22.1: a) E A (14 N/C) (0.250 m 2 ) cos 60 1.75 Nm2 C. b) As long as the sheet is flat, its shape does not matter. ci) The maximum flux occurs at an angle 0 between the normal and field. cii) The minimum flux occurs at an angle 90 between the normal and field. In part i), the paper is oriented to ―capture‖ the most field lines whereas in ii) the area is oriented so that it―captures‖ no field lines.
ˆ 22.2: a) E A EA cos θ where A An ˆ nS1 ˆ (left) S1 (4 103 N C) (0.1 m) 2 cos (90 36.9) 24 N m 2 C j
ˆ ˆ nS2 k ( top) S2 (4 10 3 N C) (0.1 m) 2 cos 90 0 ˆ nS3 ˆ (right) S3 (4 103 N C) (0.1 m) 2 cos (90 36.9) 24 N m 2 C j ˆ ˆ nS4 k (bottom) S4 (4 103 N C) (0.1 m) 2 cos 90 0 ˆ ˆ nS5 i(front) S5 (4 103 N C) (0.1 m) 2 cos 36.9 32 N m 2 C ˆ ˆ nS6 i (back) S6 (4 10 3 N C) (0.1 m) 2 cos 36.9 32 N m 2 C
b) The total flux through the cube must be zero; any flux entering the cube must also leave it.
ˆ ˆ 22.3: a) Given that E Bi Cˆ Dk , E A, edge length L, and j ˆ ˆ nS1 ˆ 1 E AnS1 CL2 . j ˆ ˆ ˆ nS2 k 2 E AnS2 DL2 . ˆ ˆ nS3 ˆ 3 E AnS3 CL2 . j ˆ ˆ ˆ nS4 k 4 E AnS4 DL2 . ˆ ˆ ˆ nS5 i 5 E AnS5 BL2 . ˆ ˆ ˆ nS6 i 6 E AnS6 BL2 .
b) Total flux i1 i 0
E A (75.0 N C) (0.240 m 2 ) cos 70 6.16 Nm2 C.
22.5: a) E A
( 6.00106 C/m) (0.400m) ε0 2.71 105 Nm2 C.
b) We would get the same flux as in (a) if the cylinder’s radius was made larger—the field lines must still pass through the surface. c) If the length was increased to l 0.800 m, the flux would increase by a factor of two: 5.42 105 Nm2 C.
22.6: a) S1 q1 ε0 (4.00 109 C) ε0 452 Nm2 C. b) S2 q2 ε0 (7.80 109 C) ε0 881 Nm2 C. c) S3 (q1 q2 ) ε0 ((4.00 7.80) 109 C) ε0 429 Nm2 C. d) S4 (q1 q2 ) ε0 ((4.00 2.40) 109 C) ε0 723 Nm2 C. e) S5 (q1 q2 q3 ) ε0 ((4.00 7.80 2.40) 109 C) ε0 158 Nm2 C. f) All that matters for Gauss’s law is the total amount of charge enclosed by the surface, not its distribution within the surface.
22.7: a) q ε0 (3.60 106 C) ε0 4.07 105 Nm2 C. b) qε0 q ε0 ε0 (780 Nm2 C) 6.90 109 C. c) No. All that matters is the total charge enclosed by the cube, not the details of where the charge is located.
22.8: a) No charge enclosed so 0 q 6.00 10 9 C 2 678 Nm2 C. b) 12 2 2 ε0 8.85 10 C Nm c)
q1 q2 (4.00 6.00) 10 9 C 226 Nm2 C. 12 2 2 ε0 8.85 10 C Nm
22.9: a) Since E is uniform, the fluxthrough a closed surface must be zero. That is: E dA εq0 ε10 ρdV 0 ρdV 0. But because we can choose any volume we
want, ρ must be zero if the integral equals zero. b) If there is no charge in a region of space, that does NOT mean that the electric field is uniform. Consider a closed volume close to, but not including, a point charge. The field diverges there, but there isno charge in that region.
22.10: a) If ρ 0 and uniform, then q inside any closed surface is greater than zero. 0 E dA 0 and so the electric field cannot be uniform, i.e., since an arbitrary surface of our choice encloses a non-zero amount of charge, E must depend on position. b) However, inside a small bubble of zero density within the material with density ρ , the field CANbe uniform. All that is important is that there be zero flux through the surface of the bubble (since it encloses no charge). (See Exercise 22.61.)
22.11: 6sides q ε0 (9.60 106 C) ε0 1.08 106 Nm2 C. But the box is symmetrical, so for one side, the flux is: 1 side 1.80 105 Nm2 C . b) No change. Charge enclosed is the same.
22.12: Since the cube is empty, there is no net...
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