329sum11hw2sol - ECE 329 Homework 2 Solution Due: June 21,...

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ECE 329 Homework 2 — Solution Due: June 21, 2011, 5PM 1. Gauss’s law for electric Feld E states that ˛ S $ E · d $ S = 1 ± o ˆ V ρdV, over any closed surface S enclosing a volume V where electric charge density is speciFed by ρ ( x,y,z ) C m 3 .He re ,w ew i l lcompu tethee lec t r ic±ux ¸ S $ E · d $ S over the surface of a cube of volume V = L 3 that is centered at the origin. L 2 (1 , 1 , 1) L 2 ( - 1 , 1 , 1) L 2 ( - 1 , 1 , - 1) L 2 (1 , 1 , - 1) L 2 (1 , - 1 , - 1) L 2 (1 , - 1 , 1) L 2 ( - 1 , - 1 , 1) L 2 ( - 1 , - 1 , - 1) x y z d ± S 2 d ± S 1 d ± S 3 a) If ρ ( )= - 2 C / m 3 (within the cube) and L =1 m ,thetota le lectr can be computed as follows ˛ S $ E · d $ S = 1 ± o ˆ V ρdV = - 2 ± o ˆ V dV = - 2 ± o L/ 2 ˆ - L/ 2 dx L/ 2 ˆ - L/ 2 dy L/ 2 ˆ - L/ 2 dz = - 2 ± o V · m . b) If ρ ( x 2 + y 2 + z 2 C / m 3 (within the cube) and L m ,theto ta l 1
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electric fux can be computed as Follows ˛ S $ E · d $ S = 1 ± o ˆ V ( x 2 + y 2 + z 2 ) dV = 1 ± o L/ 2 ˆ - L/ 2 L/ 2 ˆ - L/ 2 L/ 2 ˆ - L/ 2 ( x 2 + y 2 + z 2 ) dxdydz = L 2 ± o L/ 2 ˆ - L/ 2 x 2 dx + L/ 2 ˆ - L/ 2 y 2 dy + L/ 2 ˆ - L/ 2 z 2 dz = 3 L 2 ± o L/ 2 ˆ - L/ 2 x 2 dx = 3 L 2 ± o ± x 3 3 ² L/ 2 - L/ 2 = L 5 4 ± o V · m . Given that L =1 m , we obtain ˛ S $ E · d $ S = 1 4 ± o V · m . c) In part (b), the electric fux Φ i on the i -th Face oF the cube V is given by Φ i = ˆ S i $ E · d $ S i , such that ˛ S $ E · d $ S = 6 ³ i =1 Φ i = 1 4 ± o V · m . Taking advantage oF the symmetry oF the charge distribution, it can be easily veri±ed that the electric fux through each oF the six Faces oF the cube must be equal, i.e., Φ 1 2 = ... 6 ,there Fore , Φ i = 1 6 × 1 4 ± o = 1 24 ± o V · m . 2. Two unknown charges, Q 1 and Q 2 are located at (1 , 0 , 0) and ( - 1 , 0 , 0) , respectively. The displacement fux ´ yz -plane $ D · ˆ xdydz through the -plane in the + ˆ x direction is 2 C, thus, Q 2 2 - Q 1 2 =2 . Also, the displacement fux through the plane y in the + ˆ y direction is - 3 C which implies that Q 1 2 + Q 2 2 = - 3 .
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This note was uploaded on 11/06/2011 for the course ECE 329 taught by Professor Kim during the Summer '08 term at University of Illinois, Urbana Champaign.

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329sum11hw2sol - ECE 329 Homework 2 Solution Due: June 21,...

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