16. Let ρv = (x +2y +3z) C/m³ in the cubical region 0 ≤ x,y,z ≤1 mm, and ρv = 0 outside the cube. (a) What is the total charge contained within the cube? (b) Set up the volume integral that will give E(x,0,0) for x 1 mm. Do not integrate. COULOMB'S LAW AND ELECTRIC FIELD INTENSITY493. A regular tetrahedron has vertices at P,(2,0,0), P₂(-1,√√3,0), P3(-1,-√√3,0), andPA(0,0,2√2). Charges of 1 mC are located at each of the four vertices. If the configura-tion is located. in free space: (a) find the magnitude of the force on each charge, and(b) the angle between any two edges of the tetrahedron.=4. A point charge, Q₁ 10 μC, is located at P₁(1,2,3) in free space, while Q₂ = -5 μC isat P₂(1,2,10). (a) Find the vector force exerted on 2₂ by 2₁. (b) Find the coordinates ofP3 at which a point charge Q3 experiences no force.5. In free space, let Q₁ = 10 nC be at P₁(0, -4,0), and Q₂ = 20 nC be at P₂(0,0,4). (a) FindE at the origin. (b) Where should a 30-nC point charge be located so that E = 0 at theorigin?6. A 5-nC point charge is located at A(2,-1,-3) in free space. (a) Find E at the origin.(b) Plot E(x,0,0)| vs. x, -1^x≤ 10 m. (c) What is E(x,0,0)|max?7. A point charge, QA = 1 µC; is at A(0,0,1), and QB = -1 μC is at B(0,0,-1). Find E., E.,and E, at P(1,2,3).8. A 20-nC point charge is located at P(2,4,-3) in free space. (a) Find E(r). (b) Find E atA(-3,2,0). (c) Find the locus of all points at which E, = 1 V/m.9. Eight 25-nC point charges in free space are located symmetrically on a circle of radius0.2 m centered at the origin in the z = 0 plane. (a) At what point on the z axis is El amaximum? (b) What is Emax?10. Identical 1-μC point charges are locate in free space at (0,0,1) and (0,0,-1). Prepare aplot showing |E| vs. z along the line x = 0, y = 2, for |z| ≤ 4.11. Eight point charges of 1 nC each are located at the corners of a cube in free space thatis 1 m on a side. Find |E| at the center of: (a) the cube; (b) a face; (c) an edge.12. A point charge Q₁, located at the origin in free space, produces a field E = 10a, V/m forr = 1 m. (a) Find Q₁. (b) Using this value of Q₁, find E at P(2,-1,5) in cartesian compo-nents. (c) Using the same value of Q₁, find E at C(2,2,2) in cylindrical components.13. For x, y, and z positive, let p, = 40xyz C/m³. Find the total charge within the regiondefined by: (a) 0 ≤ x,y,z ≤ 2; (b) x = 0, y = 0, 0≤ 2x + 3y ≤ 10,0 ≤ z ≤ 2.14. Volume charge density within the operating region of a parallel-plate vacuum diode isgiven by P,= - -()Vod-4/³x-2/³, where the cathode is located at x = 0 and theanode at x = d. The voltage between anode and cathode is Vo. If the total charge in a1-m² cross section is -20 nC and: (a) d 8 mm,find Vo; (b) Vo = 24 V, find d. (c) Ifd = 8 mm and Vo = 24 V, sketch p, vs. x.=15. Volume charge density is given by Pu 10e 1000pe-100% C/m³. (a) Find the maximumvalue of p, in the region 0 ≤p ≤ 0.01 m, 0 ≤

Question

16. Let ρv = (x +2y +3z) C/m³ in the cubical region 0 ≤ x,y,z ≤1 mm, and ρv = 0 outside the cube. (a) What is the total charge contained within the cube? (b) Set up the volume integral that will give E(x,0,0) for x > 1 mm. Do not integrate. COULOMB'S LAW AND ELECTRIC FIELD INTENSITY493. A regular tetrahedron has vertices at P,(2,0,0), P₂(-1,√√3,0), P3(-1,-√√3,0), andPA(0,0,2√2). Charges of 1 mC are located at each of the four vertices. If the configura-tion is located. in free space: (a) find the magnitude of the force on each charge, and(b) the angle between any two edges of the tetrahedron.=4. A point charge, Q₁ 10 μC, is located at P₁(1,2,3) in free space, while Q₂ = -5 μC isat P₂(1,2,10). (a) Find the vector force exerted on 2₂ by 2₁. (b) Find the coordinates ofP3 at which a point charge Q3 experiences no force.5. In free space, let Q₁ = 10 nC be at P₁(0, -4,0), and Q₂ = 20 nC be at P₂(0,0,4). (a) FindE at the origin. (b) Where should a 30-nC point charge be located so that E = 0 at theorigin?6. A 5-nC point charge is located at A(2,-1,-3) in free space. (a) Find E at the origin.(b) Plot E(x,0,0)| vs. x, -1^x≤ 10 m. (c) What is E(x,0,0)|max?7. A point charge, QA = 1 µC; is at A(0,0,1), and QB = -1 μC is at B(0,0,-1). Find E., E.,and E, at P(1,2,3).8. A 20-nC point charge is located at P(2,4,-3) in free space. (a) Find E(r). (b) Find E atA(-3,2,0). (c) Find the locus of all points at which E, = 1 V/m.9. Eight 25-nC point charges in free space are located symmetrically on a circle of radius0.2 m centered at the origin in the z = 0 plane. (a) At what point on the z axis is El amaximum? (b) What is Emax?10. Identical 1-μC point charges are locate in free space at (0,0,1) and (0,0,-1). Prepare aplot showing |E| vs. z along the line x = 0, y = 2, for |z| ≤ 4.11. Eight point charges of 1 nC each are located at the corners of a cube in free space thatis 1 m on a side. Find |E| at the center of: (a) the cube; (b) a face; (c) an edge.12. A point charge Q₁, located at the origin in free space, produces a field E = 10a, V/m forr = 1 m. (a) Find Q₁. (b) Using this value of Q₁, find E at P(2,-1,5) in cartesian compo-nents. (c) Using the same value of Q₁, find E at C(2,2,2) in cylindrical components.13. For x, y, and z positive, let p, = 40xyz C/m³. Find the total charge within the regiondefined by: (a) 0 ≤ x,y,z ≤ 2; (b) x = 0, y = 0, 0≤ 2x + 3y ≤ 10,0 ≤ z ≤ 2.14. Volume charge density within the operating region of a parallel-plate vacuum diode isgiven by P,= - -()Vod-4/³x-2/³, where the cathode is located at x = 0 and theanode at x = d. The voltage between anode and cathode is Vo. If the total charge in a1-m² cross section is -20 nC and: (a) d 8 mm,find Vo; (b) Vo = 24 V, find d. (c) Ifd = 8 mm and Vo = 24 V, sketch p, vs. x.=15. Volume charge density is given by Pu 10e 1000pe-100% C/m³. (a) Find the maximumvalue of p, in the region 0 ≤p ≤ 0.01 m, 0 ≤ <2, 0≤ z ≤ 0.01 m. (b) Find the totalcharge contained in the first octant, where x, y, and z are positive. (c) Find b if the totalcharge found in the volume 0 ≤p ≤ b, 0≤ ≤ π/2, z ≥ 0, is half the value found in part(b) above.0 out-=16. Let p,= (x + 2y + 3z) C/m³ in the cubical region 0 < x,y,z ≤ 1 mm, and poside the cube. (a) What is the total charge contained within the cube? (b) Set up thevolume integral that will give E(x,0,0) for x> 1 mm. Do not integrate.

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