Chapter 3
Electric Flux Density, Gauss’s Law,
and Divergence
3.1 Electric Flux Density
• Faraday’s Experiment
Electric Flux Density, D
• Units: C/m2
• Magnitude: Number of flux lines (coulombs)
crossing a surface normal to the lines divided by
the surface area.
• Direction: Direction of flux lines (same direction
as E).
• For a point charge:
• For a general charge distribution,
D3.1
Given a 60-uC point charge located at the origin, find the
total electric flux passing through:
(a) That portion of the sphere r = 26 cm bounded by
0 < theta < Pi/2 and 0 < phi < Pi/2
D3.2
Calculate D in rectangular coordinates at point P(2,-3,6)
produced by : (a) a point charge QA = 55mC at Q(-2,3,-6)
 2

P   3

 6




 0  8.854  10
D 
 2
 3
Q 

 6
 12
QA
4    R
r
2




R  P  Q
QA  55  10
r 
 6.38  10  6


D   9.57  10  6


 5
 1.914  10






P  Q
P  Q
 3
(b) a uniform line charge pLB = 20 mC/m on the x axis
(c) a uniform surface charge density pSC = 120 uC/m2 on
the plane z = -5 m.
Gauss’s Law
• “The electric flux passing through any
closed surface is equal to the total charge
enclosed by that surface.”
• The integration is performed over a closed
surface, i.e. gaussian surface.
• We can check Gauss’s law with a point
charge example.
Symmetrical Charge Distributions
•
Gauss’s law is useful under two
conditions.
1. DS is everywhere either normal or
tangential to the closed surface, so that
DS.dS becomes either DS dS or zero,
respectively.
2. On that portion of the closed surface for
which DS.dS is not zero, DS = constant.
Gauss’s law simplifies the task of finding D near an
infinite line charge.
Infinite coaxial cable:
Differential Volume Element
• If we take a small enough closed surface,
then D is almost constant over the surface.
D3.6a
 8  x y  z4


2 4
D ( x  y  z) 
4 x  z


2
3
16

x

y

z





3
1




2
0
D ( x  y  2 )  10
2






 12
d x d y  1.365  10
 9
D3.6b
 8  x y  z4


2 4
D ( x  y  z) 
4 x  z


2
3
16

x

y

z

 0  8.854  10
 2

P   1

 3






  10 



12
 12
E 
D ( 2   1  3)
0
  146.375
 146.375
E 

  195.166




Divergence
Divergence is the outflow of flux from a small
closed surface area (per unit volume) as
volume shrinks to zero.
-Water leaving a bathtub
-Closed surface (water itself) is essentially incompressible
-Net outflow is zero
-Air leaving a punctured tire
-Divergence is positive, as closed surface (tire) exhibits net
outflow
Mathematical definition of divergence
div  D 

lim 
v  0 

D
v
dS
Surface integral as the volume element (v) approaches zero
D is the vector flux density
div  D 
D
D 
 D x
y
z




y
z 
 x
- Cartesian
Divergence in Other Coordinate Systems
Cylindrical
div  D 
1



  D



1

D



D

z
z
Spherical
div  D 
 D r

1  r

2
r
r
2


1
r sin   


 D   sin   



1
r sin   

D 

Divergence at origin for given vector flux density A
A
div  A 
div  A 
 e  x  sin ( y )

  e  x  cos ( y )

2z


x





 e  x  sin ( y )  
e
x
 sin ( y )  e

y
x

 e
x

 cos ( y ) 
 sin ( y )  2

z
 ( 2  z)
3-6: Maxwell’s First Equation



.



.
A dS
Q
Gauss’ Law…
Q
…per unit volume
S
A dS
S
v
v
.

 A dS

Volume shrinks to zero
lim
v  0
S
v
lim
Q
v  0  v
Electric flux per unit volume is equal to the volume charge density
Maxwell’s First Equation
.

 A dS

S
lim
v  0
v
div  D 
lim
Q
v  0  v
 v
Sometimes called the point form of Gauss’ Law
Enclosed surface is reduced to a single point
3-7:  and the Divergence Theorem
  del operator
What is


 
 ax
x

del?
 
 ay
y

 
 az
z
’s Relationship to Divergence
div  D 
D
V
True for all coordinate systems
Other  Relationships
Gradient – results from  operating on a function
Represents direction of greatest change
Curl – cross product of  and
Relates to work in a field
If curl is zero, so is work
Examination of  and flux
Cube defined by 1 < x,y,z < 1.2
D
2
2
2
2x ya x  3x y a y
Calculation of total flux
.
Q

 D dS
S
 total



.
 v dv

vol
 left   right   front   back
z2

 x1  
z
1
z2

 x2  
z
1
y2


y
z2
2
 2  x1  y d y d z
1
y2


y
1
z2
2
2  x2  y d y d z
1

 y1  
z

 y2  
z
1
x1  1
x2  1.2
y 1  1
y 2  1.2
z1  1
z2  1.2
x2


x
2
2
 3  x  y 1 dx dz
1
x2


x
2
2
3  x  y 2 dx dz
1
 total   x1   x2   y1   y2
 total  0.103
Evaluation ofV
 Dat center of cube
div  D 
d
 2  x2  y 

dx
div  D 
d
 3  x2  y 2 
dy
2
4  x y  6  x  y
2
divD  4  ( 1.1 )  ( 1.1 )  6  ( 1.1 )  ( 1.1 )
divD  12.826
Non-Cartesian Example
Equipotential Surfaces – Free Software
Semiconductor Application - Device Charge Field Potential
Vector Fields
Potential Field
Applications of Gauss’s Law