lecture_4
DESCRIPTION
Lecture_4TRANSCRIPT
Dr. Ahmed Said Eltrass
Electrical Engineering Department
Alexandria University, Alexandria, Egypt
Fall 2015
Office hours: Sunday (10:00 to 12:00 a.m )
4th floor, Electrical Engineering Building
ELECTROMAGNETICS
Lecture 4
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|
• 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
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
Ideas a- Using Gauss’s Law, find the electric flux density everywhere
x
y
z
v
a
x
y
z
a
b
1v
2v
s
uniform are and , , 21 svv ρρ
cr
brr
ρ
ρ
s
v
vv
at uniform is
a , 1
uniform are and
2
31
x
y
z
a
b
1v
2v3v
s
c
Ideas
b- Using Gauss’s Law, find the electric flux density everywhere
Ideas
C- Find the electric flux density everywhere if you have a point
charge Q at the origin and uniform charge density ( ) for a<r<b
x
y
z
a
b
v
Q
v
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
Ideas
a- Using Gauss’s Law, find the electric flux density everywhere
cρ
ρ
v
vs
b , 1
uniform are and
2
1
x
y
z
a
b
1v2v
s
c