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Prof. David R. JacksonECE Dept.
Fall 2014
Notes 9
ECE 2317 Applied Electricity and Magnetism
1
Flux Density
q
E
20
ˆ4
qE r
r
Define:
“flux density vector”
2
0D E
120 F/m8.854187818 10
From Coulomb’s law:
We then have2
2ˆ [C/m ]
4
qD r
r
Flux Through Surface
q
D
n̂
S
ˆ CS
D n dS
Define flux through a surface:
3
The electric flux through a surface is analogous to the current flowing through a surface.
ˆ AS
I J n dS The total current (amps) through a surface is the “flux of the current density.”
In this picture, flux is the flux crossing the surface in the upward sense.
Cross-sectional view
Flux Through Surface (cont.)
4
Analogy with electric current
ˆ AS
I J n dS
A small electrode in a conducting medium spews out current equally in all directions.
I
n̂
S
J2
ˆ4
IJ r
r
Cross-sectional view
Flux Through Surface (cont.)
5
22
ˆ [A/m ]4
IJ r
r2
2ˆ [C/m ]
4
qD r
r
J
I
D
q
Analogy
Example
2
2
ˆ
ˆ
ˆ ˆ4
4
S
S
S
S
D n dS
D r dS
qr r dS
r
qdS
r
(We want the flux going out.)
ˆ ˆn r
Find the flux from a point charge going out through a spherical surface.
6
x
y
z
q
D
S
r
Example (cont.)
22
20 0
2
0 0
0
sin4
sin4
2 sin4
2 24
qr d d
r
qd d
qd
q
C[ ]q
7
We will see later that his has to be true from Gauss’s law.
Water Analogy
x
y
z
w
8
Each electric flux line is like a stream of water.
w = flow rate of water [liters/s]
Each stream of water carries w / Nf [liters/s].
Water nozzle
Nf streamlinesWater streams
Note that the total flow rate through the surface is w.
Water Analogy (cont.)
9
Here is a real “flux fountain” (Wortham fountain, on Allen Parkway).
Flux in 2D Problems
l
D
n̂
C
ˆ C/ml
C
D n dl
The flux is now the flux per meter in the z direction.
10
We can also think of the flux through a surface S that is the contour C extruded h in the z direction.
ˆ CS
D n dl h [m]
S
l
C ml h
Flux Plot (2D)
1) Lines are in direction of D
2) #
D flux lines
length
LND
L
L = small length perpendicular to the flux vector
NL = # flux lines through L
l0
DL
x
y
Rules:
11
Rule #2 tells us that a region with a stronger electric field will have flux lines
that are closer together. Flux lines
Example
0
0
V/mˆ2
lE
#D
L
lines
Draw flux plot for a line charge
20 C/mˆ2
lD
0 0
1 1 1
#
2 2lD
C C C
linesconstantHencel0 [C/m]
Nf lines
x
y
L
12
1 1
1
# #
#
D C CL
C
lines lines
lines
Rule 2:
#
lines
constant
l0 [C/m]
13
Example (cont.)
This result implies that the number of flux lines coming
out of the line charge is fixed, and flux lines are thus never
created or destroyed.
This is actually a consequence of Gauss’s law.
(This is discussed later.)
Nf lines
x
y
L
Example (cont.)
Note: If Nf = 16, then each flux line represents l0 / 16 [C/m]
Choose Nf = 16 l0 [C/m]
x
y
Note: Flux lines come out of positive charges and
end on negative charges.
Flux line can also terminate at infinity.
14
#
lines
constant
Flux Property
NC is the number of flux lines through C.C
ˆl C
C
D n dl N
The flux (per meter) l through a contour is proportional to the number of flux lines that cross the contour.
15
Note: In 3D, we would have that the total flux through a surface is proportional to the number of flux lines crossing the surface.
Please see the Appendix for a proof.
Example
Nf = 16
l0 = 1 [C/m]
x
y
C ˆl
C
D n dl
Graphically evaluate
lines /line1
4 C/m16l
C/m1
4l
16
Note: The answer will be more accurate if we use a plot with more flux lines!
Line charge example
l0
D
x
y
17
Equipotential Contours
Equipotential contours CV
= 1 [V]
= -1 [V]
= 0 [V]
Flux lines
The equipotential contour CV is a contour on which the potential is constant.
Equipotential Contours (cont.)
D CV
18
Property:
The flux line are always perpendicular to the equipotential contours.
CV: ( = constant )
CV
D
(proof on next slide)
Equipotential Contours (cont.)
Proof:
0
0
0
E r
D r
D r
On CV :
19
The r vector is tangent (parallel) to the contour CV.
CV
D
A
B
r
B
AB
A
V E dr
E r
Two nearby points on an equipotential contour are considered.
Proof of perpendicular property:
Method of Curvilinear Squares
Assume a constant voltage difference V between adjacent equipotential lines in a
2D flux plot.
2D flux plot
Note: Along a flux line, the voltage always decreases as we go in the direction of the flux line.
0B B B
AB
A A A
V V E dr E dr E dl
20
Note: It is called a curvilinear “square” even though the shape may be rectangular.
(If we integrate along the flux line, E is in the same direction as to dr.)“Curvilinear square”
CV
DV
AB+
-
Method of Curvilinear Squares (cont.)Theorem: The shape (aspect ratio) of the “curvilinear squares” is preserved throughout the plot.
L
Wconstant
CV
DV CV
WL
V
21
Assumption: V is constant throughout plot.
Proof of constant aspect ratio property
B
AB
A
V E dr V
If we integrate along the flux line, E is in the same direction as dr.
so
B
A
E dr VB
A
E dl V E L VTherefore
Method of Curvilinear Squares (cont.)
Hence,
22
CV
D
WL
A
B
V+
-
Also,
VL
E
1 1
1L LN ND C C
L L W
1CW
D
Hence,
01 1 1
D DL V V V
W E C C E C
constant
so
Method of Curvilinear Squares (cont.)
23
Hence,
(proof complete)
CV
D
WL
AB
V+
-
Summary of Flux Plot Rules
24
1) Lines are in direction of D .
2) Equipotential contours are perpendicular to the flux lines.
3) We have a fixed V between equipotential contours.
4) L / W is kept constant throughout the plot.
#D
flux lines
length
If all of these rules are followed, then we have the following:
ExampleLine charge
The aspect ratio L/W has been chosen to be unity in this plot.
This distance between equipotential contours (which defines W) is proportional to the radius (since the distance between flux lines is).
25
l0
D
x
y
L
W
Note how the flux lines get closer as we approach the
line charge: there is a stronger electric field there.
Example
http://www.opencollege.com
26
A parallel-plate capacitor Note: L / W 0.5
Example
Figure 6-12 in the Hayt and Buck book.
Coaxial cable with a square inner conductor
1L
W
27
28
http://en.wikipedia.org/wiki/Electrostatics
Flux lines are closer together where the field is stronger.
The field is strong near a sharp conducting corner.
Flux lines begin on positive charges and end on negative charges.
Flux lines enter a conductor perpendicular to it.
Some observations:
Flux Plot with Conductors (cont.)Conductor
Example of Electric Flux Plot
29
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a Genetronics Inc., 11199 Sorrento Valley Rd., San Diego, CA 92121, USAb Research Institute for Plastic, Cosmetic and Reconstructive Surgery Inc., 3399 First Ave., San Diego, CA 92103, USA.
Note: In this example, the aspect ratio L/W is
not held constant.
Example of Magnetic Flux PlotSolenoid near a ferrite core (cross sectional view)
Flux plots are often used to display the
results of a numerical simulation, for either the
electric field or the magnetic field.
30
Magnetic flux lines
Ferrite core Solenoid windings
Appendix: Proof of Flux Property
31
Proof of Flux Property
ˆ cosl D n L D L
NC : flux lines
Through C
cosl D L
l D L
so
or
32
NC : # flux lines
C
D
n̂
L
One small piece of the contour(the length is L)
C
LC
L
D
L
Flux Property Proof (cont.)
Also,
l D L
CND
L
(from the definition of a flux plot)
Hence, substituting into the above equation, we have
Cl C
ND L L N
L
l CN Therefore,
33
L
D
(proof complete)