Dr. Ahmed Said Eltrass

Electrical Engineering Department

Alexandria University, Alexandria, Egypt

Fall 2015

Part-I: Electro-Static

Lecture 4

Office hours: Sunday (10:00 to 12:00 a.m )

4th floor, Electrical Engineering Building

Chapter 3

Electric Flux Density, Gauss’s

Law, and Divergence

Electrical Flux Lines

• Electric flux lines are imaginary lines drawn to show the direction of

the electric field and their density is proportional to the magnitude of

the electric field intensity |E|

Applications: Lightning rods - Microwave oven Examples of Course Project

• Electric flux lines start on (+) charges and end on (-) charges,

and are unbroken

• The electric flux lines have the same direction of the electric

field E

• The flux density is proportional to the magnitude of the electric

field |E|

• The total electric flux (ψ) produced by a charge (Q) is the

number of lines in Coulombs

Ψ= |Q| Coulomb

Example: 1- Given a point charge (Q) placed at the origin. Find the amount of flux

passing through:

-A sphere of radius (r)

- A closed cylinder of radius a and length L

- An infinite sheet (Z=constant)

Electrical Flux Density (D)

• It is a vector defined to be in the same direction of the electric field

intensity (E) and it is given by:

• D is the Electric flux per unit area of S (C/m2) • ψ is the total flux passing perpendicular to an area S (Coulomb)

2C/m ED

S area thelar toperpendicuflux only the Counting

cosD

constantD &direction general ain is D If 2

D

constantD & S area D If 1

S

S

S

S

D

D

S

S

Sd

Sd

dD

dD

cosdD

(function) surface on the variableis D &direction general ain is D If -3

Sds

sd

D

Gauss’s Law The electric flux passing through any closed surface is equal to

the total charge enclosed by that surface.

S

Sd

Sd

Sd

Sd

dD

dD

cosdD

d crossingflux

enclosedenclosed ChargedD QSS

Gauss’s law

charge Volume4

charge Surface3

charge Line2

chargespoint Several1

:enclosed charge of Types

V

dvQ

dsQ

dlQ

QQ

v

S

s

ll

n

/εDED

Q

Q/SDD

D

SDDSS

SDDS

D

D

D

D

calculatecan weThen, .Get 5

choosedyou surface closed by the Get 4

)( integral theoutside bringcan we

surface,Gaussian thealongconstant is As-3

)d( surface// if dD dD

)d//( surface if 0 dD

surface. closed the toor tangent normaleither

ofdirection theknow should We2

it. alongconstant is away that such in carefully

surface)(Gaussian (S) surface a choose toneed we, find To1

:Req on.distributi chargecertain a :Given

enclosed

Applications of Gauss’s law

S

SQ

QD

dD

:Req . surface closed theand :Given

enclosed

enclosed

This application is

the most important

vs

vs

or by chargedcylinder Infinite

)or ( charge of Sphere

charges ofsheet Infinite

charge line Infinite

chargePoint

: todue or find toused is law Gauss Mainly,

ED

Examples

2- Using Gauss’s Law, find the electric flux density and the electric

field intensity due to a point charge (Q) placed at the origin.

x

y

z

Q

Figures Satisfy a high

degree of symmetry

3- Using Gauss’s Law, find the electric flux density and the

electric field intensity due to an infinite line charge on the z-

axis

x

y

z

4- Using Gauss’s Law, find the electric flux density due to an

infinite sheet of charge on the xy-plane

x

y

z

5- Using Gauss’s Law, find the electric flux density everywhere

due to a sphere of radius (a) and uniform charge density ( ) v

x

y

z

v

a

6- Using Gauss’s Law, find the electric flux density everywhere

due to the infinite cylinder shown

x

y

z

a

b

1v

2vs

b

bρ

aρ

s

v

v

at

)(a cylinders twoebetween th

)(cylinder inner theinside

2

1