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notes8 2317

# notes8 2317 - ECE 2317 Applied Electricity and Magnetism...

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Prof. David R. Jackson ECE Dept. Spring 2011 Notes 8 ECE 2317 ECE 2317 Applied Electricity and Magnetism Applied Electricity and Magnetism Notes prepared by the EM Group University of Houston

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Example Example Find E ( 0, 0, z ) x y z a R y ρ s = ρ s 0 [C/m 2 ] r = (0, 0, z ) b x d φ ´ dS´ ( 29 2 2 2 2 ˆ ˆ ' ˆ ˆ ˆ dS d d R zz R z zz R z ρ ρ φ ρ ρ ρ ρ ρ ρ = = - + = + ′ ′ - + = +
Example (cont.) Example (cont.) ( 29 ( 29 ( 29 ( 29 2 0 2 0 1 2 2 2 2 0 0 2 2 0 3 2 2 0 0 2 ˆ 4 ˆ ˆ ' 1 4 ' ' ˆ ˆ ' 4 ' s S b s a b s a r E R dS R zz E d d z z zz d d z π π ρ πε ρ ρ ρ ρ ρ φ πε ρ ρ ρ ρ ρ ρ ρ φ πε ρ = - + = + + - + = + ∫ ∫ ∫ ∫ Note: The upper limits are larger than the lower limits, in order to keep the differential lengths (defining the differential area) positive.

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Example (cont.) Example (cont.) ( 29 2 0 3 2 2 0 0 2 ˆ ˆ 4 ' b s a zz E d d z π ρ ρ ρ ρ ρ φ πε ρ ′ ′ - + = + ∫ ∫ Switch integration order and do φ ' integral first: ( 29 2 0 3 2 2 0 0 2 ˆ ˆ 4 b s a zz E d d z π ρ ρ ρ ρ φ ρ πε ρ ′ ′ - + = + ∫ ∫ 2 0 ˆ 0 d π ρ φ = Note that By doing the φ integral first and realizing that it is zero for the first term, we save a lot of work.
Example (cont.) Example (cont.) ( 29 0 3 2 2 0 2 ˆ 2 4 b s a E z z d z ρ ρ π ρ πε ρ = + ( 29 2 0 3 2 2 0 0 2 ˆ 4 b s a z z E d d z π ρ ρ φ ρ πε ρ

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notes8 2317 - ECE 2317 Applied Electricity and Magnetism...

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