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07_Poisson_2019b_mk.pdf

  1. 1. Engineering Electromagnetics Poisson’s & Laplace’s Equations Nguyễn Công Phương
  2. 2. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Contents I. Introduction II. Vector Analysis III. Coulomb’s Law & Electric Field Intensity IV. Electric Flux Density, Gauss’ Law & Divergence V. Energy & Potential VI. Current & Conductors VII. Dielectrics & Capacitance VIII.Poisson’s & Laplace’s Equations IX. The Steady Magnetic Field X. Magnetic Forces & Inductance XI. Time – Varying Fields & Maxwell’s Equations XII. Transmission Lines XIII. The Uniform Plane Wave XIV. Plane Wave Reflection & Dispersion XV. Guided Waves & Radiation 2
  3. 3. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Poisson’s & Laplace’s Equations 1. Poisson’s Equation 2. Laplace’s Equation 3. Uniqueness Theorem 4. Examples of the Solution of Laplace’s Equation 5. Examples of the Solution of Poisson’s Equation 6. Product Solution of Laplace’s Equation 7. Numerical Methods 3
  4. 4. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Poisson's Equation (1) 0 D E ε = Gauss’s Law: v ρ ∇ = .D ( ) ( ) v V ε ε ρ → ∇ = ∇ = −∇ ∇ = .D . E . V = −∇ E Gradient: v V ρ ε → ∇ ∇ = − . (Poisson's Equation) x y z V V V V x y z ∂ ∂ ∂ ∇ = + + ∂ ∂ ∂ a a a y x z A A A x y z ∂ ∂ ∂ ∇ = + + ∂ ∂ ∂ .A 2 2 2 2 2 2 . y x z V V V V V V V x x y y z z x y z ∂   ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂     → ∇∇ = + + = + +       ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂       4
  5. 5. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Poisson's Equation (2) v V ρ ε ∇ ∇ = − . Define 2 ∇ ∇ = ∇ . 2 2 2 2 2 2 . V V V V x y z ∂ ∂ ∂ ∇ ∇ = + + ∂ ∂ ∂ 2 2 2 2 2 2 2 v V V V V x y z ρ ε ∂ ∂ ∂ ∇ = + + = − ∂ ∂ ∂ (rectangular) 2 2 2 2 2 1 1 v V V V z ρ ρ ρ ρ ρ ρ ϕ ε   ∂ ∂ ∂ ∂ + + = −   ∂ ∂ ∂ ∂   2 2 2 2 2 2 2 1 1 1 sin sin sin v V V V r r r r r r ρ θ θ θ θ θ ϕ ε ∂ ∂ ∂ ∂ ∂     + + = −     ∂ ∂ ∂ ∂ ∂     (cylindrical) (spherical) 5
  6. 6. Poisson’s Equation (3) Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Ex. Find the Laplacian of the following scalar fields: 2 3 3 ) 2 cos2 ) 20sin ) a A xy z b B c C r ϕ ρ θ = = = 6
  7. 7. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Poisson’s & Laplace’s Equations 1. Poisson’s Equation 2. Laplace’s Equation 3. Uniqueness Theorem 4. Examples of the Solution of Laplace’s Equation 5. Examples of the Solution of Poisson’s Equation 6. Product Solution of Laplace’s Equation 7. Numerical Methods 7
  8. 8. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Laplace’s Equation 0 v ρ = 2 2 2 2 2 2 2 v V V V V x y z ρ ε ∂ ∂ ∂ ∇ = + + = − ∂ ∂ ∂ (Laplace’s equation, rectangular) 2 2 2 2 2 1 1 0 V V V z ρ ρ ρ ρ ρ ϕ   ∂ ∂ ∂ ∂ + + =   ∂ ∂ ∂ ∂   2 2 2 2 2 2 2 1 1 1 sin 0 sin sin V V V r r r r r r θ θ θ θ θ ϕ ∂ ∂ ∂ ∂ ∂     + + =     ∂ ∂ ∂ ∂ ∂     (cylindrical) (spherical) 2 2 2 2 2 2 2 0 V V V V x y z ∂ ∂ ∂ ∇ = + + = ∂ ∂ ∂ Poisson's Equation: 8
  9. 9. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Poisson’s & Laplace’s Equations 1. Poisson’s Equation 2. Laplace’s Equation 3. Uniqueness Theorem 4. Examples of the Solution of Laplace’s Equation 5. Examples of the Solution of Poisson’s Equation 6. Product Solution of Laplace’s Equation 7. Numerical Methods 9
  10. 10. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Uniqueness Theorem (1) 2 2 2 2 2 2 2 0 V V V V x y z ∂ ∂ ∂ ∇ = + + = ∂ ∂ ∂ Assume two solutions V1 & V2, : 2 1 0 V ∇ = 2 2 0 V ∇ = 2 1 2 ( ) 0 V V → ∇ − = Assume the boundary condition Vb 1 2 b b b V V V → = = ( ) ( ) ( ) . D .D D. V V V ∇ = ∇ + ∇ 1 2 1 2 1 2 1 2 1 2 1 2 [( ) ( )] ( )[ ( )] ( ) ( ) . . . V V V V V V V V V V V V → ∇ − ∇ − = − ∇ ∇ − + +∇ − ∇ − 1 2 V V V = − 1 2 ( ) D V V = ∇ − 10
  11. 11. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Uniqueness Theorem (2) 1 2 1 2 1 2 1 2 1 2 1 2 [( ) ( )] ( )[ ( )] ( ) ( ) . . . V V V V V V V V V V V V ∇ − ∇ − = − ∇ ∇ − +∇ − ∇ − 1 2 1 2 1 2 1 2 1 2 1 2 [( ) ( )] ( )[ ( )] ( ) ( ) V V V V V V V dv V V V V dv V V V V dv → ∇ − ∇ − = − ∇ ∇ − + + ∇ − ∇ −    . . . . . S V d dv = ∇   D S D Divergence theorem: 1 2 1 2 1 2 1 2 [( ) ( )] [( ) ( )] b b b b V S V V V V dv V V V V d → ∇ − ∇ − = − ∇ −   . . S 1 2 b b b V V V = = 1 2 1 2 [( ) ( )] 0 V V V V V dv → ∇ − ∇ − =  . 1 2 1 2 1 2 1 2 0 ( )[ ( )] ( ) ( ) V V V V V V dv V V V V dv → = − ∇ ∇ − + ∇ − ∇ −   . . 11
  12. 12. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Uniqueness Theorem (3) 1 2 1 2 1 2 1 2 ( )[ ( )] ( ) ( ) 0 V V V V V V dv V V V V dv − ∇ ∇ − + ∇ − ∇ − =   . . 2 1 2 1 2 ( ) ( ) 0 . V V V V ∇ ∇ − = ∇ − = 1 2 1 2 ( ) ( ) 0 V V V V V dv → ∇ − ∇ − =  . [ ] 2 1 2 ( ) V V V dv = ∇ −  [ ] 2 1 2 ( ) 0 V V → ∇ − = [ ] 2 1 2 ( ) 0 V V ∇ − ≥ 1 2 const V V → − = 1 2 ( ) 0 V V → ∇ − = x y z V V V V x y z ∂ ∂ ∂ ∇ = + + ∂ ∂ ∂ a a a At boundary V1 = Vb1, V2 = Vb2 → const = Vb1 – Vb2 = 0 1 2 b b b V V V = = V1 = V2 12
  13. 13. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Poisson’s & Laplace’s Equations 1. Poisson’s Equation 2. Laplace’s Equation 3. Uniqueness Theorem 4. Examples of the Solution of Laplace’s Equation 5. Examples of the Solution of Poisson’s Equation 6. Product Solution of Laplace’s Equation 7. Numerical Methods 13
  14. 14. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Examples of the Solution of Laplace’s Equation (1) Assume V = V(x) 2 2 2 2 2 2 2 0 V V V V x y z ∂ ∂ ∂ ∇ = + + = ∂ ∂ ∂ 2 2 0 d V dx → = V Ax B → = + 1 1 x x V V = = 2 2 x x V V = = 1 2 1 2 2 1 1 2 1 2 V V A x x V x V x B x x −  =  −  →  −  =  −  1 2 2 1 1 2 ( ) ( ) V x x V x x V x x − − − → = − 0 0 x V = = 0 x d V V = = 0 V x V d → = Ex. 1 14
  15. 15. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Examples of the Solution of Laplace’s Equation (2) Conductor surface Conductor surface x = d x = 0 x V = V(x) 0 0 x V = = 0 x d V V = = 0 V x V d → = E V = −∇ 0 E ax V d → = − D E ε = 0 D ax V d ε → = − 0 0 D D a S x x V d ε = → = = − 0 N V D d ε → = − 0 S N V D d ρ ε → = = − 0 V S d ε = − 0 S S S V Q dS dS d ε ρ − → = =   0 Q C V → = S d ε = Ex. 1 15
  16. 16. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Examples of the Solution of Laplace’s Equation (3) Assume V = V(ρ) (cylindrical) 2 2 2 2 2 2 1 1 0 V V V V z ρ ρ ρ ρ ρ ϕ   ∂ ∂ ∂ ∂ ∇ = + + =   ∂ ∂ ∂ ∂   1 0 V ρ ρ ρ ρ   ∂ ∂ → =   ∂ ∂   1 0 d dV d d ρ ρ ρ ρ   → =     0 d dV d d ρ ρ ρ   → =     dV A d ρ ρ → = ln V A B ρ → = + 0 ln a V A a B V ρ = = + = ln 0 ( ) b V A b B b a ρ = = + = > 0 0 ln ln ln ln ln V A a b V b B a b  =   − →   = −  −  0 ln( / ) ln( / ) b V V b a ρ → = Ex. 2 16
  17. 17. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Examples of the Solution of Laplace’s Equation (4) Assume V = V(ρ) (cylindrical) 2 2 2 2 2 2 1 1 0 V V V V z ρ ρ ρ ρ ρ ϕ   ∂ ∂ ∂ ∂ ∇ = + + =   ∂ ∂ ∂ ∂   0 ln( / ) ln( / ) b V V b a ρ → = 0 ln( / ) E a V V b a ρ ρ → = −∇ = 0 ( ) ln( / ) N a S V D a b a ρ ε ρ = → = = 0 2 ln( / ) S S V aL Q dS a b a ε π ρ → = =  0 2 ln( / ) Q L C V b a ε π → = = Ex. 2 17
  18. 18. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Examples of the Solution of Laplace’s Equation (5) Assume V = V(φ) (cylindrical) 2 2 2 2 2 2 1 1 0 V V V V z ρ ρ ρ ρ ρ ϕ   ∂ ∂ ∂ ∂ ∇ = + + =   ∂ ∂ ∂ ∂   2 2 2 1 0 V ρ ϕ ∂ → = ∂ 2 2 0 V ϕ ∂ → = ∂ V A B ϕ → = + 0 0 V B ϕ= = = 0 V A B V ϕ α α = = + = 0 0 B V A α =   →  =   0 V V ϕ α → = 0 E a V V ϕ αρ → = −∇ = − Ex. 3 z α Air gap 18
  19. 19. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Examples of the Solution of Laplace’s Equation (6) Ex. 4 Assume V = V(θ) (spherical) 2 2 2 2 2 2 2 2 1 1 1 sin 0 sin sin V V V V r r r r r r θ θ θ θ θ ϕ ∂ ∂ ∂ ∂ ∂     ∇ = + + =     ∂ ∂ ∂ ∂ ∂     2 1 sin 0 sin V r θ θ θ θ ∂ ∂   → =   ∂ ∂   sin 0 V θ θ θ ∂ ∂   → =   ∂ ∂   sin dV A d θ θ → = sin d dV A θ θ → = ln tan 2 A B θ   = +     sin d V A B θ θ → = +  Assume r ≠ 0; θ ≠ 0; θ ≠ π 19
  20. 20. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Examples of the Solution of Laplace’s Equation (7) Ex. 4 Assume V = V(θ) ln tan 2 V A B θ   → = +     0 ln tan 2 ln tan 2 V V θ α       → =       / 2 0 V θ π = = 0 ( / 2) V V θ α α π = = < V = 0 V = V0 α Air gap 0 1 sin ln tan 2 V V V r r θ θ α θ θ ∂ → = −∇ = − = − ∂       E a a 0 sin ln tan 2 S N V D E r ε ρ ε α α → = = = −       20
  21. 21. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Examples of the Solution of Laplace’s Equation (8) Ex. 4 Assume V = V(θ) 0 0 2 ln tan 2 V dr πε α ∞ − =        0 sin ln tan 2 S S S V Q dS dS r ε ρ α α → = = −         2 0 0 0 sin sin ln tan 2 V r d dr Q r π ε α ϕ α α ∞ − → =         sin dS r d dr α ϕ = 0 sin ln tan 2 S V r ε ρ α α → = −       1 0 2 ln cot 2 Q r C V πε α → =       ≐ V = 0 V = V0 α Air gap 21
  22. 22. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Poisson’s & Laplace’s Equations 1. Poisson’s Equation 2. Laplace’s Equation 3. Uniqueness Theorem 4. Examples of the Solution of Laplace’s Equation 5. Examples of the Solution of Poisson’s Equation 6. Product Solution of Laplace’s Equation 7. Numerical Methods 22
  23. 23. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Examples of the Solution of Poisson’s Equation (1) 2 0 2 2 sech th v d V x x a a dx ρ ε → = − 2 ( sech ; th ) x x x x x x e e x x e e e e − − − − = = + + 0 2 sech th v v x x a a ρ ρ = x dV E dx = − 2 v V ρ ε ∇ = − Poisson’s Equation : 0 1 2 sech v a dV x C dx a ρ ε → = + 0 1 2 sech v x a x E C a ρ ε → = − − –5 –4 –3 –2 –1 1 2 3 4 5 0,5 1 –0,5 –1 / x a 0 v v ρ ρ p-type n-type v ρ –5 –4 –3 –2 –1 1 2 3 4 5 –0,5 –1 / x a 0 2 x v E a ε ρ x E 1 0 C → = 0 2 sech v x a x E a ρ ε → = − If x → ± ∞ then Ex → 0 23
  24. 24. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 2 / 0 4 arctg 4 x a v a V e ρ π ε   → = −     0 2 sech th v v x x a a ρ ρ = 2 v V ρ ε ∇ = − 2 / 0 2 4 arctg x a v a V e C ρ ε → = + 0 2 sech v x a x E a ρ ε → = − –5 –4 –3 –2 –1 1 2 3 4 5 –0,5 –1 / x a 0 2 x v E a ε ρ V –5 –4 –3 –2 –1 1 2 3 4 5 0,25 0,5 –0,25 –0,5 / x a 2 0 2 v V a πε ρ x E Poisson’s equation : 0 0 x V = = Supp. 2 0 2 4 0 4 v a C ρ π ε → = + Examples of the Solution of Poisson’s Equation (2) –5 –4 –3 –2 –1 1 2 3 4 5 0,5 1 –0,5 –1 / x a 0 v v ρ ρ p-type n-type v ρ 24
  25. 25. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 2 / 0 4 arctg 4 x a v a V e ρ π ε   = −     2 0 0 2 v x x a V V V πρ ε → ∞ → −∞ = − = 0 0 0 0 2 sech th 2 sech th 2 v v v v V V x x x x Q dv dv S dx aS a a a a ρ ρ ρ ρ ∞ = = = =    0 0 2 v V Q S ρ ε π → = 0 2 sech th v v x x a a ρ ρ = 0 0 dV dQ dQ I C C dt dt dV = = → = 0 0 2 2 v S C S V a ρ ε ε π π → = = Examples of the Solution of Poisson’s Equation (3) –5 –4 –3 –2 –1 1 2 3 4 5 0,5 1 –0,5 –1 / x a 0 v v ρ ρ p-type n-type v ρ 25
  26. 26. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Poisson’s & Laplace’s Equations 1. Poisson’s Equation 2. Laplace’s Equation 3. Uniqueness Theorem 4. Examples of the Solution of Laplace’s Equation 5. Examples of the Solution of Poisson’s Equation 6. Product Solution of Laplace’s Equation 7. Numerical Methods 26
  27. 27. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Product Solution of Laplace’s Equation (1) • Previous examples assumed that V varies with one of the three coordinates • The product solution can be used to solve for V(x, y) • Assume V = XY, X = X(x), Y = Y(y) 2 2 2 2 2 2 2 0 V V V V x y z ∂ ∂ ∂ ∇ = + + = ∂ ∂ ∂ 2 2 2 2 0 V V x y ∂ ∂ → + = ∂ ∂ ( , ) V V x y = 2 2 2 2 0 X Y Y X x y ∂ ∂ → + = ∂ ∂ 2 2 2 2 0 d X d Y Y X dx dy → + = 27
  28. 28. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Product Solution of Laplace’s Equation (2) 2 2 2 2 0 d X d Y Y X dx dy + = 2 2 2 2 1 1 0 d X d Y X dx Y dy → + = 2 2 2 2 1 1 d X d Y X dx Y dy → = − 2 2 1 d X X dx involves no y 2 2 1 d Y Y dy − involves no x 2 2 2 2 2 2 1 1 d X X dx d Y Y dy α α  =   →  − =   28
  29. 29. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Product Solution of Laplace’s Equation (3) 2 2 2 2 1 d dR R d d d d ρ ρ α ρ ρ α ϕ    =       →  Φ − =  Φ  ( , ) ( ) ( ) V V R ρ ϕ ρ ϕ = = Φ 2 2 2 2 2 1 1 0 V V V z ρ ρ ρ ρ ρ ϕ   ∂ ∂ ∂ ∂ + + =   ∂ ∂ ∂ ∂   2 2 1 0 R R ρ ρ ρ ρ ϕ   ∂ ∂ ∂ Φ → + =   ∂ ∂ Φ ∂   29
  30. 30. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Product Solution of Laplace’s Equation (4) 2 2 2 2 2 2 ( 1) 1 1 ( 1) tg R R n n R R n n ρ ρ ρ ρ θ θ θ  ∂ ∂ + = +  ∂ ∂  →  ∂ Θ ∂Θ  + = − + Θ ∂ Θ ∂  ( , ) ( ) ( ) V V R ρ θ ρ θ = = Θ 2 2 2 2 2 2 1 1 0 tg R R R R ρ ρ ρ ρ θ θ θ     ∂ ∂ ∂ Θ ∂Θ → + + + =     ∂ ∂ Θ ∂ Θ ∂     2 2 2 2 2 2 2 1 1 1 sin 0 sin sin V V V r r r r r r θ θ θ θ θ ϕ ∂ ∂ ∂ ∂ ∂     + + =     ∂ ∂ ∂ ∂ ∂     30
  31. 31. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Product Solution of Laplace’s Equation (5) Ex. 2 2 2 2 1 d dR R d d d d ρ ρ α ρ ρ α ϕ    =       →  Φ − =  Φ  ( , ) ( ) ( ) V V R ρ ϕ ρ ϕ = = Φ ( ) ( ); ( ) ( ) V V V V ϕ ϕ ϕ π ϕ = − = − − ( ) cos sin , Assume p p A p B p p ϕ ϕ ϕ α Φ = + = ±   1 ( ) cos , 1 A ϕ ϕ α → Φ = = Solve for the potential & EFI in the vicinity of a conducting cylinder (posses an infinite length) in a uniform EFI E0. The permittivities of the environment & the cylinder are ε1 & ε2 respectively. EFI is perpendicular to the cylinder’s axis. 31
  32. 32. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Product Solution of Laplace’s Equation (6) 2 2 2 2 1 d d d dR R d d α ϕ ρ ρ α ρ ρ  Φ − =  Φ      =       2 2 k k k k k B d dR R d d B ρ ρ ρ α ρ ρ ρ   → = =     1 α = ( ) Assume k k R B ρ ρ = 1 ( ) cos , 1 A ϕ ϕ α → Φ = = 1 k → = ± 1 1 1 ( ) R B B ρ ρ ρ + − − → = + Ex. 32 Solve for the potential & EFI in the vicinity of a conducting cylinder (posses an infinite length) in a uniform EFI E0. The permittivities of the environment & the cylinder are ε1 & ε2 respectively. EFI is perpendicular to the cylinder’s axis.
  33. 33. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Product Solution of Laplace’s Equation (7) 2 2 1 2 2 1 1 1 1 ( ) cos ( ) d A d d dR R B B R d d α ϕ ϕ ϕ ρ ρ α ρ ρ ρ ρ ρ + − −  Φ − = → Φ =  Φ      = → = +       1 1 1 1 1 cos cos V A B A B ρ ϕ ρ ϕ + − − → = + ( , ) ( ) ( ) V V R ρ ϕ ρ ϕ = = Φ 1 cos cos C C ρ ϕ ρ ϕ + − − = + Ex. 33 Solve for the potential & EFI in the vicinity of a conducting cylinder (posses an infinite length) in a uniform EFI E0. The permittivities of the environment & the cylinder are ε1 & ε2 respectively. EFI is perpendicular to the cylinder’s axis.
  34. 34. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Product Solution of Laplace’s Equation (8) 1 1 1 1 cos cos Exterior: V C C ρ ϕ ρ ϕ + − − = + 0 x V E x −∞ →−∞ = 1 2 2 2 cos cos Interior: V C C ρ ϕ ρ ϕ + − − = + 1 1 , x V V C x θ π ρ + −∞ = →−∞ →−∞ = = − x y E0 ρ θ ε1 ε2 1 0 C E + → = − 2 2 0 0 gèc täa ®é C V V ρ ρ ρ − → → = = → ∞ 2 0 C− → = EFI is finite at the origin Ex. 34 Solve for the potential & EFI in the vicinity of a conducting cylinder (posses an infinite length) in a uniform EFI E0. The permittivities of the environment & the cylinder are ε1 & ε2 respectively. EFI is perpendicular to the cylinder’s axis.
  35. 35. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Product Solution of Laplace’s Equation (9) 1 1 1 1 cos cos Exterior: V C C ρ ϕ ρ ϕ + − − = + 1 2 2 2 cos cos Interior: V C C ρ ϕ ρ ϕ + − − = + x y E0 ρ θ ε1 ε2 1 0 2 , 0 C E C + − = − = 1 1 0 1 2 2 cos cos cos V E C V C ρ ϕ ρ ϕ ρ ϕ − − +  = − +  →  =   Ex. 35 Solve for the potential & EFI in the vicinity of a conducting cylinder (posses an infinite length) in a uniform EFI E0. The permittivities of the environment & the cylinder are ε1 & ε2 respectively. EFI is perpendicular to the cylinder’s axis.
  36. 36. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Product Solution of Laplace’s Equation (10) 1 1 0 1 cos cos V E C ρ ϕ ρ ϕ − − = − + 2 2 cos V C ρ ϕ + = x y E0 ρ θ ε1 ε2 1 2 a a V V ρ ρ = = = 1 0 1 2 ( )cos cos E a C a C a ϕ ϕ − − + → − + = a 1 2 1 2 a a V V ρ ρ ε ε ρ ρ = = ∂ ∂ = ∂ ∂ 2 1 0 1 2 2 ( )cos cos E C a C a ε ϕ ε ϕ − − + → − − = Ex. 36 Solve for the potential & EFI in the vicinity of a conducting cylinder (posses an infinite length) in a uniform EFI E0. The permittivities of the environment & the cylinder are ε1 & ε2 respectively. EFI is perpendicular to the cylinder’s axis.
  37. 37. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Product Solution of Laplace’s Equation (11) 1 0 1 2 ( )cos cos E a C a C a ϕ ϕ − − + − + = 2 1 0 1 2 2 ( )cos cos E C a C a ε ϕ ε ϕ − − + − − = 2 1 2 1 1 0 2 0 1 2 1 2 2 , C E a C E ε ε ε ε ε ε ε − + − → = − = − + + 2 1 2 1 0 2 1 2 1 2 0 1 2 1 cos , 2 cos as as a V E a V E a ε ε ρ ϕ ρ ε ε ρ ε ρ ϕ ρ ε ε    − = − − ≥    +    →   = − ≤  +  1 1 0 1 cos cos V E C ρ ϕ ρ ϕ − − = − + 2 2 cos V C ρ ϕ + = Ex. 37 Solve for the potential & EFI in the vicinity of a conducting cylinder (posses an infinite length) in a uniform EFI E0. The permittivities of the environment & the cylinder are ε1 & ε2 respectively. EFI is perpendicular to the cylinder’s axis.
  38. 38. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Product Solution of Laplace’s Equation (12) 2 1 2 1 0 2 1 2 1 2 0 1 2 1 cos , 2 cos as as a V E a V E a ε ε ρ ϕ ρ ε ε ρ ε ρ ϕ ρ ε ε    − = − − ≥    +      = − ≤  +  2 1 2 1 2 0 2 0 1 2 1 2 2 2 cos , sin V V E E E E ρ ϕ ε ε ϕ ϕ ρ ε ε ϕ ε ε ∂ ∂ = − = = − = − ∂ + ∂ + 2 2 1 1 2 1 1 2 1 0 1 0 2 2 1 2 1 2 1 cos , 1 sin V V a a E E E E ρ ϕ ε ε ε ε ϕ ϕ ρ ε ε ρ ϕ ε ε ρ     ∂ − ∂ − = − = − = − = − −     ∂ + ∂ +     1 2 2 0 1 2 2 z E E E ε ε ε → = = + Ex. 38 Solve for the potential & EFI in the vicinity of a conducting cylinder (posses an infinite length) in a uniform EFI E0. The permittivities of the environment & the cylinder are ε1 & ε2 respectively. EFI is perpendicular to the cylinder’s axis.
  39. 39. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Poisson’s & Laplace’s Equations 1. Poisson’s Equation 2. Laplace’s Equation 3. Uniqueness Theorem 4. Examples of the Solution of Laplace’s Equation 5. Examples of the Solution of Poisson’s Equation 6. Product Solution of Laplace’s Equation 7. Numerical Methods a. Finite Difference Method b. Finite Element Method 39
  40. 40. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Finite Difference Method (1) x y h h V0 V1 V2 V3 V4 a b c d 2 2 2 2 2 2 2 0 V V V V x y z ∂ ∂ ∂ ∇ = + + = ∂ ∂ ∂ 2 2 2 2 0 V V x y ∂ ∂ → + = ∂ ∂ ( , ) V V x y = 1 0 a V V V x h − ∂ ≈ ∂ 2 1 0 0 3 2 2 V V V V V x h − − + ∂ → ≈ ∂ 0 3 c V V V x h − ∂ ≈ ∂ 2 2 a c V V V x x x h ∂ ∂ − ∂ ∂ ∂ ≈ ∂ 40
  41. 41. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Finite Difference Method (2) x y h h V0 V1 V2 V3 V4 2 2 2 2 0 V V x y ∂ ∂ + = ∂ ∂ 2 2 0 0 4 2 2 V V V V V y h − − + ∂ ≈ ∂ 2 1 0 0 3 2 2 V V V V V x h − − + ∂ ≈ ∂ 2 2 1 2 3 4 0 2 2 4 0 V V V V V V V x y h + + + − ∂ ∂ → + ≈ = ∂ ∂ ( ) 0 1 2 3 4 1 4 V V V V V ≈ + + + → 41
  42. 42. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Finite Difference Method (3) Ex. 1 V = 100 V = 0 V = 0 V = 0 Air gap Air gap ( ) 0 1 2 3 4 1 4 V V V V V = + + + 42 Ve Vc Va Vb Vd Vf Vg Vi Vh 4 100 0 a d b V V V = + + + 4 100 b a e c V V V V = + + + 4 0 100 c b f V V V = + + + 4 0 d g e a V V V V = + + + 4 e b d h f V V V V V = + + + 4 0 f c e i V V V V = + + + 4 0 0 g d h V V V = + + + 4 0 h e g i V V V V = + + + 4 0 0 i f h V V V = + + + 42.86V 52.68V 42.86V 18.75V 25.00V 18,75V 7.14V 9.82 V 7.14V a b c d e f g h i V V V V V V V V V  =  =   =  =   → =   =   =  =   =  Method 1
  43. 43. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Finite Difference Method (4) Ex. 1 V = 100 V = 0 V = 0 V = 0 Air gap Air gap ( ) 0 1 2 3 4 1 4 V V V V V = + + + ( ) 1 0 100 0 0 25 4 + + + = ( ) 1 100 50 0 25 43.8 4 + + + = ( ) 1 0 25 0 0 6.2 4 + + + = ( ) 1 43.8 100 43.8 25 53.2 4 + + + = ( ) 1 25 43.8 0 6.2 18.8 4 + + + = ( ) 1 6.2 25 6.2 0 9.4 4 + + + = 25 43.8 43.8 53.2 18.8 18.8 6.2 6.2 9.4 Step 1 43 Method 2
  44. 44. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Finite Difference Method (5) V = 100 V = 0 V = 0 V = 0 Air gap Air gap ( ) 0 1 2 3 4 1 4 V V V V V = + + + ( ) 1 53.2 100 0 18.8 43 4 + + + = ( ) 1 100 50 0 25 43.8 4 + + + = ( ) 1 43 100 43 25 52.8 4 + + + = ( ) 1 43.8 100 43.8 25 53.2 4 + + + = ( ) 1 25 43.8 0 6.2 18.8 4 + + + = 25 43.8 43.8 53.2 18.8 18.8 6.2 6.2 9.4 ( ) 1 25 43 0 6.2 18.6 4 + + + = 43 43 52.8 18.6 18.6 Step 2 Ex. 1 44 Method 2
  45. 45. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Finite Difference Method (6) V = 100 V = 0 V = 0 V = 0 Air gap Air gap ( ) 0 1 2 3 4 1 4 V V V V V = + + + 25 43.8 43.8 53.2 18.8 18.8 6.2 6.2 9.4 43 43 52.8 18.6 18.6 ( ) 1 0 100 0 0 25 4 + + + = ( ) 1 0 25 0 0 6.2 4 + + + = ( ) 1 6.2 25 6.2 0 9.4 4 + + + = ( ) 1 18.6 52.8 18.6 9.4 24.9 4 + + + = ( ) 1 9.4 18.6 0 0 7.0 4 + + + = 24.9 7.0 7.0 ( ) 1 7.0 25 7.0 0 9.8 4 + + + = 9.8 Step 2 Ex. 1 45 Method 2
  46. 46. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Finite Difference Method (7) V = 100 V = 0 V = 0 V = 0 Air gap Air gap ( ) 0 1 2 3 4 1 4 V V V V V = + + + 25 43.8 43.8 53.2 18.8 18.8 6.2 6.2 9.4 43 43 52.8 18.6 18.6 24.9 7.0 7.0 9.8 ( ) 1 52.8 100 0 18.6 42.9 4 + + + = ( ) 1 42.9 100 42.9 24.9 52.7 4 + + + = ( ) 1 24.9 42.9 0 7.0 18.7 4 + + + = ( ) 1 18.7 52.7 18.7 9.8 25.0 4 + + + = ( ) 1 9.8 18.7 0 0 7.1 4 + + + = ( ) 1 7.1 25 7.1 0 9.8 4 + + + = 42.9 42.9 52.7 18.7 18.7 25.0 7.1 7.1 9.8 Step 3 Ex. 1 46 Method 2
  47. 47. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Finite Difference Method (8) Ex. 2 0 V 0 V 0 V 0 V 10 V 20 V 4 1 2 3 5 6 7 8 9 10 47 1 2 4 4 10 0 V V V = + + + 2 1 5 3 4 10 V V V V = + + + 3 2 6 4 20 10 V V V = + + + 4 1 5 4 0 0 V V V = + + + 5 2 4 7 6 4V V V V V = + + + 6 3 5 8 4 20 V V V V = + + + 7 5 9 8 4 0 V V V V = + + + 8 6 7 10 4 20 V V V V = + + + 9 7 10 4 0 0 V V V = + + + 10 8 9 4 0 20 V V V = + + + 1 2 3 4 5 6 7 8 9 10 5.6423V 9.1735V 13.1111V 3.3957 V 7.9405V 13.2710 V 5.9219 V 12.0324V 3.7147 V 8.9368V V V V V V V V V V V =   =   =  =   =  →  =   =  =   =   =  Method 1
  48. 48. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Finite Difference Method (9) Ex. 2 0 V 0 V 0 V 0 V 10 V 20 V 4 1 2 3 5 6 7 8 9 10 (0) (0) (0) 1 2 10 ... 0 V V V = = = = ( ) (1) (0) (0) 1 2 4 1 10 0 2.5000V 4 V V V = + + + = ( ) (1) (0) (1) (0) 2 3 1 5 1 10 3.1250V 4 V V V V = + + + = ( ) (1) (0) (1) (0) 7 8 5 9 ... 1 0 0.2344V 4 ... V V V V = + + + = ( ) (1) (1) (1) 10 8 9 1 20 0 6.7358V 4 V V V = + + + = 48 Method 2
  49. 49. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Finite Difference Method (10) Ex. 2 0 V 0 V 0 V 0 V 10 V 20 V 4 1 2 3 5 6 7 8 9 10 k ( ) 1 (V) k V ( ) 2 (V) k V ( ) 3 (V) k V ( ) 4 (V) k V ( ) 5 (V) k V ( ) 6 (V) k V ( ) 7 (V) k V ( ) 8 (V) k V ( ) 9 (V) k V ( ) 10 (V) k V 0 1 0 0 0 0 0 0 0 0 0 0 2.5000 3.1250 8.2813 0.6250 0.9375 7.3047 0.2344 6.8848 0.0586 6.7358 23 5.6429 9.1735 13.1111 3.3957 7.9405 13.2710 5.9219 13.0324 3.7147 8.9368 24 5.6429 9.1735 13.1111 3.3957 7.9405 13.2710 5.9219 13.0324 3.7147 8.9368 49 Method 2
  50. 50. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Poisson’s & Laplace’s Equations 1. Poisson’s Equation 2. Laplace’s Equation 3. Uniqueness Theorem 4. Examples of the Solution of Laplace’s Equation 5. Examples of the Solution of Poisson’s Equation 6. Product Solution of Laplace’s Equation 7. Numerical Methods a. Finite Difference Method b. Finite Element Method 50
  51. 51. Finite Element Method (1) Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn http://w ww.cosy.sbg.ac.at/~held/projects/mesh/mesh.html 51 http://imagine.inrialpes.fr/people/Francois.Faure/htmlCourses/FiniteElements.html 2 0 V ∇ = ( , ) V x y 1( , ) V x y 2 V 3 V 4 V
  52. 52. Finite Element Method (2) Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn http://illustrations.marin.ntnu.no/structures/analysis/FEM/theory/index.html 52 http://imagine.inrialpes.fr/people/Francois.Faure/htmlCourses/FiniteElements.html 2 0 V ∇ = ( , ) V x y 1( , ) V x y 2 V 3 V 4 V
  53. 53. Finite Element Method (3) • Discretizing the solution region into a finite number of elements, • Obtaining governing equations for a typical elements, • Combining all elements in the solution, & • Solving the system of equations obtained. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 53 http://imagine.inrialpes.fr/people/Francois.Faure/htmlCourses/FiniteElements.html 2 0 V ∇ = ( , ) V x y 1( , ) V x y 2 V 3 V 4 V
  54. 54. Finite Element Method (4) Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 54 1 e V 2 e V 3 e V 1 1 ( , ) x y 2 2 ( , ) x y 3 3 ( , ) x y 1 2 3 x y 1 ( , ) ( , ) N e e V x y V x y = =  ( , ) e V x y a bx cy = + + 1 1 1 2 2 2 3 3 3 1 1 1 e e e V x y a V x y b V x y c             =                   1 1 1 1 2 2 2 3 3 3 1 1 1 e e e a x y V b x y V c x y V −             → =                   [ ] 1 2 3 3 2 3 1 1 3 1 2 2 1 1 2 3 3 1 1 2 2 1 1 3 2 1 3 2 1 3 2 2 3 3 ( ) ( ) ( ) 1 1 ( ) ( ) ( ) 1 ( ) ( ) ( ) 1 1 e e e e x y x y x y x y x y x y V V x y y y y y y y V x y x x x x x x V x y x y − − − −         → = − − −         − − −    
  55. 55. Finite Element Method (5) Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 55 [ ] 1 2 3 3 2 3 1 1 3 1 2 2 1 1 2 3 3 1 1 2 2 1 1 3 2 1 3 2 1 3 2 2 3 3 ( ) ( ) ( ) 1 1 ( ) ( ) ( ) 1 ( ) ( ) ( ) 1 1 e e e e x y x y x y x y x y x y V V x y y y y y y y V x y x x x x x x V x y x y − − − −         = − − −         − − −     3 1 ( , ) e i ei i V x y V α = =  1 2 3 3 2 2 3 3 2 2 3 1 1 3 3 1 1 3 3 1 2 2 1 1 2 2 1 2 1 3 1 3 1 2 1 1 [( ) ( ) ( ) ], 2 1 [( ) ( ) ( ) ], 2 1 [( ) ( ) ( ) ], 2 1 [( )( ) ( )( )] 2 x y x y y y x x x y A x y x y y y x x x y A x y x y y y x x x y A A x x y y x x y y α α α = − + − + − = − + − + − = − + − + − = − − − − −
  56. 56. Finite Element Method (6) Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 56 2 0 V ∇ = 0 ax b + = 2 0 2 d ax bx c dx   + + =     2 1 2 e e S W E dS ε =  V = −∇ E 2 1 2 e e S W V dS ε → = ∇  3 1 ( , ) e i ei i V x y V α = =  3 1 e ei i i V V α = → ∇ = ∇  3 3 1 1 1 ( )( ) 2 e ei i j ej S i j W V dS V ε α α = =   → = ∇ ∇    
  57. 57. Finite Element Method (7) Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 57 3 3 1 1 1 ( )( ) 2 e ei i j ej S i j W V dS V ε α α = =   = ∇ ∇     ( ) ( )( ) e ij i j S C dS α α = ∇ ∇  [ ] 1 2 3 e e e e V V V V     =       ( ) ( ) ( ) 11 12 13 ( ) ( ) ( ) ( ) 21 22 23 ( ) ( ) ( ) 31 32 33 e e e e e e e e e e C C C C C C C C C C       =         [ ] [ ] ( ) 1 2 T e e e e W V C V ε   → =  
  58. 58. Finite Element Method (8) Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 58 ( ) ( )( ) e ij i j S C dS α α = ∇ ∇  ( ) 12 1 2 ( )( ) e S C dS α α = ∇ ∇  1 2 3 3 2 2 3 3 2 2 3 1 1 3 3 1 1 3 2 1 3 1 3 1 2 1 1 [( ) ( ) ( ) ], 2 1 [( ) ( ) ( ) ], 2 1 [( )( ) ( )( )] 2 x y x y y y x x x y A x y x y y y x x x y A A x x y y x x y y α α = − + − + − = − + − + − = − − − − − ( ) 12 2 3 3 1 3 2 1 3 1 [( )( ) ( )( )] 4 e C y y y y x x x x A → = − − + − −
  59. 59. Finite Element Method (9) Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 59 ( ) 12 2 3 3 1 3 2 1 3 1 [( )( ) ( )( )] 4 e C y y y y x x x x A = − − + − − ( ) 13 2 3 1 2 3 2 2 1 ( ) 23 3 1 1 2 1 3 2 1 ( ) 2 2 11 2 3 3 2 ( ) 2 2 22 3 1 1 3 ( ) 2 2 33 1 2 2 1 ( ) ( ) ( ) ( ) 21 12 31 13 32 1 [( )( ) ( )( )] 4 1 [( )( ) ( )( )] 4 1 [( ) ( ) ] 4 1 [( ) ( ) ] 4 1 [( ) ( ) ] 4 , , e e e e e e e e e C y y y y x x x x A C y y y y x x x x A C y y x x A C y y x x A C y y x x A C C C C C = − − + − − = − − + − − = − + − = − + − = − + − = = ( ) ( ) 23 e e C = 1 2 3 2 3 1 3 1 2 1 3 2 2 1 3 3 2 1 , , , , P y y P y y P y y Q x x Q x x Q x x = − = − = − = − = − = − ( ) 2 3 3 2 1 ( ) 4 2 e ij i j i j C PP QQ A PQ PQ A → = + − =
  60. 60. Finite Element Method (3) • Discretizing the solution region into a finite number of elements, • Obtaining governing equations for a typical elements, • Combining all elements in the solution, & • Solving the system of equations obtained. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 60 http://imagine.inrialpes.fr/people/Francois.Faure/htmlCourses/FiniteElements.html 2 0 V ∇ = ( , ) V x y 1( , ) V x y 2 V 3 V 4 V
  61. 61. Finite Element Method (10) Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 61 [ ] [ ] ( ) 1 2 T e e e e W V C V ε   =   [ ] [ ][ ] 1 1 2 N T e e W W V C V ε = = =  [ ] 1 2 3 n V V V V V         =         ⋮
  62. 62. Finite Element Method (11) Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 62 3 2 1 1 1 1 2 2 2 3 3 3 1 2 4 3 5 [ ] [ ][ ] 1 2 T W V C V ε = [ ] 11 12 13 14 15 21 22 23 24 25 31 32 33 34 35 41 42 43 44 45 51 52 53 54 55 C C C C C C C C C C C C C C C C C C C C C C C C C C         =         (1) (2) 11 11 11 C C C = + (1) 22 33 C C = (1) (2) (3) 44 22 33 33 C C C C = + + (1) (2) 14 41 12 13 C C C C = = + 23 32 0 C C = = [ ] (1) (2) (1) (2) (1) (2) 11 11 13 12 12 13 (1) (1) (1) 31 33 32 (2) (2) (3) (2) (3) (3) 21 22 11 23 13 12 (1) (2) (1) (2) (3) (1) (2) (3) (3) 21 31 23 32 31 22 33 33 32 (3) (3) (3) 21 23 22 0 0 0 0 0 0 C C C C C C C C C C C C C C C C C C C C C C C C C C C C   + +       = + +   + + + +      
  63. 63. Finite Element Method (3) • Discretizing the solution region into a finite number of elements, • Obtaining governing equations for a typical elements, • Combining all elements in the solution, & • Solving the system of equations obtained. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 63 http://imagine.inrialpes.fr/people/Francois.Faure/htmlCourses/FiniteElements.html 2 0 V ∇ = ( , ) V x y 1( , ) V x y 2 V 3 V 4 V
  64. 64. Finite Element Method (12) Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 64 0 ax b + = 2 0 2 d ax bx c dx   + + =     2 0 V ∇ = [ ] [ ][ ] 1 2 T W V C V ε = 1 2 0 0, 1, 2,..., n k W W W W k n V V V V ∂ ∂ ∂ ∂ = = = = ↔ = = ∂ ∂ ∂ ∂ ⋯ 1, 1 n k i ki i i k kk V VC C = ≠ → = − 
  65. 65. Finite Element Method (13) Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 65 Ex. 0 V = 1 2 3 x y 4 100 V = 2 1 1 2 3 1 2 4 3 2 4 1 2 3 Node 1 2 3 4 x 0.5 3.1 5.0 2.8 y 1.0 0.4 1.7 2.0 ( ) 2 3 3 2 1 2 3 2 3 1 3 1 2 1 3 2 2 1 3 3 2 1 1 ( ), 4 2 , , , , e ij i j i j P Q PQ C PP QQ A A P y y P y y P y y Q x x Q x x Q x x − = + = = − = − = − = − = − = − Element 1: 1 2 3 0.4 2.0 1.6; 2.0 1.0 1.0; 1.0 0.4 0.6 P P P = − = − = − = = − = 1 2 3 2.8 3.1 0.3; 0.5 2.8 2.3; 3.1 0.5 2.6 Q Q Q = − = − = − = − = − = 1.0 2.6 0.6( 2.3) 1.99 2 A × − − = = (1) 1 2 1 2 12 ( 1.6)1.0 ( 0.3)( 2.3) 0.1143 4 1.99 4 1.99 PP Q Q C + − + − − = = = − × × (1) 0.3329 0.1143 0.2186 0.1143 0.7902 0.6759 0.2186 0.6759 0.8945 C − −       = − −       − −  
  66. 66. Finite Element Method (14) Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 66 Ex. 0 V = 1 2 3 x y 4 100 V = Node 1 2 3 4 x 0.5 3.1 5.0 2.8 y 1.0 0.4 1.7 2.0 2 1 1 2 3 1 2 4 3 2 4 1 2 3 ( ) 2 3 3 2 1 2 3 2 3 1 3 1 2 1 3 2 2 1 3 3 2 1 1 ( ), 4 2 , , , , e ij i j i j P Q PQ C PP QQ A A P y y P y y P y y Q x x Q x x Q x x − = + = = − = − = − = − = − = − Element 2: 1 2 3 1.7 2.0 0.3; 2.0 0.4 1.6; 0.4 1.7 1.3 P P P = − = − = − = = − = − 1 2 3 2.8 5 2.2; 3.1 2.8 0.3; 5.0 3.1 1.9 Q Q Q = − = − = − = = − = 1.6 1.9 ( 1.3)0.3 1.715 2 A × − − = = (2) 2 2 2 2 22 1.6 1.6 0.3 0.3 0.3863 4 1.715 4 1.715 P P Q Q C + × + × = = = × × (2) 0.7187 0.1662 0.5525 0.1662 0.3863 0.2201 0.5525 0.2201 0.7726 C − −       = − −       − −  
  67. 67. Finite Element Method (15) Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 67 Ex. 0 V = 1 2 3 x y 4 100 V = (1) (2) 0.3329 0.1143 0.2186 0.1143 0.7902 0.6759 0.2186 0.6759 0.8945 0.7187 0.1662 0.5525 0.1662 0.3863 0.2201 0.5525 0.2201 0.7726 C C − −       = − −       − −   − −       = − −       − −   [ ] (1) (1) (1) 11 12 13 (1) (1) (2) (2) (1) (2) 21 22 11 12 23 13 (2) (2) (2) 21 22 23 (1) (1) (2) (2) (1) (2) 31 32 31 32 33 33 0 0 C C C C C C C C C C C C C C C C C C C     + +   =     + +     1 2 3 1 2 3 0.3329 0.1143 0 0.2186 0.1143 1.5089 0.1662 1.2284 0 0.1662 0.3863 0.2201 0.2186 1.2284 0.2201 1.6671 − −     − − −   =   − −   − − −  
  68. 68. Finite Element Method (16) Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 68 Ex. 0 V = 1 2 3 x y 4 100 V = [ ] 0.3329 0.1143 0 0.2186 0.1143 1.5089 0.1662 1.2284 0 0.1662 0.3863 0.2201 0.2186 1.2284 0.2201 1.6671 C − −     − − −   =   − −   − − −   1, 1 n k i ki i i k kk V VC C = ≠ = −  1 2 3 1 2 3 2 1 12 3 32 4 42 22 4 1 14 2 24 3 34 44 1 ( ) 1 ( ) V VC V C V C C V VC V C V C C  = − + +   →   = − + +   ( 1) ( ) ( ) 2 4 4 ( 1) ( ) ( ) 4 2 2 1 [(0( 0.1143) 100( 0.1662) ( 1.2284)] 11.0146 0.8141 1.5089 1 [0( 0.2186) ( 1.2284) 100( 0.2201)] 13.2026 0.7368 1.6671 k k k k k k V V V V V V + +  = − − + − + − = +   →   = − − + − + − = +  
  69. 69. Finite Element Method (17) Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 69 Ex. 0 V = 1 2 3 x y 4 100 V = ( 1) ( ) 2 4 ( 1) ( ) 4 2 11.0146 0.8141 13.2026 0.7368 k k k k V V V V + +  = +   = +   (0) (0) 2 4 0 100 50 2 V V + = = = 1 2 3 1 2 3 (1) (0) 2 4 (1) (0) 4 2 11.0146 0.8141 11.0146 0.8141 50 51.7196 13.2026 0.7368 13.2026 0.7368 50 50.0426 V V V V  = + = + × =   = + = + × =   (2) (1) 2 4 (2) (1) 4 2 11.0146 0.8141 11.0146 0.8141 50.0426 51.7543 13.2026 0.7368 13.2026 0.7368 51.7196 51.3096 V V V V  = + = + × =   = + = + × =   (3) (2) 2 4 (3) (2) 4 2 11.0146 0.8141 11.0146 0.8141 51.3096 52.7857 13.2026 0.7368 13.2026 0.7368 51.7543 51.3352 V V V V  = + = + × =   = + = + × =  
  70. 70. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 70 Q 1 2 2 4 R Q Q R πε = F a 2 4 R Q R πε = E a ε = D E . W Q d = − E L . V d = −E L Q C V = dQ I dt = V R I = 2 v V ρ ε ∇ = −

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