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