3. A regular_ tetrahedron has vertices at P₁(2,0,0), P₂(-1,√3,0), P3(-1,-√√3,0), and P4(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.

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COULOMB'S LAW AND ELECTRIC FIELD INTENSITY 49
3. A regular tetrahedron has vertices at P₁(2,0,0), P₂(-1,√3,0), P3(-1,-√3,0), and
P4(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 is
at P₂(1,2,10). (a) Find the vector force exerted on 2₂ by 2₁. (b) Find the coordinates of
P3 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) Find
E at the origin. (b) Where should a 30-nC point charge be located so that E = 0 at the
origin?
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, Q₁ = 1 µC; is at A(0,0,1), and QB
and E, at P(1,2,3).
-1 μC is at B(0,0,-1). Find E., E.,
8. A 20-nC point charge is located at P(2,4,-3) in free space. (a) Find E(r). (b) Find E at
A(-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 radius
0.2 m centered at the origin in the z = 0 plane. (a) At what point on the z axis is El a
maximum? (b) What is Emax?
10. Identical 1-μC point charges are located in free space at (0,0,1) and (0,0,-1). Prepare a
plot 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 that
is 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
=
r=
10a, V/m for
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 region
defined 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 is
given by p= -()eo Vod-4/3x-2/3, where the cathode is located at x = 0 and the
anode at x = d. The voltage between anode and cathode is Vo. If the total charge in a
1-m² cross section is -20 nC and: (a) d = 8 mm, find Vo; (b) Vo = 24 V, find d. (c) If
d = 8 mm and Vo = 24 V, sketch p, vs. x.
15. Volume charge density is given by p. = 10e-1000pe-100% C/m³. (a) Find the maximum
value of p, in the region 0 ≤ p ≤ 0.01 m, 0 ≤ ¢ < 2π, 0 ≤ z ≤ 0.01 m. (b) Find the total
charge contained in the first octant, where x, y, and z are positive. (c) Find b if the total
charge found in the volume 0 ≤ p ≤ b, 0≤ ≤ π/2, z ≥ 0, is half the value found in part
(b) above.
and Pv
0 out-
16. Let p,= (x + 2y + 3z) C/m³ in the cubical region 0≤ x,y,z ≤ 1 mm,
side the cube. (a) What 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.
=
Transcribed Image Text:COULOMB'S LAW AND ELECTRIC FIELD INTENSITY 49 3. A regular tetrahedron has vertices at P₁(2,0,0), P₂(-1,√3,0), P3(-1,-√3,0), and P4(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 is at P₂(1,2,10). (a) Find the vector force exerted on 2₂ by 2₁. (b) Find the coordinates of P3 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) Find E at the origin. (b) Where should a 30-nC point charge be located so that E = 0 at the origin? 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, Q₁ = 1 µC; is at A(0,0,1), and QB and E, at P(1,2,3). -1 μC is at B(0,0,-1). Find E., E., 8. A 20-nC point charge is located at P(2,4,-3) in free space. (a) Find E(r). (b) Find E at A(-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 radius 0.2 m centered at the origin in the z = 0 plane. (a) At what point on the z axis is El a maximum? (b) What is Emax? 10. Identical 1-μC point charges are located in free space at (0,0,1) and (0,0,-1). Prepare a plot 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 that is 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 = r= 10a, V/m for 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 region defined 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 is given by p= -()eo Vod-4/3x-2/3, where the cathode is located at x = 0 and the anode at x = d. The voltage between anode and cathode is Vo. If the total charge in a 1-m² cross section is -20 nC and: (a) d = 8 mm, find Vo; (b) Vo = 24 V, find d. (c) If d = 8 mm and Vo = 24 V, sketch p, vs. x. 15. Volume charge density is given by p. = 10e-1000pe-100% C/m³. (a) Find the maximum value of p, in the region 0 ≤ p ≤ 0.01 m, 0 ≤ ¢ < 2π, 0 ≤ z ≤ 0.01 m. (b) Find the total charge contained in the first octant, where x, y, and z are positive. (c) Find b if the total charge found in the volume 0 ≤ p ≤ b, 0≤ ≤ π/2, z ≥ 0, is half the value found in part (b) above. and Pv 0 out- 16. Let p,= (x + 2y + 3z) C/m³ in the cubical region 0≤ x,y,z ≤ 1 mm, side the cube. (a) What 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. =
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