Prof. David R. Jackson ECE Dept. Fall 2014 Notes 21 ECE 2317 Applied Electricity and Magnetism 1.
Prof. D. Wilton ECE Dept. Notes 13 ECE 2317 Applied Electricity and Magnetism Notes prepared by the...
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Transcript of Prof. D. Wilton ECE Dept. Notes 13 ECE 2317 Applied Electricity and Magnetism Notes prepared by the...
Prof. D. WiltonECE Dept.
Notes 13
ECE 2317 ECE 2317 Applied Electricity and MagnetismApplied Electricity and Magnetism
Notes prepared by the EM group,
University of Houston.
Divergence -- Physical ConceptDivergence -- Physical Concept
r < a:
2 30
20
44
3
C/m3
encl
S
r v
vr
D n dS Q
r D r
rD
x
y
z
v = v0
a
r
Start by considering a sphere of uniform volume charge density
The electric field is calculated using Gauss's law:
r > a: 2 30
320
2
44
3
C/m3
r v
vr
r D a
aD
r
Divergence -- Physical Concept (cont.)Divergence -- Physical Concept (cont.)
x
y
z
v = v0
ar
2ˆ 4 r
S
D n dS r D
30
4
3 va (r > a)
(r < a)
Flux through a spherical surface:
30
4
3 vr
20 C/m3v
r
rD
320
2C/m
3v
r
aD
r
Divergence -- Physical Concept (cont.)Divergence -- Physical Concept (cont.)
(r < a)
(r > a)
More flux lines are added as the radius increases (as long as we stay inside the charge region).
0S
D n dS
Observation:
V
S
Divergence -- Physical Concept (cont.)Divergence -- Physical Concept (cont.)
0S
D n dS
The net flux out of a small volume V inside the charge region is not zero.
Divergence is a mathematical way of describing this.
0 0
1ˆdiv lim lim
V VS
D D n dSV V
V
Definition of divergence:
Gauss’s Law -- Differential FormGauss’s Law -- Differential Form
Note: the limit exists independent of the shape of the volume (proven later).
0
1div lim
VS
D D n dSV
V
v (r)
encl v
S
D n dS Q r V
0
div lim encl
V
v
QD r
Vr
Apply divergence definition to small volume inside a region of charge
Gauss’s Law -- Differential FormGauss’s Law -- Differential Form
Gauss’s Law -- Differential Form (cont.)Gauss’s Law -- Differential Form (cont.)
div vD r r
The electric Gauss law: This is one of Maxwell’s equations.
0 0
0
avg
0
div lim lim
1lim
lim
encl
V V
vVV
vV
v
QD r
V V
dVV
r
Alternatively,
ExampleExample
Choose V to be small sphere of radius r:
x
y
z
v = v0
a
r
0
1div lim
VS
D D n dSV
34
3V r
3
2 200
44 4
3 3v
r v
S
r rD n dS D r r
30
0 00 03
41div lim lim4 3
3
vv v
V V
rD
r
Verify that the differential form of Gauss’s law gives the correct result at the origin for the example of a sphere of uniform volume charge density.V
Calculation of DivergenceCalculation of Divergence
,0,02
,0,02
0, ,02
0, ,02
0,0,2
0,0,2
x
S
x
y
y
z
z
xD n dS D y z
xD y z
yD x z
yD x z
zD x y
zD x y
Assume point of interest is at the origin for simplicity.
The integrals over the 6 faces are approximated by “sampling” the integrand at the centers of the faces.
0
1div lim
VS
D D n dSx y z
x
y
z
(0,0,0)
x
y
z
Calculation of Divergence (cont.)Calculation of Divergence (cont.)
1
,0,0 ,0,02 2
0, ,0 0, ,02 2
0,0, 0,0,2 2
S
x x
y y
z z
D n dSx y z
x xD D
xy y
D D
y
z zD D
z
,0,02
,0,02
0, ,02
0, ,02
0,0,2
0,0,2
x
S
x
y
y
z
z
xD n dS D y z
xD y z
yD x z
yD x z
zD x y
zD x y
0
1div lim
VS
D D n dSx y z
Calculation of Divergence (cont.)Calculation of Divergence (cont.)
0
,0,0 ,0,02 2
div lim
0, ,0 0, ,02 2
0,0, 0,0,2 2
x x
V
y y
z z
x xD D
Dx
y yD D
y
z zD D
z
Hence div yx zDD D
Dx y z
For arbitrary origin, just add x,y,z to coordinate quantities in parentheses!
Calculation of Divergence (cont.)Calculation of Divergence (cont.)
div yx zDD D
Dx y z
0
1div lim
VS
D D n dSV
The divergence of a vector is its “flux per unit volume”
““del operator”del operator”
x y zx y z
: sin cos
: sin cos
sin sin cos
ˆ ˆsin sin cos
d dx x
dx dxd d
x x x x xdx dx
d dx x x x x x x
dx dx
d dx y x z x z x
dx dx
(1)
(2)
(3)
Examples of derivative operators:
scalar
vector
The “del” operator is a vector differential operator
scalar -> vector
vector->scalar
vector->vector
scalar -> scalar
ExampleExample
, , sin 3V x y z x x y y z xy
Find V
, , sin 3V x y z x y z x x y y z xyx y z
, , sin 3V x y z x y xyx y z
, , cos 3 0 3 cosV x y z x x
““del operator” (cont.)del operator” (cont.)Now consider:
x y z
yx z
D x y z xD yD zDx y z
DD Dx x y y z z
x y z
yx zDD D
Dx y z
divD D
Hence
sovD
Gauss's law :
Note: No unit vectors appear!
Summary of Divergence FormulasSummary of Divergence FormulasRectangular:
Cylindrical:
Spherical:
yx zDD D
Dx y z
1 1 zD D
D Dz
22
1 1 1sin
sin sinr
DD r D D
r r r r
Note the dot the del
operator is important; any
symbol following it tells
us how to use and read it:
"gradient"
"divergence"
"curl"
after
The divergence of a vector is its “flux per unit volume”
ExampleExample
x
y
z
v = v0
a
r < a :
0vD Note: This agrees with the electric Gauss law.
22
2 02
202
1
1
3
1
r
v
v
D r Dr r
rr
r r
rr
r
Evaluate the divergence of the electric flux vector inside and outside a sphere of uniform volume charge density, and verify that the answer is what is expected from the electric Gauss law.
0
3v r
D r
Example (cont.)Example (cont.)
x
y
z
v = v0
a
r > a :3
023
v aD r
r
0D
Note: This agrees with the electric Gauss law.
32 0
2 2
30
2
1
3
10
3
v
v
aD r
r r r
a
r r
Maxwell’s EquationsMaxwell’s Equations
0v
BE
tD
H Jt
D
B
Faraday’s law
Ampere’s law
electric Gauss law
magnetic Gauss law
Divergence TheoremDivergence Theorem
V
S
V S
A dV A n dS
The volume integral of “flux per unit volume” equals the total flux!
( , , )A A x y z an arbitrary
vector function of position
In words, for a vector :A
Divergence Theorem (cont.)Divergence Theorem (cont.)
Proof:
V
0
1
limn
N
rVnV
A dV A V
rn is the center of cube n
From the definition of divergence:
0
1lim
1
n
n
n
r VS
S
A A n dSV
A n dSV
Hence:
0 0
1 1
lim limn
n
N N
rV Vn nV S
AdV V A A n dS
Divergence Theorem (cont.)Divergence Theorem (cont.)
Hence: the surface integral cancels on all INTERIOR faces.
0
1
limn
N
VnV S
AdV A n dS
2n
1n
1 2
A n
Consider two adjacent cubes:
is opposite on the two faces
V
Divergence Theorem (cont.)Divergence Theorem (cont.)
01
0outsidefaces
lim
lim
n
n
N
VnV S
VS
A dr A n dS
A n dS
V S
AdV A n dS
Hence: 0
outsidefaces
limn
VV S S
A dV A n dS A n dS
Therefore: (proof complete)
V
Divergence Theorem (cont.)Divergence Theorem (cont.)
n
n
The vol. integral of the “flux per unit volume” is the “flux”
ExampleExample
x
y
z
1 [m]
2 [m]3 [m]
3A x x
Verify the divergence theorem using the box.
3 3 2 3 0 2
18S
A n dS x x x x
3 3
yx zAA A
Ax y z
xx
3 3 3 1 2 3 18V V
A dV dV V
Given:
From the divergence theorem:
Validity of Divergence DefinitionValidity of Divergence Definition
0
1div lim
SV
D D dSV
n
V
S
0
0
1div lim
1lim
r
VV
rV
DV
D
dV
DV
D
V
n̂
Is this limit independent of the shape of the volume?
r
Hence, the limit is the same regardless of the shape of the limiting volume.
This is valid for any volume, so let V V (a small volume inside the original volume)
Gauss’s Law Gauss’s Law (Conversion between forms)(Conversion between forms)
V
enclQ
S
encl
S
D n dS Q
V 0
v
v
V V
v
D dV dV
D V V
Divergence theorem:
Hence:
v
V S
encl v
V
D dV D n dS
Q dV
vD