2-1
Engineering ElectromagneticsW.H. Hayt, Jr. and J. A. Buck
Chapter 2Coulomb’s Law and Electric Field Intensity
2-2
2.1 Coulomb’s Experimental Law
+ +Q1 Q2
R F
where
Assumption: “Static (or time-invariant) electric field”
Force of repulsion, F, occurs when charges have the same sign.Charges attract when of opposite sign.
2-3
with which the Coulomb force becomes:
Free space permittivity
Scalar,Not vector
2-4
122120
21212
120
2121 44
aR
QQaR
QQFF
πεπε−==−=
( )50,2,@ C10 , 3)2,@(1, C103 4241 −− −=×= QQEx.]
( ) ( )
12 2 1
12
4 4
29
(2 1) (0 2) (5 3) 2 22 2 1 ( 2 2 )
31 4 43 10 10 2 2 2 2
301 3 34 10 936
10 20 20 [N]
x y z x y z
x y zx y z
x y z x y z
x y z
R r r a a a a a aa a a
a a a a
a a a a a aF N
a a a
ππ
− −
−
= − = − + − + − = − +
− += = − +
+ +
× ⋅ − − + − + = = −
× × × /
= − + −
The force on a charge in the presence of several other chargesis the sum of the forces on a charge due to each of the othercharges acting alone. (Superposition Theorem)
2-6
2.2 Electric Field Intensity
Consider the force acting on a test charge, Qt , arising from charge Q1:
where a1t : unit vector directed from Q1 to Qt
Electric field intensity : Force per unit test charge
A more convenient unit for electric field is V/m, as will be shown.
N/C
2-7
Electric Field of a Charge Off-Origin
2-8
Electric Field of a Charge at Origin
zyx azayaxrR
++==
Q is at the origin and a test charge (1 C) is at (x, y, z).
222 zyx
azayaxaa zyxrR
++
++==
+++
+++
++++= zyx
o
azyx
zazyx
yazyx
xzyx
QE
222222222222 )(4πε
2-9
Superposition of Fields from Two Point Charges
For n charges:
* Assumption : Individual charge must be independent to each other. Superposition theorem
2-10
Example
Find E at P, using
First, find the vectors:
3 nC @ P1(1, 1, 0), P2(-1, 1, 0), P3(-1, -1, 0), P4(1, -1, 0) 1)1,(1,@ ? PE =
zyzyx
zxz
aarrRaaarrR
aarrRarrR
+=−=++=−=
+=−==−=
2 22
2
4433
2211
2-11
Example (continued)
2-12
2.3 Field arising from a Continuous VolumeCharge Distribution
2.3.1 Volume Charge Density Definition
Volume charge density (ρv [C/m3]): Distribution of very small particles with a smooth continuous distribution
Small amount of charge ∆Q within a small volume ∆v :
vQ v∆=∆ ρ
Total charge :
r2sinθdrdθdϕ
2-13
Ex. 2-3] Find the charge contained within a 2-cm length of the electron beam shown below.
Charge density : [ ][ ]3106-
310
m/C105
m/C55
5
z
zv
e
eρ
ρ µρ−
−
××−=
−=
( )
( )
5
5
5
0.04 2 0.01 -6 10
0.02 0 0
0.04 0.01 -5 10
0.02 0
0.04-50.01 1050
0.02
5 10
10
10 10
z
z
z
zz
z
Q e d d dz
e d dz
e d
π ρ
φ ρ
ρ
ρ
ρ ρ φ
π ρ ρ
π ρ ρρ
−
∆ ∆ ∆
−
=
−
=
= − × ×
= − ×
−= / − /
∫ ∫ ∫
∫ ∫
∫
2-14
( ) ( ) ( )
( ) ( ) ( ) 368.0)4000(1
400020001
200010
4000200010
))(10())(10(
14020
10
01.0
0
4000200010
01.0
0
400020001001.0
0
2000400010
≈←
−
+−
−−
−−
−=
−
−−
−=
−−=−=
−−−
−
−−−
−−−−−− ∫∫
eee
ee
deedee
π
π
ρπρπ
ρρ
ρρρρ
( ) ( ) [ ]
]pC[40
C104040
120110
40001
2000110 121210
π
πππ
−=
×
−=
−−=
−−≈ −−−
2-15
2.3.2 Electric Field from Volume Charge Distributions
Sum all contributions throughout a volume and take the limit as ∆v approaches zero
The incremental contribution to the electric field intensity at produced by an incremental charge ΔQ at :
rr ′
( ) ( )
( ) ( ) ( )∫ −−=∴
−−
∆=
−−
⋅−
∆=∆
volv
v
dvrrrr
rrE
rrrr
vrrrr
rrQrE
''|'|4
'
'|'|4|'|
'|'|4
30
30
20
περ
περ
πε
2-16
2.4 Line Charge Electric Field A filamentlike distribution of volume charge density such as a very
fine, sharp beam Line charge density ρL [C/m]
Straight-line charge extending along the z-axis Move around the line charge, varying
while keeping ρ and zconstantly.
Unit charge:
Unit electric field intensity:
'dzdQ Lρ=
where
( ) ( )|'|
'|'|4
'|'|4''
20
30 rr
rrrr
dzrrrrdzEd LL
−−
−=
−−
=πε
ρπερ
z
zy
azarrazraayr
'-'''
ρ
ρ
ρ
ρ
=−
===
2-17
By dEz cancellation,
( )( )( ) 2/3220
22220
'
'-4
'
'
'-'4
'
z
azadz
z
azaz
dzEd
zL
zL
+=
++=∴
ρ
ρπερ
ρ
ρρπερ
ρ
ρ
2-18
Appling , ( )∫ +=+ 2222/322 xaax
xadx
( )( )
ρρρ
ρ
ρπερ
ρπερ
ρπερρ
ρρρ
περ
aaEE
zzE
L
L
L
L
0
0
20
2220
2
2
1114
''1
4
==
=
−−⋅⋅
=
+⋅=
∞
∞−
2-19
Another method
[ ]
ρ
ππρ
ρ
ρπερ
ρπερ
θρπε
ρθθρπε
ρ
ρπεθρθ
θρπερ
θθρθρ
ρπερ
περρθρ
θθρθρθ
θρ
aE
dE
dθd
dRdzdE
R
ddzddzz
L
L
LL
LL
LL
0
0
0
0
0
0
00
233
03
0
22
2
2
cos4
sin4
4
sincsc4
csc)(csc44
'csc
csc' csc'cot'
=
=
=−=
−=−=
−==
=
−=∴−=
=
∫
선 전 하 가 있 을 때 임 의 의위치에서의 field는 측정 점에서제일 가까운 위치에 있는 선전하밀도로부터 거리에 반비례하고선전하 밀도에 비례
2-20
2.4.2 Field of an Off-Axis Line Charge
Electric field in case of line displaced to (6,8)
where
Finally:
2-21
2.5 Field of a Sheet Charge
Surface charge density ρs[C/m2]: infinite sheet of charge having auniform density
Consider the field of the infinite line charge by dividing theinfinite sheet into differential-width (dy’) strips
2-22
( )
( )[ ]
general)(in 2
2
22secsec
2
sec' tan' '2
'2
cos2
''
'
0
0
0
2
20
2
222
22
0
22'2
0
2'20
2'20
22
Ns
xs
sss
sx
ssx
sL
aE
aE
dxx
dxdyxydyyx
xE
dyyx
xyx
dydE
yxR
dy
ερερ
ερθ
περθ
θθ
περ
θθθπε
ρ
περθ
περ
ρρ
π
π
π
π
=∴
=
===
==⇐+
=
+=
+=∴
+=
=
−−
∞
∞−
∫
∫
대칭성에 의해 dEy성분은 Canceling-out 됨.
1) charge 평면에 normal 한방향.
2)거리에 무관.
2-23
2.5.3 Capacitor Model in conditions of ρS @ x = 0 and -ρS @ x = a
1) In the region of x > a,
2) In the region of x < 0,
3) In the region of 0 < x < a,
?=E
xs
xs aEaE
00 2
2 ερ
ερ −
== −+
0=+=∴ −+ EEE
( ) ( )xsxs aEaE
−−
=−= −+00 2
2 ε
ρερ
0=+=∴ −+ EEE
)(2
2 00
xs
xs aEaE
−
−== −+ ε
ρερ
xo
s aEEE
ερ
=+=∴ −+0 a x
ρS -ρS𝐸𝐸− 𝐸𝐸+
2-24
2.6 Streamline and Sketches of Fields: One picture is worth about a thousand words if we just knew
what picture to draw. (百聞不如一見)
Electric field due to line charge:
a) Symmetric property is not explained.
b) Although symmetry property isexplained, the longest lines must be drawnin the most crowded region.
ρρπερ aE L
02=
φ
φ
2-25
c) Although symmetry property is explained, the stronger fieldmust be explained with the thicker line, especially at origin.
d) The spacing of the lines is inversely proportional to the strength of the field.
φ
d’d
d < d’
2-26
Methodology of Streamline Construction
Sketch the field of the point charge in 2-dimension (xy-plane).
@ Ez = 0
This equation will enable us to obtain the equations of the streamlines.
2-27
Ex.]
in cylindrical coordinateρ
ρ
ρ
περπερρ
a
aE LL
1
2 2
=
=⇐=
2 2 2 2 2 2
2 2 2 2
1 1( ) ( ) ( ) cos sin
1
: in Catesian coordinate
x x y y z z x yz
x y
x y x x y y
E a a a a a a a a a a a
x ya ax y x y x y
x ya a E a E ax y x y
ρ ρ ρ φ φρ ρ = ⋅ + ⋅ + ⋅ = +
= + + + +
= + = ++ +
( )( )
CxyCxCxyCxy
xdx
ydy
xy
yxxyxy
EE
dxdy
x
y
==+=+=∴
==++
==
.lnlnlnln or lnln
or //
22
22
If this stream line pass through P(-2, 7, 10), C = -3.5
Engineering Electromagnetics�W.H. Hayt, Jr. and J. A. Buck2.1 Coulomb’s Experimental Law슬라이드 번호 3슬라이드 번호 4슬라이드 번호 52.2 Electric Field IntensityElectric Field of a Charge Off-OriginElectric Field of a Charge at OriginSuperposition of Fields from Two Point ChargesExampleExample (continued)슬라이드 번호 12슬라이드 번호 13슬라이드 번호 142.3.2 Electric Field from Volume Charge Distributions2.4 Line Charge Electric Field슬라이드 번호 17슬라이드 번호 18슬라이드 번호 192.4.2 Field of an Off-Axis Line Charge2.5 Field of a Sheet Charge슬라이드 번호 22슬라이드 번호 232.6 Streamline and Sketches of Fields:슬라이드 번호 25Methodology of Streamline Construction슬라이드 번호 27
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