Download Notes 9 3318 Flux

ECE 3318
Applied Electricity and Magnetism
Spring 2016
Prof. David R. Jackson
ECE Dept.
Notes 9
1
Flux Density
From Coulomb’s law:
E
q
E
4 0 r
2
rˆ
 0  8.854187818 1012
F/m
Define:
q
D  0 E
We then have
D
q
4 r 2
“flux density vector”
rˆ
[C/m 2 ]
2
Flux Through Surface
D
Define flux through a surface:
n̂
   D  nˆ dS
 C
S
q
S
In this picture, flux is the flux crossing
the surface in the upward sense.
The electric flux through a surface is analogous
to the current flowing through a surface.
Cross-sectional view
I   J  nˆ dS
A
S
The total current (amps) through a surface is the “flux of the current density.”
3
Current Analogy
Conducting medium
Analogy with electric current
n̂
I   J  nˆ dS
A
S
I
S
J
A small electrode in a conducting
medium spews out current equally
in all directions.
J
I
4 r
2
rˆ
Cross-sectional view
4
Current Analogy (cont.)
D
I
q
4 r 2
rˆ
   D  nˆ dS
S
Conducting medium
Free-space
q
D
J
[C/m 2 ]
 C
J
I
4 r 2
rˆ
I   J  nˆ dS
[A/m 2 ]
A
S
5
Water Analogy
z
N streamlines
Each electric flux line is
like a stream of water.
Water streams
Each stream of water carries
F / N [liters/s].
F
y
x
Water nozzle
F = flow rate of water [liters/s]
Note that the total flow rate through the surface is F.
6
Water Analogy (cont.)
Here is a real “flux fountain” (Wortham fountain, on Allen Parkway).
7
Example
z
Find the flux from a point charge
going out through a spherical surface.
D

r

D  nˆ dS
S
q
y

S
x
D  rˆ dS
S


S
nˆ   rˆ
(We want the flux going out.)



S
 q

rˆ   rˆ dS

2
 4 r 
q
dS
2
4 r
8
Example (cont.)

2 

0
 q  2
0  4 r 2  r sin  d d
q

4
2 
  sin  d d
0 0

q

2  sin  d
4
0
q

 2  2 
4
  q [ C]
We will see later that his has to be true for any surface
that surrounds the charge, from Gauss’s law.
9
D
Flux in 2D Problems
The flux is now the flux per meter in the
z direction.
n̂
 l   D  nˆ dl
C/m
C
l
C
We can also think of the flux through
a surface S that is the contour C
extruded h in the z direction.
l
   D  nˆ dl
S
C 
S
h [m]
C
   l   h  m 
10
Flux Plot (2D)
Rules:
1) Lines are in direction of D
2) D 
# flux lines
 length
NL
D
L
L = small length perpendicular to the flux vector
NL = # flux lines through L
y
D
L
x
s
Flux lines
l0
Rule #2 tells us that a region with a
stronger electric field will have flux lines
that are closer together.
L
s  spacing 
 1/ D
NL
11
Example
Draw flux plot for a line charge
 l 0 
E  ˆ 

2


0


 V/m

  
D  ˆ  l 0 
 2 
C/m 2 
y

Rule 2:
Nf lines
D  C1
L
x
D
# lines
L
# lines
# lines
 C1
L
 
C1  # lines 

   
Hence
l0 [C/m]
l 0 C1  # lines 
l 0
# lines
 


 constant

2    

2 C1
12
Example (cont.)
# lines
 constant

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.
y

Nf lines
This is actually a consequence
of Gauss’s law.
L
(This is discussed later.)
x
l0 [C/m]
13
Example (cont.)
y
Choose Nf = 16
Note: Flux lines come out
of positive charges and
end on negative charges.
l0 [C/m]
Flux lines can also
terminate at infinity.
x
# lines
 constant

Note: If Nf = 16, then each flux line represents l0 / 16 [C/m]
14
Flux Property
The flux (per meter) l through a contour is proportional to the
number of flux lines that cross the contour.
NC is the number of flux lines through C.
C
 l   D  nˆ dl  NC
C
Please see the Appendix for a proof.
Note: In 3D, we would have that the total flux through a surface is proportional to
the number of flux lines crossing the surface.
15
Example
l0 = 1 [C/m]
y
Choose: Nf = 16
Graphically evaluate
 l   D  nˆ dl
C
C
x
 1

 C/m  /line  
 16

 l   4 lines  
Note: The answer will be more accurate
if we use a plot with more flux lines!
1
l 
4
C/m
16
Equipotential Contours
The equipotential contour CV is a contour on which the potential is constant.
y
Line charge example
D
Flux lines
l0
x
 = -1 [V]
Equipotential contours CV
 = 1 [V]
 = 0 [V]
17
Equipotential Contours (cont.)
Property:
D  CV
CV
D
The flux line are always perpendicular
to the equipotential contours.
(proof on next slide)
CV: ( = constant )
18
Equipotential Contours (cont.)
Proof of perpendicular property:
Proof:
Two nearby points on an equipotential
contour are considered.
B
  VAB   E  dr
On CV :
  0
A
B
 E   dr
CV
E  r  0
A
D
 E  r
B
D  r  0
r
A
D  r
The r vector is tangent (parallel) to the contour CV.
19
Method of Curvilinear Squares
2D Flux Plot
V
+
CV
-
B
D
Assume a constant voltage difference V
between adjacent equipotential lines in a
2D flux plot.
A
Note: Along a flux line, the voltage always
decreases as we go in the direction of the flux line.
B
B
B
A
A
A
V  VAB   E  d r   E d r   E dl  0
“Curvilinear square”
(If we integrate along the flux line, E is in the same direction as to dr .)
Note: It is called a curvilinear “square” even though the shape may be rectangular.
20
Method of Curvilinear Squares (cont.)
Theorem: The shape (aspect ratio) of the “curvilinear
squares” is preserved throughout the plot.
Assumption: V is constant throughout plot.
CV
V
D
CV
W
L
V
L
 constant
W
21
Method of Curvilinear Squares (cont.)
Proof of constant aspect ratio property
V -
CV
B
VAB   E  d r  V
D
+
A
W
A
L
B
If we integrate along the flux line, E is in the same direction as dr .
B
Hence,
E
d r  V
A
B
so
E
 dl  V
Therefore
E L  V
A
22
Method of Curvilinear Squares (cont.)
V -
CV
Hence,
D
+
L
V
L
E
Also,
W
L
NL
NL
1
D
 C1
 C1
L
L
W
B
A
so
C1
W
D
Hence,
L  V   D


W  E   C1
 V D V

 0  constant

C1
 C1 E
(proof complete)
23
Summary of Flux Plot Rules
Rule 1: Lines are in direction of D.
Rule: 2 Equipotential contours are perpendicular to the flux lines.
Rule 3: L / W is kept constant throughout the plot.
If all of these rules are followed, then we have the following properties:
Property 1:
D
# flux lines
 length
Property 2: We have a fixed V between equipotential contours.
24
Example
Line charge
y
Note how the flux lines get
closer as we approach the
line charge: there is a
stronger electric field there.
D
x
L
W
l0
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
Example
A parallel-plate capacitor
Note: L / W  0.5
http://www.opencollege.com
26
Example
Coaxial cable with a square inner conductor
L
1
W
Figure 6-12 in the Hayt and Buck book.
27
Flux Plot with Conductors (cont.)
Conductor
Some observations:
 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.
http://en.wikipedia.org/wiki/Electrostatics
28
Example of Electric Flux Plot
Note: In this example,
the aspect ratio L/W is
not held constant.
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29
Example of Magnetic Flux Plot
Solenoid 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.
Magnetic flux lines
Ferrite core
Solenoid windings
30
Appendix:
Proof of Flux Property
31
Proof of Flux Property
NC : flux lines
Through C
C
C
L
D

n̂
L
One small piece of the contour
(the length is L)
NC : # flux lines
C
 l   D  nˆ  L  D cos L
so
 l  D
or
D
 L cos 
 l  D  L


L
L

32
Flux Property Proof (cont.)
 l  D
 L 
D
Also,

N C
D 
L
L
(from the definition of a flux plot)
Hence, substituting into the above equation, we have
 N C 
 l  D L  
  L   N C
 L 
Therefore,
 l  NC
(proof complete)
33