329sum11hw2sol

# 329sum11hw2sol - ECE 329 Homework 2 Solution Due 5PM 1...

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ECE 329 Homework 2 — Solution Due: June 21, 2011, 5PM 1. Gauss’s law for electric field E states that ˛ S E · dS = 1 o ˆ V ρ dV, over any closed surface S enclosing a volume V where electric charge density is specified by ρ ( x, y, z ) C m 3 . Here, we will compute the electric flux ¸ S E · dS 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 dS 2 dS 1 dS 3 a) If ρ ( x, y, z ) = - 2 C / m 3 (within the cube) and L = 1 m , the total electric flux can be computed as follows ˛ S E · dS = 1 o ˆ V ρ dV = - 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, y, z ) = x 2 + y 2 + z 2 C / m 3 (within the cube) and L = 1 m , the total 1

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electric flux can be computed as follows ˛ S E · dS = 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 L/ 2 - L/ 2 = L 5 4 o V · m . Given that L = 1 m , we obtain ˛ S E · dS = 1 4 o V · m . c) In part (b), the electric flux Φ i on the i -th face of the cube V is given by Φ i = ˆ S i E · dS i , such that ˛ S E · dS = 6 i =1 Φ i = 1 4 o V · m . Taking advantage of the symmetry of the charge distribution, it can be easily verified that the electric flux through each of the six faces of the cube must be equal, i.e., Φ 1 = Φ 2 = ... = Φ 6 , therefore, 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 flux ´ yz -plane D · ˆ xdydz through the yz -plane in the + ˆ x direction is 2 C, thus, Q 2 2 - Q 1 2 = 2 . Also, the displacement flux through the plane y = 1 in the + ˆ y direction is - 3 C which implies that Q 1 2 + Q 2 2 = - 3 .
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