329sp09hw2sol - ECE-329 Spring 2009 Homework 2 — Solution...

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Unformatted text preview: ECE-329 Spring 2009 Homework 2 — Solution January 29, 2009 1. Charge Q 1 = 8 π o C is placed at r 1 = (- 1 , , 0) and generates E 1 ( x,y,z ) = Q 1 4 π o r- r 1 | r- r 1 | 3 = 2 ( x + 1) ˆ x + y ˆ y + z ˆ z (( x + 1) 2 + y 2 + z 2 ) 3 / 2 V m . Charge Q 2 =- 4 π o C is placed at r 2 = (1 , , 0) and generates E 2 ( x,y,z ) = Q 2 4 π o r- r 2 | r- r 2 | 3 =- ( x- 1) ˆ x + y ˆ y + z ˆ z (( x- 1) 2 + y 2 + z 2 ) 3 / 2 V m . Then, the total electric field generated by these two charges is E ( x,y,z ) = E 1 ( x,y,z ) + E 2 ( x,y,z ) = 2 ( x + 1) ˆ x + y ˆ y + z ˆ z (( x + 1) 2 + y 2 + z 2 ) 3 / 2- ( x- 1) ˆ x + y ˆ y + z ˆ z (( x- 1) 2 + y 2 + z 2 ) 3 / 2 V m . Using this result, we obtain that the field at point r 3 = (0 , 1 , 0) is E ( r 3 ) = 3ˆ x + ˆ y 2 √ 2 V m , and the field at point r 4 = (0 , , 1) is E ( r 4 ) = 3ˆ x + ˆ z 2 √ 2 V m . (- 1 , , 0) (1 , , 0) ~ r 3 = (0 , 1 , 0) Q 1 Q 2 x y ~ E 1 ~ E 2 ~ E ( ~ r 3 ) z (- 1 , , 0) (1 , , 0) ~ r 4 = (0 , , 1) Q 1 Q 2 x z ~ E 1 ~ E 2 ~ E ( ~ r 4 ) y 2. The infinitesimal electric field at position ( x o , , 0) generated by an infinite charge strip along the z-axis with an infinitesimal width dx is found to be d E = ˆ x ρ S 2 π o x o dx, where ρ S is the uniform surface charge density. We can use this result to compute the electric field produced at the same location by a charge strip of finite width W that extends from x = 0 to x = W < x o (see the next figure). 1 ECE-329 Spring 2009 dx x z W d ~ E x o x o- x x Since the contribution of an infinitesimal strip that is placed at a distance x from the origin is simply d E = ˆ x ρ S 2 π o ( x o- x ) dx, the total field produced by the strip can be easily obtained by integrating d E from x = 0 to x = W as follows E = ˆ x W ˆ ρ S 2 π o ( x o- x ) dx = ˆ x ρ S 2 π o [- ln( x o- x )] W = ˆ x ρ S 2 π o (- ln( x o- W ) + ln( x o )) = ˆ x ρ S 2 π o ln x o x o- W ....
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This note was uploaded on 11/16/2009 for the course ECE 329 taught by Professor Franke during the Spring '08 term at University of Illinois at Urbana–Champaign.

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329sp09hw2sol - ECE-329 Spring 2009 Homework 2 — Solution...

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