Chapter 3 Electric Flux Density, Gauss’s Law, and Divergence.
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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,
D3.1Given 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 by0 < theta < Pi/2 and 0 < phi < Pi/2
D3.2Calculate D in rectangular coordinates at point P(2,-3,6) produced by : (a) a point charge QA = 55mC at Q(-2,3,-6)
P
2
3
6
QA 55 103
Q
2
3
6
0 8.8541012 R P Q r
P QP Q
DQA
4 R 2r D
6.38 106
9.57 106
1.914 105
(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.
D x y z( )
8 x y z4
4 x2 z
4
16 x2 y z
3
1
3
y
0
2
xD x y 2( )2
1012
d
d 1.365 109
D3.6a
D x y z( )
8 x y z4
4 x2 z
4
16 x2 y z
3
1012
0 8.8541012
ED 2 1 3( )
0
P
2
1
3
E
146.375
146.375
195.166
D3.6b
DivergenceDivergence 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 D x
x
D y
y
D z
z
- Cartesian
div D 0v
SD
v
dlim
Surface integral as the volume element (v) approaches zero
D is the vector flux density
Cylindrical
Spherical
div D 1
D 1
D
Dz
z
div D 1
r2
D r r2
r
1
r sin
D sin
1
r sin
D
Divergence in Other Coordinate Systems
A
e x sin y( )
e x cos y( )
2 z
div A
xe x sin y( )
ye x cos y( )
z2 z( )
div A e x sin y( ) e x sin y( ) 2
Divergence at origin for given vector flux density A
3-6: Maxwell’s First Equation
S
.
SA
d Q
S
.
SA
d
v
Q
v
Gauss’ Law…
…per unit volume
Volume shrinks to zero 0v
S
.
SA
d
vlim 0v
Q
vlim
Electric flux per unit volume is equal to the volume charge density
Maxwell’s First Equation
div D v
0v
S
.
SA
d
vlim 0v
Q
vlim
Sometimes called the point form of Gauss’ Law
Enclosed surface is reduced to a single point
3-7: and the Divergence Theorem
del operator
ax
x
ay y
az
z
What is del?
’s Relationship to Divergence
div D VDTrue 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 x2 y a x 3 x2 y2 a y
QS
.
SD
dvol
.
v v
d
Calculation of total flux
total left right front back
x1 1 x2 1.2
y1 1 y2 1.2
z1 1 z2 1.2
x1z1
z2
zy1
y2
y2 x12 y
d
d y1z1
z2
zx1
x2
x3 x2 y12
d
d
x2z1
z2
zy1
y2
y2 x22 y
d
d y2z1
z2
zx1
x2
x3 x2 y22
d
d
total x1 x2 y1 y2
total 0.103
Evaluation of at center of cube VD
div D x
2 x2 y d
d y3 x2 y2 d
d
div D 4 x y 6 x2 y
divD 4 1.1( ) 1.1( ) 6 1.1( )2 1.1( )
divD 12.826
Non-Cartesian Example
Equipotential Surfaces – Free Software
Semiconductor Application - Device Charge Field Potential
Applications of Gauss’s Law
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