notes21 2317

notes21 2317 - Boundary Value Problem ECE 2317 Applied...

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Prof. Filippo Capolino ECE Dept. Fall 2006 Notes 21 ECE 2317 ECE 2317 Applied Electricity and Magnetism Applied Electricity and Magnetism Notes prepared by the EM group, University of Houston . (used by Dr. Jackson, spring 2006) Boundary Value Problem Boundary Value Problem ( ) 2 ,, 0 xyz ∇Φ = ( ) B Φ=Φ ( ) Φ Uniqueness principle: is unique On boundary: Goal: Solve for the potential function inside of a region, given the value of the potential function on the boundary. Example Example Solve for Φ ( x , y , z ) Assume ( ) ( ) x Φ= Φ + - ε r h R x V 0 Φ = 0 at R ( x = h ) Example (cont.) Example (cont.) () 222 2 2 1 12 0 0 x x xC x C ∂Φ ∂Φ ∂Φ ++= ∂∂∂ ∂Φ = + Hence: 2 0 ∇Φ= Solution:
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Example (cont.) x = h : ( ) ( ) () 12 21 0 reference point 0 h Ch C Ch C Φ= += =− x = 0: 0 0 20 0 0 V CC V CV = Hence: 0 1 V C h = [] 0 0 ,, V xy zx V V h ⎛⎞ + ⎜⎟ ⎝⎠ Example (cont.) Example (cont.) Check: ± ± ± 0 0 V/m E x x x V Ex h V h =−∇Φ ∂Φ ⎡⎤ =− − ⎢⎥ ⎣⎦ = Previous method: 0 0 B A h x x VE d r E dx Eh =+ = = ± 0 0 V/m x V E h V h = = Example Example + - V 0 0 V Φ = 0 Φ = 0 φ R 2 22 2 0 11 0 z ρ ρρ ρ φ ∇Φ= ∂∂
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notes21 2317 - Boundary Value Problem ECE 2317 Applied...

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