EE140 MT1B Spring 2022 Dr Ray Kwok 1 35 pts An infinite wire running parallel to the z axis at a distance u along the x axis is carrying a uniform charge density as shown in the figure The yz plane is an infinite ground plane a Find the electric field at any location x y z in front of the ground plane b Find the electric potential at the same location x y z z u x E2 y E1 x y z r2 u x r1 From Gauss s Law E field due to an infinite line of charge with uniform charge density is 2 2 2 2 1 2 2 2 2 0 2 2 2 0 1 2 To find V we use the ground plane as a reference point Integrate E d from 0 y z to x y z 0 0 0 2 2 2 2 2 2 0 2 2 1 2 ln ln 2 2 ln 2 2 ln 2 2 0 2 2 2 2 2 0 2 2 2 2 2 2 D N U O R G EE140 MT1B Spring 2022 Dr Ray Kwok 2 30 pts A dielectric sphere of radius a r1 2 is carrying a volume charge density of r2 It is completely enclosed by another sphere of outer radius b r2 4 and has a charge density 5r Find E r in all regions including the air region outside r b No plot is needed r2 4 r1 2 a b For r a 3 2 3 5 3 3 10 4 2 3 2 4 2 5 12 5 0 For b r a 4 2 5 5 4 2 5 5 4 4 12 5 12 5 3 5 5 2 5 4 4 4 2 4 3 5 20 2 5 4 4 16 2 For r b 4 2 5 5 4 4 12 5 3 5 5 2 5 4 4 4 2 3 5 5 2 5 4 4 4 2 MT1B Spring 2022 Dr Ray Kwok 35 pts A thin disc with radius R and a uniform charge density is laying flat on the xy plane with its center at the origin 0 0 0 Derive the electric field at a distance z away from the center of the disc namely at 0 0 z Use this result to find the electric field at the tip of a right circular cone with a base radius R height h and a uniform charge density as shown in the figure h R s z dE 2 2 0 1 2 2 2 r z h 1 2 0 2 2 2 0 1 1 2 2 2 2 2 EE140 3 r R 22220222 302 32232212 1121242cosRzzzRzzzErzzuduzrzrdrzEsdrrdzkdEdEsdrrdksdqkdEoodiscRooRoz
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