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Week 5: Electric Flux and Gauss's

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Week 5: Electric Flux and Gauss's 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.
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