Chapte 3. The Electic oce Chapte 3. The Electic oce Coulomb's Law 3-1. Two balls each having a chage of 3 C ae sepaated by mm. What is the foce of epulsion between them? (9 x 1 N m /C )(3 x 1 C)(3 x 1 C) ; = N -3 ( x 1 m) 3-. Two point chages of -3 and +4 C ae 1 mm apat in a vacuum. What is the electostatic foce between them? (9 x 1 N m /C )( 3 x 1 C)(4 x 1 C) ; = 75 N, attaction -3 (1 x 1 m) 3-3. An alpha paticle consists of two potons ( e = 1.6 x 1-19 C) and two neutons (no chage). What is the epulsive foce between two alpha paticles sepaated by nm? = (1.6 x 1-19 C) = 3. x 1-19 C 9-19 -19 (9 x 1 N m /C )(3. x 1 C)(3. x 1 C) ; =.3 x 1-1 N -9 (. x 1 m) 3-4. Assume that the adius of the electon's obit aound the poton in a hydogen atom is appoximately 5. x 1-11 m. What is the electostatic foce of attaction? 9-19 -19 (9 x 1 N m /C )(1.6 x 1 C)( 1.6 x 1 C) ; = 8.5 x 1-8 N -11 (5. x 1 m) 3-5. What is the sepaation of two -4 C chages if the foce of epulsion between them is N? k ' (9 x 1 N m /C )( 4 x 1 C) N 9-6 ; = 6.8 mm 316
Chapte 3. The Electic oce 3-6. Two identical chages sepaated by 3 mm expeience a epulsive foce of 98 N. What is the magnitude of each chage? k (98 N)(.3 m) 9 x 1 N m /C ; k 9 ; = 9.9 C *3-7. A 1 C chage and a -6 C chage ae sepaated by 4 mm. What is the foce between them. The sphees ae placed in contact fo a few moments and then sepaated again by 4 mm. What is the new foce? Is it attactive o epulsive? (9 x 1 N m /C )( 6 x 1 C)(1 x 1 C) ; = 338 N, attaction -3 (4 x 1 m) When sphees touch, 6 C of chage ae neutalized, -6 C 1 C.8 m leaving 4 C to be shaed by two sphees, o + C on each sphee. Now they ae again sepaated. (9 x 1 N m /C )( x 1 C)( x 1 C) ; (.8 m) C C C C.8 m = 5.6 N, epulsion *3-8. Two point chages initially attact each othe with a foce of 6 N. If thei sepaation is educed to one-thid of its oiginal distance, what is the new foce of attaction? 1 ; ; 1 1 1 1 1 1 ; 1 = 3 3 9 ; 1 1 = 54 N 317
Chapte 3. The Electic oce The Resultant Electostatic oce 3-9. A +6 C chage is placed 6 mm to the left of a + C chage. What is the esultant foce on a -35 C chage placed midway between the two chages? (9 x 1 N m /C )(6 x 1 C)(35 x 1 C) 13-3 (3 x 1 m) 13 =.1 x 1 4 N, diected to the left (9 x 1 N m /C )( x 1 C)(35 x 1 C) 3-3 (3 x 1 m) 6 mm +6 C -35 C + C 1 3 1 ; 13 =.1 x 1 4 N, diected to ight. R = 13 + 3 = (-.1 x 1 4 N) + (.7 x 1 4 N); R = -1.4 x 1 4 N, left. 3-1. A point chage of +36 C is placed to the left of a second point chage of - C. What foce is exeted on thid chage of +1 C located at the midpoint? (9 x 1 N m /C )(36 x 1 C)(1 x 1 C) 13-3 (4 x 1 m) 13 = 5 N, diected to the ight +36 C 1 1 C - C 1 3 (9 x 1 N m /C )( x 1 C)(1 x 1 C) 3-3 (4 x 1 m) ; 13 = 138 N, diected to ight. R = 13 + 3 = 5 N + 138 N; R = 36 N, left. 3-11. o Poblem 3-1, what is the esultant foce on a thid chage of +1 C placed between the othe chages and located 6 mm fom the +36 C chage? (9 x 1 N m /C )(36 x 1 C)(1 x 1 C 13-3 (6 x 1 m) (9 x 1 N m /C )( x 1 C)(1 x 1 C) 3-3 ; ( x 1 m) +36 C 1 3 = 1 C - C 6 mm 1 Both to ight, so R = 13 + 3 = 18 N + 594 N; = 7 N, ightwad. 318
Chapte 3. The Electic oce 3-1. A +6 C chage is 44 mm to the ight of a -8 C chage. What is the esultant foce on a - C chage that is mm to the ight of the -8 C chage? (9 x 1 N m /C )(8 x 1 C)( x 1 C) 13-3 ( x 1 m) (9 x 1 N m /C )( x 1 C)(6 x 1 C) 3-3 ; (4 x 1 m) -8 C 1 44 mm 3 = - C 4 mm mm 6 C 1 Both to ight, so R = 13 + 3 = 36 N + 187.5 N; = 548 N, ightwad *3-13. A 64- C chage is locate 3 cm to the left of a 16- C chage. What is the esultant foce on a -1 C chage positioned exactly 5 mm below the 16 C chage? s (3 mm) (5 mm) 58.3 mm 5 mm tan ; 59. 3 mm (9 x 1 N m /C )(64 x 1 C)(1 x 1 C) 13-3 (58.3 x 1 m) 1 +64 C s 3 mm 1 16 C 5 mm 13 = 33 N, 59. N of W (9 x 1 N m /C )(16 x 1 C)(1 x 1 C) 3-3 (5 x 1 m) = 691 N, upwad. 3 = -1 C x = 13 cos 59. = -(33 N) cos 59 ; x = -147 N y = 3 + 13 sin 59. = 691 N + (33 N) sin 59 ; y = 434 N 434 N ( 147 N) (434 N) 65 N; tan ; = 66.7 N of W. -147 N Resultant foce: R = 65 N, 66.7 N of W (o 113.3 ) 319
Chapte 3. The Electic oce *3-14. A chage of +6 nc is located above a -4-nC chage. What is the esultant foce on a -5-nC chage located 45 mm hoizontally to the ight of the -4-nC chage? s (45 mm) () 91.8 mm 1 +6 C tan ; 6.64 45 mm (9 x 1 N m /C )(6 x 1 C)(5 x 1 C) 13-3 (91.8 x 1 m) s 1 13 = 564 N, 6.64 N of W (9 x 1 N m /C )(4 x 1 C)(5 x 1 C) 3-3 (45 x 1 m) -4 C 45 mm = 8889 N, ightwad. 3 = -5 C x = 13 cos 6.64 + 3 = - (564 N) cos 6.64 + 8889 N ; x = 763 N y = + 13 sin 6.64 + = (564 N) sin 6.64 ; y = 35 N 35 N (763 N) (35 N) ; tan ; R = 795 N, = 16.3 N of E. 763 N *3-15. Thee point chages 1 = + 8 C, = -4 C, and 3 = + C ae placed at the cones of an euilateal tiangle, on each side as descibed by ig. 3-15. What ae the magnitude and diection of the esultant foce on the +8 C chage? (9 x 1 N m /C )(4 x 1 C)(8 x 1 C) 1-3 (8 x 1 m) 1 = 45. N, 6 S of E 1 = 8 C 6 6 1 (9 x 1 N m /C )( x 1 C)(8 x 1 C) 31-3 (8 x 1 m) 3 + C = -4 C 31 =.5 N, 6 N of E; x = (.5 N) cos 6 + (45 N) cos 6 = 33.8 N y = (.5 N) sin 6 - (45 N) sin 6 = -19.5 N 3
Chapte 3. The Electic oce *3-15. (Cont.) -19.5 N (33.8 N) ( 19.5 N) = 39. N; tan ; 3 33.8 N Resultant electic foce: R = 39. N, = 33. Challenge Poblems 3-16. What should be the sepaation of two +5-uC chages so that the foce of epulsion is 4 N? k ' (9 x 1 N m /C )(5 x 1 C) 4. N 9-6 ; = 3.7 cm 3-17. The epulsive foce between two pith balls is found to be 6 N. If each pith ball caies a chage of 8 nc, what is thei sepaation? k ' (9 x 1 N m /C )(8 x 1 C) -6 6 x 1 N 9-9 ; = 98. mm 3-18. Two identical unknown chages expeience a mutual epulsive foce of 48 N when sepaated by 6 mm. What is the magnitude of each chage? k (48 N)(.6 m) 9 x 1 N m /C ; k 9 ; = 4.38 C 3-19. One object contains an excess of 5 x 1 14 electons and anothe has a deficiency of 4 x 1 14 electons. What is the foce each exets on the othe if the objects ae 3 mm apat? Is it attaction o epulsion? ( 1e = 1.6 x 1-19 C, excess = -, deficiency = +. ) 1 = (5 x 1 14 e)(1.6 x 1-19 C/e) = -8 C; = (4 x 1 14 e)(1.6 x 1-19 C/e) = +64 C (9 x 1 N m /C )(8 x 1 C)(64 x 1 C) ; = 5.1 x 1 4 N, attaction -3 (3 x 1 m) 31
Chapte 3. The Electic oce 3-. If it wee possible to put 1 C of chage on each of two sphees sepaated by a distance of 1 m, what would be the epulsive foce in newtons. 9 (9 x 1 N m /C )(1 C)(1 C) ; = 9 x 1 9 N! (1 m) The coulomb is a vey lage unit fo electostatics applications. 3-1. How many electons must be placed on each of two sphees sepaated by 4 mm in ode to poduce a epulsive foce of one 4 N? k (4 N)(.4 m) 9 x 1 N m /C ; k 9 ; = 843 nc 1 e ; = 5.7 x 1 1 electons 843 x 1-9 C 1.6 x 1-19 C 3-. A 4-nC chage is placed 4 mm to the left of a +6-nC chage. What is the esultant foce on a 1-nC chage place 8 mm to the ight of the +6-nC chage? 9-9 -9 (9 x 1 N m /C )(4 x 1 C)(1 x 1 C) 1-3 (48 x 1 m) 3 = -1 nc 6 nc -4 nc 8 mm 4 mm 9-9 -9 (9 x 1 N m /C )(6 x 1 C)(1 x 1 C) -3 ; (8 x 1 m) 1 1 Both to ight, so R = 1 + = 1.88 mn 1.1 mn; = -8.5 mn, leftwad 3-3. A 5- C chage is placed 6 cm to the ight of a - C chage. What is the esultant foce on a 9 nc chage placed cm to the left of the - C chage? 9-9 -6 (9 x 1 N m /C )(9 x 1 C)( x 1 C) 1 - ( x 1 m) -9 nc C cm 1 6 cm 5 C 1 = +45 mn, to ight 1 3
Chapte 3. The Electic oce 9-6 -9 (9 x 1 N m /C )(5 x 1 C)(9 x 1 C) - ; = +63.3 mn, to ight (8 x 1 m) Resultant foce: R = 45 mn + 63.3 mn; R = 468 mn 3-4. An eual numbe of electons is placed on two metal sphees 3. cm apat in ai. How many electons ae on each sphee if the esultant foce is 45 N? k (45 N)(.3 m) 9 x 1 N m /C ; k 9 ; = 1. C 1 e ; = 1.33 x 1 14 electons 1. x 1-6 C 1.6 x 1-19 C 3-5. A 4-nC chage is placed on a 4-g sphee that is fee to move. A fixed 1- C point chage is 4 cm away. What is the initial acceleation of the 4- C chage? 9-9 -6 (9 x 1 N m /C )(4 x 1 C)(1 x 1 C) - ; = 5 mn (4 x 1 m) a m.5 N.4 kg a = 56. m/s *3-6. What is the esultant foce on a + C chage an eual distance of 6 mm fom each of two 4- C chages that ae apat in ai? 1 = C 4 mm tan ; 48. 6 mm (9 x 1 N m /C )(4 x 1 C)( x 1 C) -3 (6 x 1 m) =. N, 48. S of E 6 mm 3-4 C 1 4 mm 6 mm = -4 C (9 x 1 N m /C )( x 1 C)(4 x 1 C) 1-3 ; 1 =. N, 48. S of W (6 x 1 m) 33
Chapte 3. The Electic oce *3-6. (Cont.) x = ( N) cos 48. + ( N) cos 48. = 13.33 N - 13.33 N; x = y = ( N) sin 48. + ( N) sin 48. = 14.9 N + 14.9 N; y = -9.8 N Resultant foce: R = 9.8 N, downwad *3-7. Two chages of +5 and +16 C ae apat. A thid chage of +6 C is placed between the othe chages 3 mm fom the +5 C chage. ind the esultant foce on the thid chage? (9 x 1 N m /C )(6 x 1 C)(5 x 1 C) 1-3 (3 x 1 m) 1 = 15 kn, diected to the ight 5 C 3 = 6 C 1 3 mm 5 mm 16 C 1 (9 x 1 N m /C )(6 x 1 C)(16 x 1 C) -3 (5 x 1 m) ; 13 = 3.46 kn, diected to left. R = 1 + = 15 kn + 3.46 kn); R = 11.5 kn, ight. *3-8. A.-g pith ball is suspended feely. The ball is given a chage of + C and placed.6 m fom a chage of +5 C. What will be the initial acceleation of the pith ball? (9 x 1 N m /C )( x 1 C)(5 x 1 C) ; = 5. N (.6 m) a m 5. N -5 x 1 kg ; a = 1.5 x 1 6 m/s *3-9. A 4 C chage is located 6 cm fom an 8 C chage. At what point on a line joining the two chages will the esultant foce be zeo? ; k x k (6 x) 3 1 3 1 4 C x 1 = 8 C 6 cm - x 1 3 34
Chapte 3. The Electic oce ; ; x (6 x) 1 1 1 x (6 x) 1 4 C Take suae oot of both sides: x (6 - x) = (6 - x); x =.77(6 - x) 8 C Solving fo x, we obtain: x =.49 cm fom 4- C chage *3-3. A chage of +8 nc is placed 4 mm to the left of a 14 nc chage. Whee should a thid chage be placed if it is to expeience a zeo esultant foce? Consideing the sign of the chages and thei magnitudes, the chage must be to the left of the 8 nc chage as shown. 1 = 8 nc -14 nc x 4 mm 3 1 ; k x k (4 x) 3 1 3 1 ; ; x (4 x) 1 1 1 x (4 x) Take suae oot of both sides: x 1 8 nc (4 x) = (4 x); x =.756(4+ x) 14 nc Solving fo x, we obtain: x = 14 mm left of 8 nc chage. *3-31. A +16- C chage is to the ight of a +9 C. Whee should a thid chage be placed so that the esultant foce is zeo? ; k x k (8 x) 3 1 3 1 9 C 1 = +16 C x - x 1 3 ; ; x (8 x) 1 1 1 x (8 x) 1 9 C Take suae oot of both sides: x (8 - x) = (8 - x); x =.75(8 - x) 16 C Solving fo x, we obtain: x = 34.3 mm fom 9- C chage 35
Chapte 3. The Electic oce *3-3. Two 3-g sphees ae suspended fom a common point with two light silk theads of negligible mass. What chage must be placed on each sphee if thei final positions ae 5 mm apat? 5 mm cos ; = 71.8 T y = T sin = mg T -3 mg (3 x 1 kg)(9.8 m/s ) T ; T = 3.9 mn sin sin 71.8 mg 5 mm mg T x = T cos 71.8 = (3.9 mn) cos 71.8 ; T x = 9.67 mn; x = : = T x k (9.67 x 1 N)(5 x 1 m) 9 x 1 N m /C -3-3 9.67 mn; k 9 ; = 51.8 nc Citical Thinking Questions *3-33. A small metal sphee is given a chage of +4 C, and a second sphee 8 cm away is given a chage of 1 C. What is the foce of attaction between them? If the two sphees ae allowed to touch and ae then again placed 8 cm apat, what new electic foce exists between them? Is it attaction o epulsion? (9 x 1 N m /C )(1 x 1 C)(4 x 1 C) ; = 675 N, attaction (.8 m) When sphees touch, 6 C of chage ae neutalized, leaving 8 C to be shaed by two sphees, o +14 C on each sphee. Now they ae again sepaated: (9 x 1 N m /C )(14 x 1 C)(14 x 1 C) ; (.8 m) 4 C -1 C.8 m 14 C 14 C 14 C 14 C.8 m = 76 N, epulsion 36
Chapte 3. The Electic oce *3-34. The total chage on two metal sphees 5 mm apat is 8 C. If they epel each othe with a foce of 8 N, what is the chage on each sphee? 5 mm 1 + = 8 C; = 8 C 1 1 k k (8 C - ) k (8 C ) k 1 1 1 1 1 ; ; k (8 C) 1 1 (8 N)(5 x 1 m) -3 9 (9 x 1 N m /C ) (8 C) ; 1 1 1 (8 x 1-6 C) 1 + x 1-1 C = Solve the uadatic euation with a = 1, b = - 8 x 1-6, and c = x 1-1 1 = 77.1 C and 1 =.89 C Now = 8 C 1 yields the following : =.89 C and = 77.1 C Thus, one chage is 77.1 C and the othe is.89 C *3-35. ou small sphees ae each given chages of = + C and placed at the cones of a suae with sides of length 6 cm. Show that the esultant foce on each chage has a magnitude eual to 1914 N. What is the diection of the foce? What will change if the chages ae each = - C? (Like chages epel, so sign doesn t matte.) R (6 cm) (6 cm) 8.485 cm 6 cm 1 1 (9 x 1 N m /C )( x 1 C)( x 1 C) 1 - (6 x 1 m) 6 cm R 6 cm (9 x 1 N m /C )( x 1 C)( x 1 C) - (8.485 x 1 m) 6 cm 1 = 1 N; = 5 N; o a suae, the angle = 45 x = (1 N) + (5 N) cos 45 + = 1354 N; y = 1 N + (5 N) sin 45 = 1354 N (1354 N) (1354 N) ; = 1914 N, 45 away fom cente. 37
Chapte 3. The Electic oce *3-36. Two chages 1 and ae sepaated by a distance. They expeience a foce at this distance. If the initial sepaation is deceased by only 4 mm, the foce between the two chages is doubled. What was the initial sepaation? k x k ( x 4 mm) 1 1 x = (x 4 mm) 1 ; x ( x 4 mm) Take suae oot of both sides: x = 1.414(x 4 mm); x = 137 mm 1 1 x x 4 mm *3-37. Two 8-g pith balls ae suspended fom silk theads 6 cm long and attached to a common point. When the sphees ae given eual amounts of negative chage, the balls come to est 3 cm apat. Calculate the magnitude of the chage on each pith ball..15 m cos ; = 75.5.6 m T y = T sin = mg -3 mg (8 x 1 kg)(9.8 m/s ) T ; T = 81. mn sin sin 75.5 mg T.15 m.6 m mg T x = T cos 71.8 = (81. mn) cos 75.5 ; T x =.5 mn; x = : = T x k (.5 x 1 N)(.3 m) 9 x 1 N m /C -4.5 mn; k 9 ; = -45 nc 38
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