notes8 2317

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

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Prof. Filippo Capolino ECE Dept. Fall 2006 Notes 8 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) Example Example Find E ( 0, 0, z ) x y z a R y ρ s = s 0 [C/m 2 ] r = (0, 0, z ) b x d φ ´ dS´ l () l l l l 22 '' ' ' ' ' ' dS d d Rz z z z R z ρφ ρρ = =− + =+ −+ = + Example (cont.) Example (cont.) l l l l l 2 0 2 0 1 0 0 2 2 0 3 0 0 2 ' ' 4 1 ''' 4 ' ' 4 ' s S b s a b s a r ER d s R zz Ed d z z dd z π πε ρρφ = ⎛⎞ ⎜⎟ = + ⎝⎠ + = + ∫∫ Note: the upper limits are larger than the lower limits, in order to keep the lengths positive Example (cont.) Example (cont.) l l 2 0 3 0 0 2 4 ' b s a E z = + Switch integration order and do ´in te g ra l f i r s t : l l 2 0 3 0 0 2 4 ' b s a E z φρ = + l 2 0 Note that ' ' 0 d =
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Example (cont.) Example (cont.) l () 0 3 22 0 2 ' 2' 4 ' b s a Ez z d z ρ π πε ⎛⎞ = ⎜⎟ ⎝⎠ + l 2 0 3 0 0 2 '' ' 4 ' b s a zz E dd z φρ = + ∫∫ l 2 0 3 0 0 2 1 4 ' b s a E d d z ρφ = + Example (cont.) Example (cont.) l 0 0 1 2 4 ' b s a z z ⎡⎤ = ⎢⎥ + ⎣⎦ l 0 3 0 2 ' 4 ' b s a z d z = + l [] 0 0 11 2V / m 4 s z az bz =− ++ Example (cont.) Example (cont.) l 0 0 / m 4 s z Limiting case: a 0, b →∞ (sheet of charge) l 0 0 2 S z z ε = l 0 0 V/m 2 S + for z > 0 -fo r z < 0 y z x E E s0 Example Example l = l 0 [C/m] ( uniform line charge ) Find E ( , φ , 0 ) r = ( , ,0) x y z R r ´= (0, 0, z ´) -h +h l R
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Example (cont.) Example (cont.) ± () ± ( ) ± ± l cos 0 sin 0 0 ' cos sin ' ' Rx y z z y z z zz ρφ φ ρρ =− +−
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notes8 2317 - Example ECE 2317 Applied Electricity and...

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