ENE 325 Electromagnetic Fields and Waves
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Transcript of ENE 325 Electromagnetic Fields and Waves
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ENE 325Electromagnetic Fields and Waves
Lecture 2 Static Electric Fields and Electric Flux density
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Review (1)
Vector quantityMagnitudeDirection
Coordinate systemsCartesian coordinates (x, y, z) Cylindrical coordinates (r, , z)Spherical coordinates (r, , )
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Review (2) Coulomb’s law
Coulomb’s force
electric field intensity (V/m)
1 212 122
0 124
�������������� QQF a
R
121
2��������������
�������������� FE
Q
2
04
��������������R
QE a
R
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Review (3) Key variables:
Coordinate system and its corresponding differential element
charge Q a unit vector
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Outline Electric field intensity in different charge configurations
infinite line charge ring charge surface charge
Examples from previous lecture
Electric flux density
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Infinite length line of charge The derivation of and electric field at any point in space
resulting from an infinite length line of charge. (good approximation)
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Infinite length line of charge only varies with the radial distance select point P on - z axis for convenience. select a segment of charge dQ at distance –z, we then
have
��������������E
��������������
p zzE E a E a
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Infinite length line of charge Consider another segment at distance z, z components
are cancelled out, we then have
��������������
pE E a
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Infinite length line of charge
From
We can write
Total field
2
04
��������������R
QE a
R
2
04
��������������R
dQdE a
R
2
04
��������������R
dQE a
R
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Infinite length line of charge
Consider each segment
Ez components are cancelled due to symmetry.
2 2
��������������
��������������
L
z
zR
dQ dz
R a za
R a zaa
R z
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Infinite length line of charge
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Ring of charge
determine at (0,0,h)
cancels each other
��������������E
��������������dE
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Ring of charge
Consider each segment:
2 2
��������������
��������������
L
z
zR
dQ dL
R aa ha
R aa haa
R a h
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Surface charge
Surface charge density S (c/m2)
dQ = Sdxdy
��������������
x y zx y zE E a E a E a
Since this is an infinite place, Ex and Ey components are cancelled due to symmetry.
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Surface charge Consider each segment:
Devide the whole area into infinite length of line charges
02
��������������
L S
L
dy
dE a
Integrate over length y to get total electric field. Convert the radial component into cylindrical coordinates
y za ya ha
Ey components are cancelled out due to symmetry.
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Surface charge
No dependence on a distance from the sheet
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Concentrate ring (alternative approach)
Total field is integrated from = 0 to
2 2 3/ 2
0
( )
2 ( )
�������������� zSd hadE
h
for each ring
Then
2 2 3/ 20 0
0
2 ( )
.2
zS
Sz
ha dE
h
E a
��������������
��������������
h
z
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Volume charge
Volume charge density V (c/m3) plasma doped semiconductor
Complicate derivation due to so many differential elements and vectors.
2
04
��������������V V
Rd
E aR
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Ex1 Determine the distance between point P (5, 3/2, 0) and point Q (5, /2, 10) in cylindrical coordinates.
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Ex2 Determine a unit vector directed from
(0, 0, h) to (r, , 0) in cylindrical coordinates.
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Ex3 Determine a unit vector from any point on z = -5 plane to the origin.
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Ex4 Find the area between on the surface of a sphere of a radius a. Given
= 0 and = .
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Ex5 A charge Q1 = 0.35 C is located at (0, 4, 0). A charge Q2 = -0.55 C is located at (3, 0, 0). Determine at point (0, 0, 5).E
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Ex6 Determine at point (-2, -1, 4) given a line charge located at x = 2 and y = -4 with a charge density L = 20 nC/m.
E��������������
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Ex7 Determine at the origin given a square sheet of charge located at z = -3 plane. The sheet is extended from -2 x 2 and -2 y 2 with a
surface charge density S = 2(x2+y2+9)3/2 nC/m2.
E��������������
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Electric flux density
Negative charges are drawn to the outer sphere Electric flux lines are radially directed away from inner sphere to outer sphere or begin from positive charges +Q and
terminate on negative charges -Q.
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Electric flux density
Electric flux density, (C/m2)
Note: (chi) is a flux in Coulomb unit and is equal to charge Q on the sphere
24
rD ar
��������������
2
04r
QE a
r
��������������
So we have 0D E
����������������������������
where 0 = 8.854x10-12 Farad/m
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The amount of flux passing through a surface is
given by the product of and the amount of surface normal to. Same polarity charges repel one another
Note: = surface vector
Dot product:
cosD S ����������������������������
S��������������
cos ABA B A B ��������������������������������������������������������
x x y y z zA B A B A B A B ���������������������������� for Cartesian coordinates.
Dot product is a projection of A on B multiplies by B
Electric flux density
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In case the flux is varied over the surface,
Electric flux density
The flux through a surface that is an angle to the direction of flux a) is less than the flux through an equivalent surface normal to the direction of flux b)
.D dS ����������������������������
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Ex8 C/m2. Given the surface defined by = 6 m, 0 90 and -2 z
2, calculate the flux through the surface.
10 5D a a ��������������
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Ex9 A charge Q = 30 nC is located at the origin, determine the electric flux density at point (1, 3, -4) m.
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Ex10 Determine the flux through the area 1x1 mm2 on a surface of a cylinder at r = 10 m, z = 2 m, = 53.2 given 2 2(1 ) 4x y zD xa y a za ��������������
C/m2.