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Static Electric Field

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Helmholtzs Theorem

ECE 230 Electromagnetics & Wave

Propagation

A vector field is determined within an additive constant ifboth its divergence & curl are specified everywhere

0&0FieldalIrrotation&Solenoidal

FF 0&0FieldalIrrotation&Solenoidal

FFnot

0&0

FieldSolenoidal&alIrrotation

FF

not

0&0

FieldalIrrotationSolenoidal

FF

norNeither

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Representation of a vector field

ECE 230 Electromagnetics & Wave

Propagation

A vector field can be decomposed into irrotational partF

i& Soleniodal part F

s

si FFF

gF

F

i

i0

GF

F

s

s 0

GFF

gFF

s

i

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ECE 230 Electromagnetics & Wave

Propagation

If Fiis irrrotational we can define Scalar Potential

functionV

VFi

If Fsis Soleniodal we can define Vector Potential

functionA

AFs

AVFFF si

Total vector field Fis given as

Helmholtz's theorem

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Null identities

ECE 230 Electromagnetics & Wave

Propagation

Curl of the gradient of any scalar field is zero

Divergence of a the curl of any vector is zero

0 V

0 V

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Static Electric Field

ECE 230 Electromagnetics & Wave

Propagation

Coulombs law- force b/n two charges q1& q

2

separated by a distance R12

R

qqkaF

R 2

12

211212

q

FEq 0lim

qEF

Electric field intensity (E)Force per unit charge

that a very small stationary charge experiences when it

is placed in a region where an electric field exists

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ECE 230 Electromagnetics & Wave

Propagation

Total outward flow of flux of electric field intensity overany closed surface in free space = 1/0 times the totalcharge enclosed in the surface.

S

QdSE

0

20

10

E

E

Fundamental postulates of

Electrostatics in free space

41

0 dvdvE vv

Applying Divergence

theorem to (4)

Taking volume integral of (1)

Gauss Law

3002 C

dlEE

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Coulomb's Law from Gausss Law

ECE 230 Electromagnetics & Wave

Propagation

q''aroundareasphericalsmalltheis

0

dS

QdSE

S

24 rdSS

2

04 r

qaEr

2

04 r

QqaQEFr

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'

'

'4 2

0

RR

RRa

RR

qaE

qP

qPqP

If the charge is not at the centre of the coordinate system

3

0 '4

'

RR

RRqEP

2

0

2112212

4 R

qqaEqFR

System of discrete charges

n

kk

kk

R

R

R

RqE

13

'

'

04

1

ECE 230 Electromagnetics & Wave

Propagation

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Gausss Law - Applications

ECE 230 Electromagnetics & Wave

Propagation

1. Electric field intensity of an infinite long, straight,line charge of uniform charge density Lin air

S

QdSE0

LlQ

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ECE 230 Electromagnetics & Wave

Propagation

E ER+E (radial & perpendicular component)no component along the line Ez= 0

Gaussian surfacecylindrical

ERconstant over the curved surface,dS = arr d dz

r

S

l

r ElrdzdrEdSE

20

2

0

E

on the top & bottom face Ez= 0

0 S

dSE

00

2

Lr

lQElr S

QdSE

0

raEaE Lrrr

02

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ECE 230 Electromagnetics & Wave

Propagation

2. Electric field intensity of an infinite plane charge

of uniform surface charge density Sin air

ERcancels out

due to side

face

Q = A S

http://arab-training.com/vb/t7171.html

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ECE 230 Electromagnetics & Wave

Propagation

dSEdSaEadSE

dSEdSaEadSE

zzzz

zzzz

.face,circular2

.face,circular1

nd

st

AEdSEdSEz

A

z

S

22

0,2

0,2

0

0

zaEaE

zaEaE

S

zzz

S

zzz

00

2

S

zAQAE

S

QdSE0

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Electric Potential

ECE 230 Electromagnetics & Wave

Propagation

Curl free vector field can be represented as gradient of a scalar

field.VE VScalar Electric Potential

Electric potential is the work done in carrying a charge from onepoint to other.

fieldheaganist tdonework

2

1

12

P

P

dlEVVq

W