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

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Prof. Filippo Capolino ECE Dept. Fall 2006 Notes 22 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) Uniqueness Theorem Uniqueness Theorem ( ) ,, xyz 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 ! () ( k n ow n ) v B on boundary S Example Example 0 v = Guess: Φ B = V 0 = constant Check: Hollow PEC shell 0 Prove that E = 0 inside a hollow PEC shell (Faraday cage effect) ( ) ( ) 0 xyz V r V Φ= 22 0 0 0i n on S VV V ∇Φ=∇ = Therefore: the correct solution is V ( ) 0 S Example (cont.) Example (cont.) 0 v = Hollow PEC shell 0 V ( ) 0 S Hence E = 0 everywhere inside the hollow cavity. ( ) 0 0 Ex y z V =−∇Φ =−∇ = Φ B = V 0 = constant
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Image Theory Image Theory x z h Φ ( x,y,z ) q PEC Note: the electric field is zero below the ground plane ( z < 0). Φ = 0 Image Theory (cont.) Image Theory (cont.) Image picture: x z h q h -q Note: there is no ground plane in the image picture!
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notes22 2317 - ECE 2317 Applied Electricity and Magnetism...

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