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notes22 2317 - ECE 2317 ECE Applied Electricity and...

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Prof. David R. Jackson ECE Dept. Spring 2011 Notes 22 ECE 2317 ECE 2317 Applied Electricity and Magnetism Applied Electricity and Magnetism
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Uniqueness Theorem Uniqueness Theorem ( 29 , , x y z Φ Given: is unique 2 v B ρ ε ∇ Φ = - Φ = Φ inside (known charge density) on boundary (known B.C.) Note: We can guess the solution, as long as we verify that the Poisson equation and the BC’s are correctly satisfied! ( 29 , , ( ) v x y z ρ known B Φ = Φ on boundary S
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Example Example 0 v ρ = Guess: Φ B = V 0 = constant Check: Hollow PEC shell 0 ε Prove that E = 0 inside a hollow PEC shell (Faraday cage effect). ( 29 ( 29 0 , , x y z V r V Φ = ( 29 2 2 0 0 0 in on S V V V ∇ Φ = ∇ = Φ = Therefore: The correct solution is V ( 29 0 , , x y z V Φ = S It’s the only solution Φ that will satisfy both the BCs and the zero charge density requirement.
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Example (cont.) Example (cont.) 0 v ρ = Hollow PEC shell 0 ε V ( 29 0 , , x y z V Φ = S Hence E = 0 everywhere inside the hollow cavity. ( 29 0 , , 0 E x y z V = -∇Φ = -∇ = Φ B = V 0 = constant
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Image Theory Image Theory Note: The electric field is zero below the ground plane ( z < 0 ). x z h Φ ( x,y,z ) q Φ = 0 infinite PEC ground plane This can be justified by the uniqueness theorem, just as we did for the Faraday cage discussion (make a closed surface by adding a large hemisphere in the lower region).
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Image Theory (cont.) Image Theory (cont.) Image picture: Note: There is no ground plane in the image picture!
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