ ˆ [V m ] or [ N C]4R
RqE R E R
R
πε
πε= =
General Expression
2 3
0 0
( )ˆ4 4
p qpq q R RE aR R R Rπε πε
′−= =
′ ′− −1 2
20
ˆ [N ]4
q qF RRπε
=cf)
Radio Technology Laboratory 6
3 3 Coulomb’s Law
Charged Shell
3-3 Coulomb s Law
Charged Shell
1 2s ds dsdE ρ ⎛ ⎞= −⎜ ⎟2 2
0 1 2
2 21 2
4
cos
dEr r
d r d r
πε
α
= ⎜ ⎟⎝ ⎠
Ω ⋅ Ω⋅= =solid angle
1 2
cos
0ds ds
dE
α= =
∴ =
solid angle
Discrete ChargesDiscrete Charges
31
( )1 [V m]4
nk k
k
q R RER Rπε
′−=
′∑
Radio Technology Laboratory 7
104 kkR Rπε = ′−
3 3 1 Electric Field due to a System of Discrete Charges3-3.1 Electric Field due to a System of Discrete Charges
Electric dipolep
2 2d dR Rq
⎧ ⎫⎪ ⎪− +⎪ ⎪
3 30
2 24
2 2
qEd dR R
πε⎪ ⎪
= −⎨ ⎬⎪ ⎪
− +⎪ ⎪⎩ ⎭ 3 333 22 2
2
cos3
,2 2 2 4
Rd
d d d dR R R R R d R d
θ
−− −⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤− = − ⋅ − = − ⋅ +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎢ ⎥
⎢ ⎥ ⎣ ⎦⎝ ⎠ ⎝ ⎠⎣ ⎦⎡ ⎤3
23 3
2 2
31 1 (binomial expansion)2
R d R dR RR R
−
− −
⎡ ⎤⎡ ⎤ ⎢ ⎥⋅ ⋅
≅ − ≅ +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎢ ⎥
⎣ ⎦2 3( 1) ( 1)( 2)(1 ) 1
2! 3!n n n n n nx nx x x
⎣ ⎦− − −
± = ± ± ±note
33 q R dd R d
−
±
⎡ ⎤ ⎡ ⎤
Radio Technology Laboratory 8
02 3 2
3 3 312 2 4
q R dE R dR R
d R dR RR πε
− ⎡ ⎤⋅+ ≅ − ∴
⎡ ⎤⋅≅ ⋅ −⎢ ⎥
⎣⎢⎣ ⎦
⎥⎦
3 3 1 Electric Field due to a System of Discrete Charges3-3.1 Electric Field due to a System of Discrete Charges
3 2
( )
1 ˆˆˆ3 , cos sin4
p qd electric dipole moment vector
R pE R p z RR R
θ θ θ
=
⎡ ⎤= ⋅ − = −⎢ ⎥
⎣ ⎦
i
Define
then,
( )( )
3 204
cos
ˆˆ i
R R
R p Rp
R
πε
θ
θ θ θ
⎢ ⎥⎣ ⎦
=i zzθ R
( )( )3
0
cos sin
ˆˆ 2cos sin [V/4
p p R
pE RR
θ θ θ
θ θ θπε
= −
= + m]θ
θ
y
x
Radio Technology Laboratory 9
3 3 2 Electric field due to a continuous distribution of charge3-3.2 Electric field due to a continuous distribution of charge
Continuous distributionContinuous distribution
2 2'0 0
' 1ˆ ˆ ˆ', where 4 4 v
dv Rd E R E R dv RR R R
ρ ρπε πε
= ⎯→ = =⎯ ∫0 0
3'0
4 4
1 ' [V/m]4 v
R R R
RE dvR
πε πε
ρπε
=∴ ∫
) Surface & line charge density casecf
0
' 20
) Surface & line charge density case1 ˆ ' [V/m]
4s
S
cf
E R dsRρ
πε= ∫
' 20
1 ˆ ' [V/m]4
lL
E R dlRρ
πε= ∫
Radio Technology Laboratory 10
Ex 3 4 Infinitely long line chargeEx 3-4 Infinitely long line charge
Determine the electric field intensity of an infinitely long,
ˆ ˆ sol) ', ' 'l l
l
R r r z z dl dzρ ρ
ρ= − =
Determine the electric field intensity of an infinitely long, straight, line charge of a uniform density in air
30 0
ˆ ˆ' ' ' 4 4
l ldz dzR r r z zd ER
ρ ρπε πε
−= = 3
2 2 2
ˆ ˆ( ' )
'
r zr dE zdEr z
d
= ++
32 2 2
0
' where 4 ( ' )
' '
lr
r dzdEr z
z dz
ρ
περ
=+
32 2 2
0
( ' )
'
lz
l
z dzdEr z
r dzdE
ρ
περ
= −4 +
⎧⎪ 3
2 2 20
3
4 ( ' ) at ',
' '
lr
l
dEr z
z zz dzdE
ρ
περ
=⎪⎪ +⎪= ⎨ −⎪ =⎪
Radio Technology Laboratory 11
3'2 2 2
0 ( ' )z z z
dEr zπε
=⎪4 +⎪⎩
Ex 3 4 Infinitely long line chargeEx 3-4 Infinitely long line charge
⎧3
2 2 20
'
4 ( ' ) at ',
' '
lr
rdzdEr z
z zd
ρ
πε
⎧ =⎪⎪ +⎪= − ⎨⎪
3' '2 2 2
0
' '
( ' )'
lz zz z z z
z dzdE dEr z
r dz
ρ
περ ρ
=− =
∞
⎪ = = −⎪
4 +⎪⎩
∫ 32 20 02
ˆ ˆ ˆ [V/m]2
( ' ) l l
rr dzE r E r r
rr z
ρ ρπε πε
∞
−∞∴ = = =
4+
∫
HW. Integrate to have the above result cf) Symmetry Gauss law→
Radio Technology Laboratory 12
3 4 Gauss’s Law and Applications3-4 Gauss s Law and Applications
C lind ical s mmetCylindrical symmetry
( ) QE E dvρ∇ = ⇒ ∇ =∫i i( )'
0 0
By Divergence theorem
0
v
s
QE ds
ε ε
ε←⎯⎯⎯⎯⎯⎯→ =
∫
∫ i0zE =
0ε
22 ,
L l LE ds E r d dz rLEπ ρφ π= = =∫ ∫ ∫i
rE
0 00
2 ,
where, , 0
r rS
r Z
E ds E r d dz rLE
E rE E
φ πε
= =
∫ ∫ ∫
0zE =
0
2
lrE
rρπε
∴ =
Radio Technology Laboratory 13
Ex 3 7 Spherical electron cloudEx 3-7 Spherical electron cloud
for 0 R bρ ρ= ≤ ≤⎧ ˆSpherical symmetry i e E r E→ =0 , for 00 , for
R bR b
ρ ρρ=− ≤ ≤⎧
⎨ = >⎩2
Spherical symmetry i.e. i) 0 b
4
r
R R
E r ER
E ds E ds E Rπ
→ =≤ ≤
⋅ = =∫ ∫iS
30 0
4 3
iR RS
v vQ dv dv R
π
πρ ρ ρ= = − = − ⋅
∫ ∫
∫ ∫0
0
ˆ , 03
E R R R bρε
= − ⋅ ≤ ≤∴
3
ii) b4 (fixed)
R
Q bπρ
≥
= 0
30
(fixed)3
ˆ 3
Q b
bE R
ρ
ρε
=
∴
−
−= 2 , R bR
≥
Radio Technology Laboratory 14
3ε0R
3 5 Electric Potential3-5 Electric Potential
2
1
0 (Vector Identity)
[J/C or V] (Work must be done against the field)P
P
E E VW E dlq
∇× = → =−∇
=− ⋅∫2 2 2 2
1 1 1 12 1
ˆ [V] ( ) ( )P P P P
P P P P
q
V V E dl E dl V l dl dV− =− ⋅ − ⋅ =− −∇ =∫ ∫ ∫ ∫∵ i
Remember when we define gradient
ˆ ˆˆ cos ( )dV dV dn dV dV n l V ldl dn dl dn dn
α= = = = ∇i i i
( )dV V dl=∴ ∇ i
) Refernece point of potential: the zero potential point is usually at infinitycf ) Refernece point of potential: the zero potential point is usually at infinity.cf
Radio Technology Laboratory 15
3 5 1 Electric potential due to a charge distribution3-5.1 Electric potential due to a charge distribution
Point charge at origin potential at R w r t that at infinityPoint charge at origin, potential at R w.r.t. that at infinity
20
ˆ4
qE RRπε
=
20 0
ˆ ˆ( ) [V]4 4
R q qV R R dRR Rπε πε∞
⎛ ⎞=− =⎜
⎠∴ ⎟
⎝∫ i
V21 : Potential difference between point P1 and P2
2 121 P P0 2 1
1 14
qV V VR Rπε
⎛ ⎞= − = −⎜ ⎟
⎝ ⎠
Radio Technology Laboratory 16
3 5 1 Electric potential due to a charge distribution3-5.1 Electric potential due to a charge distribution
No work ˆ
ˆ
r
dl R d
F F r
φ φ=
=
iNo work
0F dl =∴ i
1 [V]n
kqV ∑ '10
[V]4
k
k k
VR Rπε =
=−
∑
Radio Technology Laboratory 17
Ex 3 8 An Electric dipoleEx 3-8 An Electric dipole
1 1q ⎛ ⎞
0
11
1 14
1 cos 1 cos2 2
qVR R
d dR RR R
πε
θ θ
+ −
−−
⎛ ⎞= −⎜ ⎟
⎝ ⎠⎧ ⎛ ⎞ ⎛ ⎞− ≅ +⎪ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎪1
1
2 2 ,
1 cos 1 cos2 2
R Rd R
d dR RR R
θ θ
+
−−
−
⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎪
⎨⎛ ⎞ ⎛ ⎞⎪ + ≅ −⎜ ⎟ ⎜ ⎟⎪ ⎝ ⎠ ⎝ ⎠⎩
2 20 0
cos 1 ˆ , where : dipole moment vector4 4
qdV p R p qdR R
E
θπε πε
= = ⋅ =
= −∇ ( )3ˆ ˆˆ ˆ2cos sin
4V V pV R RR R R
θ θ θ θθ
∂ ∂= − − = +
∂ ∂ ( )30
1 2 31 1 2 2 3 3
4
ˆ ˆ ˆ )
R R RV V Vcf V u u u
h u h u h u
θ πε∂ ∂∂ ∂ ∂
∇ = + +∂ ∂ ∂
2
Note
1. Equipotential line : cos
2 fi ld li iVR c
E R
θ
θ
=
Radio Technology Laboratory 18
2 2. field line : sinEE R c θ− =
Potential due to continuous charge
'
1 '4 v
V dvRρ
πε= ∫ [V] : volume charge distribution
0
'0
41 '
4
v
s
R
V dsR
περ
πε=
∫
∫ [V] : surface charge distribution
'
0
1 '4
lL
V dlRρ
πε= ∫ [V] : line charge distribution
Radio Technology Laboratory 19
Ex 3 9 A uniformly charged diskEx 3-9 A uniformly charged disk
Obtain a formula for the electric field intensity2 2
2
1
y
sol) ' ' ' ' and '' ' '
bs
ds r dr d R z rrV dr d
π
φρ φ
= = +
= ∫ ∫( )
( )
10 02 20 2
12 2 2
'
'b
s
z r
z r z
φπε
ρ
4+
⎡ ⎤= +⎢ ⎥
∫ ∫
( )0 0
2
z r zε
= + −⎢ ⎥⎣ ⎦
'
( )2 2 2
12 2 2
2 2
'cf) ', substitute ''
' 0
r dr z r kz r
k k
+ =+
⎫
∫
( )( )
( )1
2 2 2
2 2
2 2 2 2 2 21
2 2
' 0,
' ,
2 ' ' 2
z b
z
r k z k z
k dkr b k z b k z b dk kkd k dk
1+
⎫= = → =⎪⎪= = + → = + ⇒ = =⎬⎪⎪
∴ ∫ ∫
Radio Technology Laboratory 20
( ) 2 ' ' 2 kr dr k dk ⎪= ⎪⎭
Ex 3 9 A uniformly charged diskEx 3-9 A uniformly charged disk
1⎧ ⎡ ⎤( )1
2 2 2
0
1
ˆ 1 , 02
ˆ
sz z z b zVE V zz
ρε
−⎧ ⎡ ⎤− + >⎪ ⎢ ⎥
∂ ⎪ ⎣ ⎦= −∇ = − = ⎨∂ ⎡ ⎤⎪ ( )1
2 2 2
0
ˆ 1 , 02
sz
z z z b zρε
−∂ ⎡ ⎤⎪− + + <⎢ ⎥⎪ ⎣ ⎦⎩
220
ˆ , 04
ˆFor very large s
NoteQz z
zbz E zπεπ ρ
⎧ >⎪⎪⎨2
02
0
For very large , 4 ˆ , 0
4
sz E zQz z z
zπε
πε
= = ⎨⎪ − <⎪⎩
Radio Technology Laboratory 21
3 6 Conductors in Static Electric Field3-6 Conductors in Static Electric Field
Conductor : Free ElectronsThe electrons in the outermost shells of the atoms of conductors are free to move
Insulator or dielectricsThe electrons in the atoms of insulators are confined to their orbits: no free electronselectrons
SemiconductorsA relatively small number of freely movable charge carriersA relatively small number of freely movable charge carriers→ Allowed energy bands for electrons. → Forbidden band → Band gap
Radio Technology Laboratory 22
3 6 Conductors in Static Electric Field
Charge introduced → Field will be set up → Charges will move away
3-6 Conductors in Static Electric Field
Charge introduced Field will be set up Charges will move away from each other to reach conductor’s surface until the charge and the fieldinside vanish
: Inside a conductor 0, 0Eρ = =
Under static conditions, the E field on a conductor surface is everywhere normal to the surface → Equipotential surface (No tangential component)
0 ( E inside conductor is zero)tabcdaE dl E w= Δ =∫ i ∵
0abcda
t
sn
EsE ds E s ρ
∴ =
Δ= Δ =
∫
∫ i0
0
ns
snE
ερε
=∴
∫
Radio Technology Laboratory 23
3 6 Conductors in Static Electric Field
Boundary conditions at a conductor / Free space interface
3-6 Conductors in Static Electric Field
Boundary conditions at a conductor / Free space interface
0tEρ
=⎧⎪⎨ ( 0 )E field insideconductor=∵
0
snE ρ
ε⎨ =⎪⎩
( 0 )E field insideconductor=∵
Uncharged conductor in a static fieldElectrons move in a direction opposite to that of the field- Electrons move in a direction opposite to that of the field
- Positive charges move in the direction of the field- So that induced free charge will distribute on the conductor surface and create an
induced field in such a way to cancel the external field inside the conductor andinduced field in such a way to cancel the external field inside the conductor and tangent to its surface
Radio Technology Laboratory 24
Ex 3 11 A point charge +Q at the center of a conducting shellEx 3-11 A point charge +Q at the center of a conducting shell
Determine and as functions of the radial distanceE V R
0
Determine and as functions of the radial distance "Ungrounded"
ˆ (a) Spherical Symmetry R
E V R
R R E E R> =∵
( )
21
0
1 1 12
4
,
RS
R
R R
QE ds E R
Q QE V E dR
πε
⋅ = =
= = −∴ =
∫
∫ ( )
( )
1 1 120 0
0
,4 4
b , Ins
R R
i
R RR R R
πε πε∞
<<
∫
2
ide a conductor
0 0RE E= ∴ =
0
2
2 10 0
, whole conducting shell is an equipotential body4
R
R R
QV VRπε=
= =
An amount of negative charge equal to must be induced on the inner shell surface at in order to cancel out the field induced by positive charge at center. i
NoteQ
R R Q−
= +
Radio Technology Laboratory 25
This implies again that + 0must be induced on the outer shell surface at .Q R R=
Ex 3 11 A point charge +Q at the center of a conducting shellEx 3-11 A point charge +Q at the center of a conducting shell
( ) R R<
3 3 320 0
(c)
,4 4
i
R R
R RQ QE V E dR C C
R Rπε πε
<
= = − + = +∫3 2 0
30 0 0 0
where, at is equal to at
1 1 1 1 1 4 4
i
i i
V R R V R R
Q QC VR R R R Rπε πε
= =
⎛ ⎞ ⎛ ⎞= − ∴ = + −⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠0 0 0 0i i⎝ ⎠ ⎝ ⎠
Electric field intensity and potential variations of A point charge +Q at the center of a conducting shell
Radio Technology Laboratory 26