Transcript Divergence Theorem - Erwin Sitompul

Engineering Electromagnetics
Lecture 4
Dr.-Ing. Erwin Sitompul
President University
http://zitompul.wordpress.com
President University
Erwin Sitompul
EEM 4/1
Chapter 3
Electric Flux Density, Gauss’s Law, and DIvergence
Application of Gauss’s Law: Differential Volume Element
 We are now going to apply the methods of Gauss’s law to a
slightly different type of problem: a surface without symmetry.
 We have to choose such a very small closed surface that D is
almost constant over the surface, and the small change in D
may be adequately represented by using the first two terms of
the Taylor’s-series expansion for D.
 The result will become more nearly correct as the volume
enclosed by the gaussian surface decreases.
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Erwin Sitompul
EEM 4/2
Chapter 3
Electric Flux Density, Gauss’s Law, and DIvergence
Taylor’s Series Expansion
f ( x0  x)  f ( x)
f ( x0 )
x0
x  x0  x
A point near x0
f ( x)  f ( x0  x)
f ( x0 )
f ( x0 )
f ( x)  f ( x0 ) 
x 
(x) 2 
1!
2!
f n ( x0 )

(x) n
n!
Only the linear terms are
used for the linearization
President University
Erwin Sitompul
EEM 4/3
Chapter 3
Electric Flux Density, Gauss’s Law, and DIvergence
Application of Gauss’s Law: Differential Volume Element
 Consider any point P, located by a
rectangular coordinate system.
 The value of D at the point P may be
expressed in rectangular components:
D0  Dx0ax  Dy 0a y  Dz 0az
 We now choose as our closed surface,
the small rectangular box, centered at P,
having sides of lengths Δx, Δy, and Δz,
and apply Gauss’s law:


D  dS  Q
S
S
D  dS  
front
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
back
 
left
right
 
top
bottom
Erwin Sitompul
EEM 4/4
Chapter 3
Electric Flux Density, Gauss’s Law, and DIvergence
Application of Gauss’s Law: Differential Volume Element
 We will now consider the front surface
in detail.
 The surface element is very small, thus
D is essentially constant over this
surface (a portion of the entire closed
surface):

front
Dfront  Sfront
Dfront  yz ax
Dx,front yz
 The front face is at a distance of Δx/2 from P, and therefore:
Dx ,front
x
Dx 0 
 rate of change of Dx with x
2
x Dx
Dx 0 
2 x
President University
Erwin Sitompul
EEM 4/5
Chapter 3
Electric Flux Density, Gauss’s Law, and DIvergence
Application of Gauss’s Law: Differential Volume Element
 We have now, for front surface:

front
x Dx 

 Dx 0 
 yz
2 x 

 In the same way, the integral over the back surface can be
found as:

back
Dback  S back
Dback  (yz a x )
Dx,back yz
Dx ,back

back
x Dx
Dx 0 
2 x
x Dx 


D

 x0
 yz
2 x 

President University
Erwin Sitompul
EEM 4/6
Chapter 3
Electric Flux Density, Gauss’s Law, and DIvergence
Application of Gauss’s Law: Differential Volume Element
 If we combine the two integrals over the front and back surface,
we have:

front


back
Dx
xyz
x
 Repeating the same process to the remaining surfaces, we find:

right

top



left

bottom
Dy
y
yxz
Dz
zxy
z
 These results may be collected to yield:
 Dx Dy Dz 
 S D  dS  x  y  z  xyz
 Dx Dy Dz 
 S D  dS  Q  x  y  z  v
President University
Erwin Sitompul
EEM 4/7
Chapter 3
Electric Flux Density, Gauss’s Law, and DIvergence
Application of Gauss’s Law: Differential Volume Element
 The previous equation is an approximation, which becomes
better as Δv becomes smaller.
 For the moment, we have applied Gauss’s law to the closed
surface surrounding the volume element Δv, with the result:
 Dx Dy Dz 
Charge enclosed in volume v 


  v
y
z 
 x
President University
Erwin Sitompul
EEM 4/8
Chapter 3
Electric Flux Density, Gauss’s Law, and DIvergence
Application of Gauss’s Law: Differential Volume Element
 Example
Let D = y2z3 ax + 2xyz3 ay + 3xy2z2 az pC/m2 in free space.
(a) Find the total electric flux passing through the surface x = 3,
0 ≤ y ≤ 2, 0 ≤ z ≤ 1 in a direction away from the origin. (b) Find
|E| at P(3,2,1). (c) Find the total charge contained in an
incremental sphere having a radius of 2 μm centered at
P(3,2,1).
(a) ψ  S DS  dS
z 0
 y z a
1
2

1

2

y 0
0 0
 y
1
3
2 3
3
3
2 2

2
xyz
a

3
xy
z a z    dydz a x 
x
y
x 3
y 2 z 3dydz
2 1
0 4
z
4
1
0
 23 pC
President University
Erwin Sitompul
EEM 4/9
Chapter 3
Electric Flux Density, Gauss’s Law, and DIvergence
Application of Gauss’s Law: Differential Volume Element
(b) D = y2 z3ax  2xyz3a y  3xy2 z 2az
DP = (2)2 (1)3 ax  2(3)(2)(1)3 a y  3(3)(2)2 (1)2 az
= 4ax 12a y  36az pC m2
DP = DP  (4) 2  (12) 2  (36) 2
 38.158 pC m2
EP 
DP
0
38.158 pC m 2

8.854 1012
 4.31V m
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Erwin Sitompul
EEM 4/10
Chapter 3
Electric Flux Density, Gauss’s Law, and DIvergence
Application of Gauss’s Law: Differential Volume Element
(c)
 Dx Dy Dz 
Q 


 v
y
z 
 x
 Dx Dy Dz 
QP 


 v
y
z  P
 x
 0  2 xz
3
 6 xy 2 z  x 3 pC m3  43  (2 106 )3 m3
 0  2(3)(1)
y 2
z 1
3
 6(3)(2) 2 (1)  43  (2 106 )3 pC
2.611027 C
President University
Erwin Sitompul
EEM 4/11
Chapter 3
Electric Flux Density, Gauss’s Law, and DIvergence
Divergence
 We shall now obtain an exact relationship, by allowing the
volume element Δv to shrink to zero.

 Dx Dy Dz 





x

y

z


D  dS
S
v

 Dx Dy Dz 



  lim
y
z  v0
 x


Q
v
D  dS
S
v
Q
 lim
v 0 v
 The last term is the volume charge density ρv, so that:
 Dx Dy Dz 



  lim
y
z  v0
 x
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
D  dS
S
v
 v
Erwin Sitompul
EEM 4/12
Chapter 3
Electric Flux Density, Gauss’s Law, and DIvergence
Divergence
 Let us no consider one information that can be obtained from
the last equation:
 Dx Dy Dz 



  lim
y
z  v0
 x

D  dS
S
v
 This equation is valid not only for electric flux density D, but
also to any vector field A to find the surface integral for a small
closed surface.
 Ax Ay Az 



  lim
y
z  v0
 x
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
A  dS
S
v
Erwin Sitompul
EEM 4/13
Chapter 3
Electric Flux Density, Gauss’s Law, and DIvergence
Divergence
 This operation received a descriptive name, divergence. The
divergence of A is defined as:

Divergence of A  div A  lim
v 0
A  dS
S
v
“The divergence of the vector flux density A is the
outflow of flux from a small closed surface per unit
volume as the volume shrinks to zero.”
 A positive divergence of a vector quantity indicates a source of
that vector quantity at that point.
 Similarly, a negative divergence indicates a sink.
President University
Erwin Sitompul
EEM 4/14
Chapter 3
Electric Flux Density, Gauss’s Law, and DIvergence
Divergence
Dx Dy Dz
div D 


x
y
z
Rectangular
1 
1 D Dz
div D 
(  D ) 

 
 
z
Cylindrical
1  2
1

1 D
div D  2 (r Dr ) 
(sin  D ) 
r r
r sin  
r sin  
Spherical
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EEM 4/15
Erwin Sitompul
Chapter 3
Electric Flux Density, Gauss’s Law, and DIvergence
Divergence
 Example
If D = e–xsiny ax – e–x cosy ay + 2z az, find div D at the origin
and P(1,2,3)
Dx Dy Dz
 e x sin y  e x sin y  2
div D 


x
y
z
2
Regardles of location the divergence of D equals 2 C/m3.
President University
Erwin Sitompul
EEM 4/16
Chapter 3
Electric Flux Density, Gauss’s Law, and DIvergence
Maxwell’s First Equation (Electrostatics)
 We may now rewrite the expressions developed until now:
div D  lim

D  dS
S
v  0
div D 
v
Dy
Dx
D

 z
x
y
z
div D  v
Maxwell’s First Equation
Point Form of Gauss’s Law
 This first of Maxwell’s four equations applies to electrostatics
and steady magnetic field.
 Physically it states that the electric flux per unit volume leaving
a vanishingly small volume unit is exactly equal to the volume
charge density there.
President University
Erwin Sitompul
EEM 4/17
Chapter 3
Electric Flux Density, Gauss’s Law, and DIvergence
The Vector Operator  and The Divergence Theorem
 Divergence is an operation on a vector yielding a scalar, just
like the dot product.
 We define the del operator  as a vector operator:




ax  a y  az
x
y
z
 Then, treating the del operator as an ordinary vector, we can
write:


 
 D   a x  a y  a z   ( Dxa x  Dy a y  Dz a z )
y
z 
 x
Dx Dy Dz
D 


x
y
z
Dx Dy Dz
div D =   D =


x
y
z
President University
Erwin Sitompul
EEM 4/18
Chapter 3
Electric Flux Density, Gauss’s Law, and DIvergence
The Vector Operator  and The Divergence Theorem
 The operator does not have a specific form in other
coordinate systems than rectangular coordinate system.
 Nevertheless,
1 
1 D Dz
D 
(  D ) 

 
 
z
Cylindrical
1  2
1

1 D
  D  2 (r Dr ) 
(sin  D ) 
r r
r sin  
r sin  
Spherical
President University
EEM 4/19
Erwin Sitompul
Chapter 3
Electric Flux Density, Gauss’s Law, and DIvergence
The Vector Operator  and The Divergence Theorem
 We shall now give name to a theorem that we actually have
obtained, the Divergence Theorem:

S
D  dS  Q   v dv     Ddv
vol
vol
 The first and last terms constitute the divergence theorem:

S
D  dS     D dv
vol
“The integral of the normal
component of any vector field
over a closed surface is equal to
the integral of the divergence of
this vector field throughout the
volume enclosed by the closed
surface.”
President University
Erwin Sitompul
EEM 4/20
Electric Flux Density, Gauss’s Law, and DIvergence
Chapter 3
The Vector Operator  and The Divergence Theorem
 Example
Evaluate both sides of the divergence theorem for the field
D = 2xy ax + x2 ay C/m2 and the rectangular parallelepiped
fomed by the planes x = 0 and 1, y = 0 and 2, and z = 0 and 3.


S
S
D  dS     D dv
Divergence Theorem
vol
3
2
0
0
DS  dS  
 (D)
   (D)
x 0
3
1
0
0
3
2
0
0
 (dydz a x )  
y 0
 (dxdz a y
 (D)
)    (D)
x 1
3
1
0
0
 (dydz a x )
y 2
 (dxdz a y )
But ( Dx ) x0  0, ( Dy ) y0  ( Dy ) y2
D
S
S
 dS  
3
0

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2
0
( Dx ) x1 dydz  
3
0

2
0
2ydydz  12 C
Erwin Sitompul
EEM 4/21
Electric Flux Density, Gauss’s Law, and DIvergence
Chapter 3
The Vector Operator  and The Divergence Theorem
D =

vol


(2 xy)  ( x 2 )  2 y
x
y
 D dv  
3
2
 
1
z 0 y 0 x 0
 x0 y
1
2 2
0
(2 y)dxdydz
3
z0
 12 C
  D  dS     D dv  12 C
S
vol
President University
Erwin Sitompul
EEM 4/22
Chapter 3
Electric Flux Density, Gauss’s Law, and DIvergence
Homework 4
 D3.6.
 D3.7.
 D3.9. All homework problems from Hayt and Buck, 7th Edition.
 Deadline: 8 February 2011, at 07:30.
President University
Erwin Sitompul
EEM 4/23