Problem Set #2
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PROBLEM SET 1. A cube 2 m on a side is located in the first octant in a Cartesian coordinate system, with one of its corners at the origin. Find the total charge contained in the cube if the charge density is given by ρv = xy 2 e 2z mC/m 3 . 2. Find the total charge contained in a cone defined by r ≤ 2m and 0 ≤ θ ≤ π/4, given that ρv = 10r 2 cos 2 θ mC/m 3 . 3. A square with sides 2 m each has a charge of 40 μC at each of its four corners. Determine the electric field at a point 5 m above the center of the square. 4. Three point charges, each with Q = 3 nC, are located at the corners of a triangle in the xyplane, with one corner at the origin, another at (2 cm, 0, 0), and the third at (0, 2 cm, 0). Find the force acting on the charge located at the origin. 5. Charge Q1 = 6μC is located at (1cm, 1cm, 0) and charge Q2 is located at (0, 0, 4cm). What should Q2 be so that E at (0, 2cm, 0) has no ycomponent? 6. Electric charge is distributed along an arc located in the xyplane and defined by ρ = 2cm and 0 ≤ Φ ≤ π/4. If ρL = 5 μC/m, find E at (0, 0, z) and then evaluate it at the origin. 7. Three infinite lines of charge, ρL1 = 3 nC/m, ρL2 = 3 nC/m, and ρL3 = 3 nC/m, are all parallel to the z axis. If they pass through the respective points (0, b), (0, 0), and (0, b) in the xyplane, find the electric field at (a, 0, 0). Evaluate your result for a = 2 cm and b = 1 cm. 8. Given the electric flux density D = xy 3 z 3 ax C/m 2 , determine the total charge Q enclosed in a cube 2 m on a side, located in the first octant with three of its sides coincident with the x, y, and zaxes and one of its corners at the origin. 9. An infinitely long line of charge with uniform density ρL = 9 nC/m lies in the xyplane parallel to the yaxis at x = 2 m. Find the potential VAB at point A(3 m, 0, 4 m) in Cartesian coordinates with respect to point B(0, 0, 0). 10. An electron with charge Qe = 1.6 x 10 19 C and mass me = 9.1 x 10 31 kg is injected at a point adjacent to the negatively charged plate in the region between the plates of an airfilled parallelplate capacitor with separation of 1 cm and rectangular plates each 10 cm 2 in area. If the voltage across the capacitor is 10 V, find the time it takes the electron to reach the positively charged plate, assuming that is starts from rest. 11. Four 5 nC positive charge are located in the z = 0 plane at the corners of a square 8 mm on a side. A fifth 5 nC positive charge is located at a point 8 mm distant from each of the other charge. Determine the magnitude of the total force on the fifth charge. 12. Let a point charge 25 nC be located at P1(4, 2, 7) and a charge 60 nC be at P2(3, 4, 2). Determine the point at the yaxis when Ex = 0. 13. The volume charge density ρv =ρoe |x||y||z| exist over all free space. Find the total charge present. 14. Let D = 4xyax + 2(x 2 +z 2 )ay + 4yzaz C/m 2 . Find the total charge enclosed in the rectangular parallelepiped 0 < x < 2 m, 0 < y < 3 m, 0 < z < 5 m.
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Transcript of Problem Set #2
PROBLEM SET
1. A cube 2 m on a side is located in the first octant in a Cartesian coordinate system, with one of its corners at the origin. Find the total charge contained in the cube if the charge density is given by ρv = xy2e-‐2z mC/m3.
2. Find the total charge contained in a cone defined by r ≤ 2m and 0 ≤ θ ≤ π/4, given that ρv = 10r2cos2θ mC/m3.
3. A square with sides 2 m each has a charge of 40 μC at each of its four corners. Determine the electric
field at a point 5 m above the center of the square.
4. Three point charges, each with Q = 3 nC, are located at the corners of a triangle in the xy-‐plane, with one corner at the origin, another at (2 cm, 0, 0), and the third at (0, 2 cm, 0). Find the force acting on the charge located at the origin.
5. Charge Q1 = 6μC is located at (1cm, 1cm, 0) and charge Q2 is located at (0, 0, 4cm). What should Q2 be
so that E at (0, 2cm, 0) has no y-‐component?
6. Electric charge is distributed along an arc located in the xy-‐plane and defined by ρ = 2cm and 0 ≤ Φ ≤ π/4. If ρL = 5 μC/m, find E at (0, 0, z) and then evaluate it at the origin.
7. Three infinite lines of charge, ρL1 = 3 nC/m, ρL2 = -‐3 nC/m, and ρL3 = 3 nC/m, are all parallel to the z-‐
axis. If they pass through the respective points (0, -‐b), (0, 0), and (0, b) in the xy-‐plane, find the electric field at (a, 0, 0). Evaluate your result for a = 2 cm and b = 1 cm.
8. Given the electric flux density D = xy3z3ax C/m2, determine the total charge Q enclosed in a cube 2 m
on a side, located in the first octant with three of its sides coincident with the x-‐, y-‐, and z-‐axes and one of its corners at the origin.
9. An infinitely long line of charge with uniform density ρL = 9 nC/m lies in the xy-‐plane parallel to the
y-‐axis at x = 2 m. Find the potential VAB at point A(3 m, 0, 4 m) in Cartesian coordinates with respect to point B(0, 0, 0).
10. An electron with charge Qe = -‐1.6 x 10-‐19 C and mass me = 9.1 x 10-‐31 kg is injected at a point adjacent
to the negatively charged plate in the region between the plates of an air-‐filled parallel-‐plate capacitor with separation of 1 cm and rectangular plates each 10 cm2 in area. If the voltage across the capacitor is 10 V, find the time it takes the electron to reach the positively charged plate, assuming that is starts from rest.
11. Four 5 nC positive charge are located in the z = 0 plane at the corners of a square 8 mm on a side. A
fifth 5 nC positive charge is located at a point 8 mm distant from each of the other charge. Determine the magnitude of the total force on the fifth charge.
12. Let a point charge 25 nC be located at P1(4, -‐2, 7) and a charge 60 nC be at P2(-‐3, 4, -‐2). Determine the
point at the y-‐axis when Ex = 0.
13. The volume charge density ρv = ρoe-‐|x|-‐|y|-‐|z| exist over all free space. Find the total charge present.
14. Let D = 4xyax + 2(x2 + z2)ay + 4yzaz C/m2. Find the total charge enclosed in the rectangular parallelepiped 0 < x < 2 m, 0 < y < 3 m, 0 < z < 5 m.
