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Material Type: Notes; Professor: Wilkins; Class: Electromagnetic Theo & Appl; Subject: Electrical Engineering; University: Morgan State University; Term: Unknown 1989;
Typology: Study notes
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Electric Flux Ψ displacement charge
Flux lines leave +Q and terminate on -Q
equal but opposite charge induced
Ψ = Q coulombs
The amount of flux per unit area is the flux density
D (^) ds a n =^ d^ Ψ
Suppose D is not normal to d s?
D s
D an d
D cos = ⋅
d Ψ = ds θ = ⋅ ds
a n (^) D
ds
θ
A surface vector points in the direction normal to the surface
Total flux out of a closed surface is equal to the net charge enclosed within the surface
∫ ∫
∫
vol
v S
S
S
enc
We can use Gauss’s Law to find D as well as using the approach discussed in the previous chapter to find E and henceforth D.
Gauss’ Law
Relationship between D and E :
Point Charge Example
r
s s
s
s s
2
2
2
0 0
2
ππ
∫ ∫
∫ ∫
Gaussian Surface
r
Coaxial Cable Example
Choose gaussian surface of a cylinder for a< ρ <b
ρ
ρ π ρ πρ
ρ φ π ρ
πρ
ρ φ
π
π
s s s s
L s s
s
L s s
a a L D L D
Q ad dz a L
Thetotalch e on the inner conductor
Q D dS D d dz
∫ ∫
∫ ∫ ∫
arg :
0
2
0
2
0 0 (^) b a
GS for ρ>b
GS for a<ρ<b
Coaxial Cable Example contd.
From before:
QOuter=- 2 π a ρ sLinner 2 π bL ρ s outer= - 2 π aL ρ s inner
If the gaussian surface is chosen for ρ > b:
The total charge =0=Ds 2 πρ L ⇒ Ds = 0
souter s inner b
ρ = −a^ ρ
Differential Volume Element
•Here we assume no symmetry.
•We would like to get an idea of the spatial variation of D.
⋅ =
→
⋅ = → ( d ñvdv)
v
vdv v v
d v (^) v D S
D S 0
lim 0
lim ∆∆ ∆∆ ∆∆ ∆∆
ρ
The Left Hand Side(LHS) is the divergence of D.
The Right Hand Side (RHS) is ρ v.
The flux flowing out of face 2 is:
y z
x x ∆ ∆
∆ − ) 2
D (^) x (
The net flow is (1) -(2)
z
x y z y
x y z x
x y z
x
x y z
x y z
x
The flux flowing out of face 1 is:
y z
x x ∆ ∆
∆
D (^) x (
Differential Volume Element contd.
Differential Volume Element contd.
v
x y z
v
x y z
z
D
y
D
x
D
v z
D y
D
x
D
ρ
ρ
= ∂
∂
∂
∂
∂
=
∂
∂
∂
∂
∂
∂ = ∆∆
v
x y z
Divergence in Other Coordinate
Systems
( )
z
D y
D x
D div x y z ∂
∂
∂
∂
∂
∂ D =
( ) ( ) φ
φ θ
θ θ θ θ ∂
∂
∂
∂
∂
D r
D r
r Dr r r
div sin
1 sin sin
1 2 1 D 2
The Vector Operator Del
x ax^ y ay ∂z a^ z
The vector operator (^) ∇ Del:
In rectangular coordinates
We are already familiar with scalar operations such as the derivative:
2
Start with Gauss’ Law:
∫ ∫
∫
∫ ∫
= ∇ ⋅
= ∇ ⋅
= =
vol
vol
vol
Q
d dv
dv
d dv
D s D
D
D s ρ
The Divergence Theorem
This Theorem allows us to calculate the charge by enclosing it within a surface or integrating over a volume
