ECE 2317
Applied Electricity and Magnetism
Spring 2014
Prof. David R. Jackson
ECE Dept.
Notes 9
1
Flux Density
From the Coulomb law:
q
E 
4  0 r
2
rˆ
E
 0  8 .8 5 4 1 8 7 8 1 8  1 0
12
 F/m 
Define:
q
D  0 E
We then have
D 
“flux density vector”
q
4 r
2
rˆ
2
[C /m ]
2
Flux Through Surface
D
Define flux through a surface:
nˆ
 
 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.
Top view
I 
 J  nˆ dS  A 
S
The total current (amps) through a surface is the “flux of the current density.”
3
Flux Through Surface
Analogy with electric current
nˆ
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
Top view
4
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
4 r
2
dS
5
Example (cont.)
2 
 

0


 
 q  2
  4 r 2  r sin  d  d 

0 
q
4
q
4
q
4
2 

sin  d  d 
0 0

2   sin  d 
0
 2   2 
  q [C ]
We will see later that his has to be true from Gauss’s law.
6
Water Analogy
z
Nf streamlines
Each electric flux line is
like a stream of water.
Water streams
Each stream of water carries
qw / Nf [liters/s].
qw
y
Water nozzle
x
qw = flow rate [liters/s]
Note that the total flow rate through the surface is qw.
7
Water Analogy (cont.)
Here is a real “flux fountain” (Wortham fountain, on Allen Parkway).
8
D
Flux in 2D Problems
The flux is now the flux per meter in the
z direction.
nˆ
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 one meter in the z direction.
l
 
S
 D  nˆ dl  C 
S
1 [m]
C
   l  1  m  
9
Flux Plot (2D)
Rules:
1) Lines are in direction of D
2) D 
# flux lines
 length
L = small length perpendicular to the flux vector
NL = # flux lines through L
y
D
 L
D 
x
l0
Flux lines
NL
 L
Rule #2 tells us that a region with a
stronger electric field will have flux lines
that are closer together.
10
Example
Draw flux plot for a line charge
  l0 
E  ˆ 

2



0


 V /m 

  l0 
ˆ
D   

2



 C /m 2 


y

D 
Rule 2:
# lines
Nf lines
D  C1
 L
x
l0 [C/m]
Hence
# lines


 D
 L
# lin e s
 L
 C1
# lin e s
 
C 1  # lin e s 


   

   
 0
  l0 

 constant
C 1  2  C 1
2 C 1
11
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
G auss's law :
 D  nˆ dS
 Q encl
S
l0 [C/m]
 D  nˆ dS  0
but
Q encl  0
S
12
Example (cont.)
y
Choose Nf = 16
Note: Flux lines come out
of positive charges and
end on negative charges.
l0 [C/m]
Flux line can also
terminate at infinity.
x
# lines

 constant
Note: If Nf = 16, then each flux line represents l0 / 16 [C/m]
13
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.
14
Example
l0 = 1 [C/m]
y
Nf = 16
Graphically evaluate
l 
C
 D  nˆ dl
C
x
 1

 l   4 lines  
  C /m  /line  
 16

l 
1
4
 C /m 
15
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]
16
Equipotential Contours (cont.)
Property:
D  CV
CV
The flux line are always perpendicular
to the equipotential contours.
D
(proof on next slide)
CV: ( = constant )
17
Equipotential Contours (cont.)
Proof of perpendicular property:
Proof:
Two nearby points on an equipotential
contour are considered.
On CV :
  0
B
   V AB 
 E dr
A
 E r
CV
D
E r  0
D r  0
B
r
A
D  r
The r vector is tangent (parallel) to the contour CV.
18
Method of Curvilinear Squares
2D flux plot
CV
V
+
-
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
 V  V AB 
“Curvilinear square”
B
 E  d r 
A
A
B
E dr 

E dl  0
A
If we integrate along the flux line, E is parallel to dr.
Note: It is called a curvilinear “square” even though the shape may be rectangular.
19
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
 co n sta n t
W
20
Method of Curvilinear Squares (cont.)
Proof of constant aspect ratio property
V
-
CV
B
V AB 
D
+
 E  d r  V
A
W
A
L
B
If we integrate along the flux line, E is parallel to dr.
B
Hence,

E d r  V
A
B
so
E

dl   V
Therefore
E L  V
A
21
Method of Curvilinear Squares (cont.)
CV
V
-
Hence,
D
L 
V
+
E
Also,
W
L
B
NL
D 
A
 L
 C1
NL
 L
 C1
1
W
so
W 
C1
D
Hence,
L
W

V D
E
C1

V D
C1
E

V
C1
 0  co n sta n t
(proof complete)
22
Summary of Flux Plot Rules
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.
If all of these rules are followed, then we have the following:
D 
# flux lines
 length
23
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
W
L
x
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).
24
Example
A parallel-plate capacitor
Note: L / W  0.5
http://www.opencollege.com
25
Example
Coaxial cable with a square inner conductor
L
1
W
Figure 6-12 in the Hayt and Buck book.
26
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
27
Example of Electric Flux Plot
Note: In this example,
the aspect ratio L/W is
not held constant.
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28
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
29
Appendix:
Proof of Flux Property
30
Proof of Flux Property
NC : flux lines
Through C
C
C
L
D

nˆ
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
D
  L co s  
or   l  D
  L 

L
L

31
Flux Property Proof (cont.)
 l  D
  L 
D
Also,
D 

N C
L
 L
(from the definition of a flux plot)
Hence, substituting into the above equation, we have
 N C 
  l  D  L  
   L
  L 
Therefore,
 l  NC

N C
(proof complete)
32