Prof. D. Wilton ECE Dept. Notes 9 ECE 2317 ECE 2317 Applied Electricity and Applied Electricity and Magnetism Magnetism Notes prepared by the EM group, University of Houston.
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Prof. D. WiltonECE Dept.
Notes 9
ECE 2317 ECE 2317 Applied Electricity and MagnetismApplied Electricity and Magnetism
Notes prepared by the EM group,
University of Houston.
Electric Flux DensityElectric Flux Density
20
0
22
4
[C/m ]4
qE r
r
D E
qD r
r
Define:
“flux density vector”
q
E
Analogy with Current Flux DensityAnalogy with Current Flux Density
I
J
22
[A/m ]4
IJ r
r
current flux density vector dueto a point source of current
r
r̂ 22
[A/m ]4r
IJ
r
The same current I passes through everysphere concentric with the source, hence
Note: if I is negative, flux density vector points towards I
Current Flux Through SurfaceCurrent Flux Through SurfaceJ
n
S
S
I J n dS A I
Electric Flux Through SurfaceElectric Flux Through Surface
q
D
n
S
S
D n dS C
ExampleExample
2
2
4
4
S
S
S
S
D n dS
D r dS
qr r dS
r
qdS
r
(We want the flux going out)
n r
x
y
z
q
D
S
Find the flux from a point charge going out through a spherical surface.
Spherical Surface (cont.)Spherical Surface (cont.)
22
20 0
2
0 0
0
sin4
sin4
2 sin4
2 24
qr d d
r
qd d
qd
q
[C]q
3D Flux Plot for a Point 3D Flux Plot for a Point ChargeCharge
Flux Plot (3D)Flux Plot (3D)Rules:
1) Flux lines are in direction of D
2) #flux lines
areaD
SND
S
NS = # flux lines through S
S
D
S = small area perpendicular to the flux vector
Flux Plot (2D)Flux Plot (2D)Rules:
1) Flux lines are in direction of D
2) #flux lines in plane #flux linesalong
length
#flux lines(in plane)(since # flux linesalong )
length
xy zD
z
xyz z
LND
L
l0
DL
L = small length perpendicular to the flux vector
NL = # flux lines through L
Note: We can construct a 3D problem by
extending the contour in the z direction by one meter to create a surface.
ExampleExample
0
0
V/m2
lE
1
1
0
#lines #lines
#lines
2
D CL L
C
Draw flux plot for a line charge
20 C/m2
lD
0
1
#lines= constant
2 C
Hence
Nf lines
l0 [C/m]
x
y
L
Example (cont.)Example (cont.)
Choose Nf = 16
l0 [C/m]
x
y
Note: If Nf = 16, then each flux line represents l0 / 16 [C/m]
Flux PropertyFlux Property
NS : flux lines
Through SS
S
S
D n dS N
• The flux through a surface is proportional to the number of flux lines in the flux plot that cross the surface (3D) or contour (2D).
• Flux lines begin on positive charges (or infinity) and end on negative charges (or infinity)
Flux Property (Proof)Flux Property (Proof)
cosD n S D S
cosD S
D S
NS : # flux lines
S
D
n
NS : flux lines
Through SS
DS
S
DS
Flux Property Proof (cont.)Flux Property Proof (cont.)
Also,
S
D D S
SND
S
(from the definition of a flux plot)
Hence SN
SN Therefore,
ExampleExample
Nf = 16
l0 = 1 [C/m]
z = 1 [m] for surface S
x
y
S S
D n dS Find
14 lines C/lines
16
1
4C
Equipotential Surfaces (Contours)Equipotential Surfaces (Contours)
D CV
Proof:
0
0
0
0
PE
F dr
E dr
D dr
On CV :
CV: (V = constant )
dr
CV
D
Equipotential Surfaces (cont.)Equipotential Surfaces (cont.)
CV
D Assume a constant voltage difference V between adjacent equipotential lines in a 2D flux plot.
Theorem: shape of the “curvilinear squares” is preserved throughout the plot.
“curvilinear square”
2D flux plot
Equipotential Surfaces (cont.)Equipotential Surfaces (cont.)
Proof:
CV
D
WL
A
B
B
A
E dr V
Along flux line, E is parallel to dr
Hence,
B
A
E dr VB
A
E dr V E L VOr
Equipotential Surfaces (cont.)Equipotential Surfaces (cont.)
Also,
VL
E
1 1
1L LN ND C C
L L W
1CW
D
Hence,0
1 1
constantDL V V
W C E C
so
CV
D
WL
AB
E L V
ExampleExample
Line charge
l0
D
x
y
ExampleExampleFlux plot for two line charges
hx
y
h
R1 R2
r = (x, y)
l0-l0
0
0 2 22 22l
x x h y y x x h y yD E
x h y x h y
flux lines
- - - - - - - - - - - equipotential lines
line charges of opposite sign line charges of opposite sign
line charges of same sign line charges of same sign
Example Example
0 1 [C/m]l
32fN
Find the flux through the red surface indicated on the figure (z = 1 m)
+ -
Counting flux lines:
Example Example
0 1 [C/m]l
32fN
+ -
1/ 32 [ / flux line]
2 flux lines
C
1/16 [ ]C
Example Example
Software for calculating cross-sectional view of 3D flux plot for two point charges: http://www.xmission.com/~locutus/astro2-old/ElectricField/ElectricField.html
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