chapter 3 electric flux density, gauss’s law, and … · 2017-03-21 1 electromagnetics 1 (em-1)...
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2017-03-21
1
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE
Chapter 3
Electric Flux Density,Gauss’s Law, and
Divergence
Week 3-2 (March 22)
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE
Chapter 3.1
Electric Flux Density
2017-03-21
2
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE
Electric Flux
• Flux generated out of electric charge:
= Electric charge generates a flux
= Electric charge itself is a flux
= The # of the electric flux lines is the Faraday’s expression
equivalent to the amount of electric charges
+ Q2Q++
Double charge
Double flux
e.g. 1C charge means 1C flux
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE
Electric Flux Density (D)The meaning and deduction
24 r
Q
π=
Q
1
2
34 5
6
7
8
9
10
11
121314
15
16
4πr20ε
� Q = 16 C
The number of Flux lines = 16
Areaof surface
where flux lines
are passing through
Flux lines
Total #
Density concept D
++++++++++
+++++
2017-03-21
3
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE
Electric Field Intensity (E)vs. Electric Flux Density (D)
with example of point charge
Electric Field Intensity: Electric Flux Density:
∗ εεεε0 : permittivity of the medium (material) where the flux lines are going through
With considering
Material factor (ε0)
Without considering
Material factor (ε0)
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE
Electric Field Intensity (E)vs. Electric Flux Density (D)
with example of point charge
∗ εεεεr : relative permittivity constant
∗ εεεε0 : permittivity of the medium (material) where the flux lines are going through
D = εrε0E (general space)
2017-03-21
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Electromagnetics 1 (EM-1) with Prof. Sungsik LEE
Electric Field Intensity (E)vs. Electric Flux Density (D)
with example of point charge
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE
Another Point of View
1
2
34 5
6
7
8
9
10
11
121314
15
16
4πr20ε
Q
1
2
34 5
6
7
8
9
10
11
121314
15
16
4πr20ε
++++++++++
+++++
Q = 16C Q = 16C
+ + ++++
+++++
+++
++
Q
If Q is maintained with geometrical symmetry, flux number is not changed
� D remains the same at r
16C in small sphere 16C in larger sphere
2017-03-21
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Electromagnetics 1 (EM-1) with Prof. Sungsik LEE
Another Point of View
-Q
+Q
0ε
r = a
r = b
r
r
� D remains the same
when bra ≤≤
0ε
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE
Another Point of View
-Q
+Q
0ε
r = a
r = b
r
� D remains the same
when bra ≤≤
2017-03-21
6
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE
Faraday used thisfor his electrostatic induction experiment
Faraday’s equipment
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE
Michael Faraday
Born 22 September 1791
Newington Butts, England
Died 25 August 1867 (aged 75)
Hampton Court, Middlesex,
England
Residence United Kingdom
Known for Faraday's law of induction
electromagnetic induction
2017-03-21
7
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE
Faraday used thisfor his electrostatic induction experiment
+ + +++++
+++++++
++
+Q
+
+
+
+
+
+++
+
+
+
+
+
++ +
--
-
-
-
--
---
-
-
-
-
- -
Ground
+ + +++++
+++++++
++
Q
--
-
-
-
--
---
-
-
-
-
- -
-Q
Outer sphere was initially neutral (no charge on it)
And satisfying the charge neutrality with +Q
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE
Chapter 3.2
Gauss’s Law
2017-03-21
8
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE
Electric Flux Density (D)
24 r
Q
π=
Q
1
2
34 5
6
7
8
9
10
11
121314
15
16
4πr20ε
� Q = 16 C
The number of Flux lines = 16
Areaof surface
where flux lines
are passing through
Flux lines
Total #
Density concept D
++++++++++
+++++
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE
Gauss’s Thought= Faraday’s thought + Closed Surface
Q
1
2
34 5
6
7
8
9
10
11
121314
15
16
4πr20ε
� Q = 16 C
The number of Flux lines = 16
++++++++++
+++++
Area: S = 4πr2
E-Flux Density:
D = Q /4πr2
� DS = Total Flux
= (Q /4πr2) 4πr2
= QDS = Q
2017-03-21
9
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE
Integral Form
Q
4πr20ε
++++++++++
+++++
DS = Q QdSDS
=⇒ ∫
Closed
Surface
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE
Integral Form with Dot Product
DS ⇒⇒ ∫S
dSD
Q
0ε
++++++++++
+++++
Generalization
For any surface
2017-03-21
10
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE
Integral Form with Dot Product
DS ⇒⇒ ∫S
dSD
Generalization
For any surface
D
dSaN
dS = dSaN
dS cosθ
θ
θD = DaD
Example:
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE
Gaussian Surface
Q
0ε
++++++++++
+++++Q
0ε
++++++++++
+++++
Special: θ = 0
QdSDS
=∫
General
θ
Parallel between D and dS
∫=
S
dSD θcos
2017-03-21
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Electromagnetics 1 (EM-1) with Prof. Sungsik LEE
Gauss’s Law
DS = Q
Carl Friedrich Gauss
Born Johann Carl Friedrich Gauss
30 April 1777
Brunswick, Duchy of Brunswick-Wol
fenbüttel, Holy Roman Empire
Died 23 February 1855 (aged 77)
Göttingen, Kingdom of Hanover
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE
Gauss’s Law
Q
0ε
++++++++++
+++++Q
0ε
++++++++++
+++++
Special: θ = 0General
θ
� Q = 16 C
The number of Flux lines = 16
� Q = 16 C
The number of Flux lines = 16
2017-03-21
12
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE
Expression for Q (difference case / symmetry)
Discrete (=group of point charges)
Charge sphere
fully filled with charge
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE
Contents of the Chapter 3.
for Next (March 22)
as Week 4-1