1 Gauss’s Law For r > a Reading: Chapter 28. 2 Gauss’s Law Chapter 28.
ENE 325 Electromagnetic Fields and Waves Lecture 3 Gauss’s law and applications, Divergence, and...
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Transcript of ENE 325 Electromagnetic Fields and Waves Lecture 3 Gauss’s law and applications, Divergence, and...
ENE 325Electromagnetic Fields and Waves
Lecture 3 Gauss’s law and applications, Divergence, and Point Form of Gauss’s law
1
Review (1)Review (1)
Coulomb’s law Coulomb’s force
electric field intensity
or V/m
1 212 122
0 124
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R
121
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�������������� FE
Q
2
04
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QE a
R
2
Review (2) Electric field intensity in different charge configurations
infinite line charge
ring charge
surface charge
02LE a
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2 2 3/ 2
02 ( )L
zah
E aa h
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02S
zE a
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3
Gauss’s law and applications
“ The net electric flux t hrough any closed surf
ace is equal to the tota l charge enclosed by t
hat surface”. If we completely enclo
se a charge, then the n et flux passing throug
h the enclosing surfac e must be equal to the
charge enclosed, Qenc.
5
Gauss’s law and applications
The integral form of Gauss’s law:
Gauss’s law is useful in finding the fields for problems t hat have a high degree of symmetry by following these
steps: Determine what variables influence and what components
of are present. Select an enclosing surface, Gaussian surface , whose surface
vector is directed outward from the enclosed volume and i s everywhere either tangential to or normal to
encD dS Q ����������������������������
D��������������
D��������������
S��������������
D��������������
D��������������
6
Gauss’s law and applications
The enclosing surface must be selected in order for to be constant and to be able to pull it out of the integral.
D dS D dS ����������������������������
D��������������
7
Ex1 Determine from a charge Q located at the origin by using
Gauss’s law. 1 .
2. Select a Gaussian surface
3. Dr at a fixed distance is constant and normal to a Gaussian surface,
can be pulled out from the integral .
D��������������
( ) rrD D r a��������������
2 sin rdS r d d a ��������������
8
Ex2 Find at any point P ( , , z ) from an infinite length line of charg
e density L -on the z axis.1 . From symmetry,2. Select a Gaussian surface with ra
dius and length h.3 . D at a fixed distance is constant a
nd normal to a Gaussian surface, can be pulled out from the integra
l. ant and normal to a Gaussian su rface, can be pulled out from the i
ntegral.
D��������������
D D a��������������
9
3Ex A parallel plate capacitor has surfac e charge +S located underneath a top pl
-ate and surface charge S located on a b ottom plate. Use Gauss’ s law to find a
nd between plates. D��������������
E��������������
10
Ex5 A point charge of 0.25 C is located at r = 0 and uniform surface charge densities are located as follows: 2 mC/m2 - at r = 1 cm and 0.6 mC/m2 at r = 1.8 c
m. Calculate at D��������������
a) r = 0.5 cm
b) r = 1.5 cm
12
EE6 A uniform volume charge density of 80 C/m3 is present throughout region 8 mm < r < 10 mm. Let v = 0 for 0 < r < 8 mm,
a)Find the total charge inside the spherical surface r = 10 mm
b)Find Dr at r = 10 mm.
14
Divergence and Point form of Gauss’s law (1) Divergence of a vector field at a particular point i
n space is a spatial derivative of the field indicatin g to what degree the field emanates from the poin t. Divergence is a scalar quantity that implies whe
ther the point source contains a source or a sink o f field.
0limv
D dSdiv D
v
����������������������������
��������������
where = volume differential element
v
16
Divergence and Point form of Gauss’s law (2)
or we can write in derivative form as
.yx zDD D
Dx y z
��������������
Del operator: x y z
It is apparent that0
lim encv
v
Qdiv D
v
��������������
therefore we can write a differential or a point form of Gauss’s law as
vD ��������������
17
Divergence and Point form of Gauss’s law (3)
For a cylindrical coordinate:
1 1 zD D
D Dz
��������������
For a spherical coordinate:
22
1 1 1sin
sin sinr
DD r D D
r r rr
��������������
18
positive indicates a source of flux. (positive charge) negative indicates a sink of flux. (negative charge)
Physical example
The plunger moves up and down indicating net movement of molecules out and in,respectively.
D��������������
.D dS Ddv ������������������������������������������
An integral form of Gauss’s law can also be written as
D��������������
19