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Chapter 6 Electrostatic Boundary-Value Problems Lecture by Qiliang Li 1

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§6.1 Introduction 2

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However, we usually don’t know the charge distribution or potential profile inside the medium. In most cases, we can observe or measure the electrostatic charge or potential at some boundaries We can determine the electric field E by using the electrostatic boundary conditions 3

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§6.2 Poisson’s and Laplace’s Equations 4

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Continue 6.2 5

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§6.3 Uniqueness Theorem Uniqueness Theorem: If a solution to Laplace’s equation can be found that satisfies the boundary conditions, then the solution is unique. 6

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§6.4 General Procedures for Solving Poisson’s or Laplace’s Equations 7

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Example 6.1 (page 219) 8

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Example 6.2 Details in P222

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Example 6.3

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§6.5 Resistance and Capacitance 16

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23 (continue) z x y b a

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26 a b

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28 x d 0 V0V0 0

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Example 6.12: determine the capacitance of each of th Є r1 =4 e capacitors in Figure Take Є r2 =6, d=5mm, S=30 cm 2. Solve: (do it by yourself) 29 Є r1 Є r2 Є r1 Є r2 d/2 w/2

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§6.6 Method of Images The method of images is introduced by Lord Kelvin to determine V, E and D, avoiding Poison’s Eq. Image Theory states that a given charge configuration above an infinite grounded perfect conducting plane may be replaced by the charge configuration itself, its image, and an equipotential surface in place of the conducting plane. 31

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32 Conducting plane grounded Image charge So that the potential at the plane position = 0 V

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In applying the image method, two conditions must always be satisfied: 1.The image charge(s) must be located in the conducting region (satify Poisson’s Eq.) 2.The image charge(s) must be located such that on the conducting surface(s) the potential is zero or constant 33 Perfect conducting surface grounded Equipotential V = 0

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A.A point charge above a grounded conducting plance 34

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