Chapter 7 - Poisson's and Laplace's Equations

Chapter 7 - Poisson's and Laplace's Equations - Chapter 7...

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Chapter 7 Poisson’s and Laplace’s Equations
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Poisson's and Laplace’s Equations Point form of Gauss's Law: ∇ • D = ρ v Relationship between D and E : D = ε E Gradient relationship between V and E : E = - V Combining the three equations: ∇• D = ∇ • ( ε E ) = - ∇ • ( ε∇ V) = ρ v ∇ • ∇ V = Poisson’s Equation ε ρ - v Recall:
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Note: ∇ • ∇ is abbreviated as 2 ("del squared") If the charge density is zero, then ∇ • ∇ V = 2 V = 0 The equation is now called Laplace's equation . The 2 operation is called the Laplacian of V .
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Any expression for potential V is valid if it satisfied Laplace’s equation (when ρ s = 0) or Poisson’s equation (when ρ s 0) Laplace’s and Poisson’s equations relate the potential field to the charge density in a region.
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In cartesian coordinates: ε ρ - = + + = v 2 2 2 2 2 2 2 z V y V x V V Cylindrical coordinates: 2 2 2 2 2 2 z V V 1 V 1 V + φ
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Chapter 7 - Poisson's and Laplace's Equations - Chapter 7...

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