Week 5:
Electric Flux and Gauss’s Law
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE Chapter 3.1 Electric Flux Density
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
e.g. 1C charge means 1C flux
Double charge + Q ++ Double flux 2Q
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE Electric Flux Density (D) The meaning and deduction
4 5 Flux lines 3 6 2 ε0 Total # Q 7 4πr2 1 = +++ ++ ++ 8 2 + ++ Q 16 +++++ Area 4π 9 of surface r 15 10 where flux lines 14 11 are passing through 13 12
The number of Flux lines = 16 ‰ Q = 16 C Density concept D
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:
With considering Without considering ε ε Material factor ( 0) Material factor ( 0)
∗ εεε 0 : permittivity of the medium (material) where the flux lines are going through
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE Electric Field Intensity (E) vs. Electric Flux Density (D) with example of point charge
ε ε D = r 0E (general space)
∗ εεε r : relative permittivity constant ∗ εεε 0 : permittivity of the medium (material) where the flux lines are going through
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
16C in small sphere 16C in larger sphere
4 5 4 5 3 3 6 6 2 ε 2 ε 0 7 π 2 0 7 π 2 1 4 r 1 4 r + + + + + + + ++ +++ 8 + 8 ++ + Q + + + ++ Q + 16 +++ 16 + + + + 9 + + 9 15 15 10 10 14 11 14 11 13 12 13 12
Q = 16C Q = 16C
If Q is maintained with geometrical symmetry, flux number is not changed ‰ D remains the same at r
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE Another Point of View
ε 0 ε -Q 0
+Q r = a r r r = b
‰ D remains the same
when a ≤ r ≤ b
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE Another Point of View
ε -Q 0
+Q r = a r
r = b
‰ D remains the same
when a ≤ r ≤ b
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE Faraday used this for 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
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE Faraday used this for his electrostatic induction experiment
Outer sphere was initially neutral (no charge on it) And satisfying the charge neutrality with +Q
+ + + - + - + ------+ - -Q + - - - + + + - + + + + + + + + + + + + + - + + + - + - +Q + - Q + + + + + + + + + + + + + - - + - - + - - - + - + - - - - - + - + +
Ground
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE Chapter 3.2 Gauss’s Law
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE Electric Flux Density (D)
4 5 Flux lines 3 6 2 ε0 Total # Q 7 4πr2 1 = +++ ++ ++ 8 2 + ++ Q 16 +++++ Area 4π 9 of surface r 15 10 where flux lines 14 11 are passing through 13 12
The number of Flux lines = 16 ‰ Q = 16 C Density concept D
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE Gauss’s Thought = Faraday’s thought + Closed Surface 2 4 5 Area: S = 4πr 3 6 2 ε 0 7 π 2 1 4 r E-Flux Density: + ++ +++ 8 ++ + Q + ++ 2 16 +++ D = Q /4πr 9 15 10 14 11 13 12 ‰ DS = Total Flux 2 2 The number of Flux lines = 16 = (Q /4πr ) 4πr ‰ Q = 16 C = Q DS = Q
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE Integral Form
ε 0 4πr2 DS = Q ⇒ D dS = Q + ++ +++ ∫ + + Q ++ + ++++ S
Closed Surface
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE Integral Form with Dot Product
DS ⇒ ∫ D dS ⇒
S Generalization For any surface
ε0
+++ ++ ++ + ++ Q +++++
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE Integral Form with Dot Product
DS ⇒ ∫ D dS ⇒
S Generalization aN dS For any surface θ D dS = dSaN θ D = DaD dS cos θ Example:
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE Gaussian Surface General Special: θ = 0
ε0 θ ε0
+++ +++ ++ ++ ++ ++ + ++ Q + ++ Q +++++ +++++
∫ D dS = Q S Parallel between D and dS = ∫ Dcosθ dS S
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE Gauss’s Law
DS = Q
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 Carl Friedrich Gauss
Electromagnetics 1 (EM-1) with Prof. Sungsik LEE Gauss’s Law
ε0 θ ε0
+++ +++ ++ ++ ++ ++ + ++ Q + ++ Q +++++ +++++
General Special: θ = 0
The number of Flux lines = 16 The number of Flux lines = 16 ‰ Q = 16 C ‰ Q = 16 C
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