Magnetic Fields Due to Currents

Transcription

1 PHYS102 Previous Exam Problems CHAPTER 29 Magnetic Fields Due to Currents Calculating the magnetic field Forces between currents Ampere s law Solenoids 1. Two long straight wires penetrate the plane of the paper at two vertices of an equilateral triangle, as shown in figure 1. They each carry a 2.0-A current, out of the page. What is the magnitude of the magnetic field at the third vertex (P)? (Ans: T) 2. A long, coaxial cable, shown in cross-section in figure 3, is made of two conductors that share a common central axis, labeled C. The current in the inner conductor is 2.0 A directed into the page and that in the outer conductor is 2.5 A directed out of the page. The distance from point C to B is m. What are the magnitude 5 and direction of the magnetic field at point B? (Ans: T, counterclockwise) 3. Solenoid 2 has twice the radius and six times the number of turns per unit length as solenoid 1. Find the ratio of the magnetic field in the interior of solenoid 2 to that in the interior of solenoid 1, if the two solenoids carry the same current. (Ans: 6) 4. In figure 4, two long parallel straight wires, at a separation d = 20.0 cm, carry currents i 1 = 3.61 ma and i 2 = 3.00 ma, out of the page. At what distance from the origin on the x axis is the net magnetic field due to the currents equal zero? (Ans: 11 cm) 5. What are the strength and direction of the magnetic field at point P in figure 5, at the center of the circular arcs, if the current in the loop is 5.0 A? (Ans: 79 μt, into the page) 6. A long solid cylindrical wire, of radius R = 2.0 cm carries a uniform current of 1.9 A. Find the magnetic field strength at a distance of 1.0 cm from the center of the wire. (Ans: 9.5 μt) 7. We wish to generate a 0.10-T uniform magnetic field near the center of a 10-cm long ideal solenoid. What is the minimum number of turns needed, if the wire can carry a maximum current of 10 A? (Ans: 800) 8. What is the magnetic field at the center of the loop in figure 8? (Ans: T, into the page) What is the net force per unit length on the upper wire (# 1) in figure 9? (Ans: N/m, upward) 10. A very long wire carries a current I = 0.5 A directed along the negative x axis. Part of the wire is bent into a circular section of radius R = 2.5 cm, as shown in figure 12. What is magnetic field at point C? (Ans: 16.6 μt, into the page) 11. An ideal solenoid that is 100 cm long has a diameter of 5.0 cm and a winding of 1000 turns, and carries a current of 5.0 A. Calculate the magnitude of the magnetic field inside the solenoid. (Ans: 6.3 mt) 12. The radius R of a long current-carrying wire is 4.2 cm. If the magnetic field at r 1 = 3.0 cm is equal to four times the magnetic field at r 2, where r 2 > R, calculate the distance r 2. (Ans: 23.5 cm) Dr. M. F. Al-Kuhaili PHYS 102 Chapter 29 Page 1

2 13. Figure 13 shows a hollow cylindrical conductor, of inner radius a = 3.0 mm and outer radius b = 5.0 mm, that carries a current of 2.0 A parallel to its axis. The current is uniformly distributed over the cross section of the conductor. Find the magnitude of the magnetic field at a point that is 2.0 mm from the axis of the conductor. (Ans: zero) 14. Two long straight wires are parallel and carry currents in opposite directions. The currents are 8.0 and 12A, and the wires are separated by 0.40 cm. What is the magnitude of the magnetic field at a point midway between the wires? (Ans: T) 15. Suppose that the identical currents I in figure 15 are all out of the page. What is the magnitude of the force per unit length on the wire at the origin? [take I = 10.0 A, and a = 10-4 m] (Ans: 0.28 N/m) 16. What is the magnitude of the magnetic field at point P due to the current carrying wire shown in figure 16, if I = 2.0 A, a = 20 cm and b = 2a? (Ans: 0.8 µt) 17. Two long wires parallel to the x-axis carry currents I 1 and I 2, as shown in figure 17. If I 1 = 5 A, what is the magnitude and direction of I 2 if the net magnetic field at the origin is 0.35 µt, and directed out the page. (Ans: 1 A to the left) 18. Figure 18 shows four wires carrying currents I 1, I 2, I 3, and I 4. What is the value of the integral B s. d for the Amperian loop shown in the figure? (Ans: T.m) 19. Figure 19 shows three parallel wires carrying currents I1 = 2 A, I 2 = 6 A, and I 3. The spacing between adjacent wires is 1 cm. Calculate the value of I 3 such that the magnitude of the net force per unit length at the third wire is N/m. (Ans: 2.5 A) 20. A cylindrical ring with inner radius R1 = 2.0 mm and outer radius R 2 = 5.0 mm, carries a constant current of 8.0 A along its length. What is the current density? (Ans: A/m 2 ) 21. Figure 21 shows two concentric, circular wire loops, of radii r1 = 15 cm and r 2 = 30 cm, that are located in the xy plane. The inner loop carries a current of 8.0 A in the clockwise direction, and the outer loop carries a current of 10.0 A in the counter clockwise direction. Find the net magnetic field at the center. (Ans: T, directed into the page) 22. How strong is the magnetic field at a distance of 10.0 cm from a long straight wire, of radius 3.0 cm carrying a current of 5.0 A? (Ans: T) 23. Three long parallel wires are arranged as shown in figure 22. Wires 1 and 3 each carries a current of 5.0 A in the directions shown. If the net magnetic force on wire 3 is zero, what is the magnitude and direction of the current in wire 2? (Ans: 2.5 A, downward) 24. A segment of wire is formed into the shape shown in figure 23, and carries a current I = 1.0 A. What is the magnitude of the resulting magnetic field at point P if R = 10 cm? (Ans: 5.5 µt, into the page) 25. A solenoid has length L = 2.0 m and diameter d = 4.0 cm, and it carries a current I = 6.0 A. It consists of seven closely-packed layers, each with 90 turns along length L. What is B at its center? (Ans: T) 26. Two infinite parallel wires are separated by 2.5 cm and carry currents of 10 A and 12 A in the same direction. What is the force per unit length on each wire? (Ans: Four long conducting wires are parallel to each other. Their cross sections form the corners of a square of side a = 15 cm. The directions of the currents are as shown in figure 24. The magnitude of the current in each wire is 15 A. What is the magnitude of the magnetic field at the center of the square? (Ans: zero) -4 N/m, attractive) Dr. M. F. Al-Kuhaili PHYS 102 Chapter 29 Page 2

3 28. Two long parallel wires, D and B, are separated by 2.0 cm. The current in D is three times the current in B. If the magnitude of the force on a 2.0-m length of one of the wires is equal to 60 µn, find the current in B. (Ans: 1.0 A) 29. Consider an infinitely long straight wire carrying a current I. If the magnetic field at r 1 = 2.5 mm inside the wire and at r 2 = 10 mm outside the wire are equal, what is the radius of the wire? (Ans: 5.0 mm) 30. Three long parallel wires, shown in figure 26, are in the xy plane. Each wire carries a current of 3.0 A. The separation between adjacent wires is d = 8.0 cm. What is the magnitude of the net force per meter exerted on the central wire by the other two wires? (Ans: N/m) 31. A long solid cylindrical conductor of radius R = 4.0 mm carries a current I parallel to its axis. The current density in the wire is A/m 2. Determine the magnitude of the magnetic field at a point that is 5.0 mm from the axis of the conductor. (Ans: 40 µt) 32. A solenoid is 3.00 m long and has a circumference of 9.40 cm. It carries a current of 12.0 A. The magnetic field inside the solenoid is In the loop shown in figure 28, a = 5.0 cm, b = 10 cm, and i = 1.0 A. What is the magnetic field at point P, which is the center of the two circular arcs? (Ans: T. What is the length of the wire forming the solenoid? (Ans: 468 m) Dr. M. F. Al-Kuhaili PHYS 102 Chapter 29 Page 3-6 T, into the page) 34. Figure 30 shows two current segments. The lower segment carries current I1 = 0.50 A, and includes a circular arc of radius 5.0 cm. The upper segment carries current I 2 = 0.90 A, and includes a circular arc of radius 4.0 cm. Both arcs are centered at point P. What is magnitude of the magnetic field at P? (Ans: 1.6 µt) 35. In figure 32, a wire forms a quarter of a circle of radius R = 3.14 m and two straight segments of length 5.5 m. The wire carries a current of 20.0 A. What is the magnitude and direction of magnetic field at the center? (Ans: 1.00 µt, into the page) 36. In figure 34, the current in both very long wires is i = 5.00 A and R = 2.00 cm. What is the magnitude of the magnetic field at point P, at the center of the semicircle shown? (Ans: 28.5 μt, into the page) 37. Figure 35 shows, in cross section, two wires that are parallel, straight, and very long. The first wire, located at (0, 5.0) cm, carries a current of 6.0 A into the page, and the other wire, located at ( 3.0, 4.0) cm, carries a current of 7.5 A out of the page. What magnetic field do these two wires produce at the origin O? (Ans: 18 ĵ µt) 38. Figure 36 shows a current-carrying loop. The three circular arcs have radii 1.0, 2.0, and 3.0 cm, and subtend angles 45 o, 90 o, and 180 o, respectively. If the smallest arc alone produces a magnetic field of magnitude 12 μt at point O, what is the magnitude of the total magnetic field produced by the whole loop at point O? (Ans: 16 µt) 39. Figure 38 shows a cross section of three parallel wires, each carrying a current of 12 A. The currents in wires A and C are out of the page, while that in wire B is into the page. If the distance R = 4.0 mm, what is the magnitude of the magnetic force on a 3.0-m length of wire B? (Ans: 11 mn) 40. Figure 41 shows two long, thin wires that carry currents in the positive z direction. Both wires are parallel to the z axis. The 50-A wire is in the xz plane and is 5.0 m from the z axis. The 40-A wire is in the yz plane and is 4.0 m from the z axis. What is the magnitude of the magnetic field at the origin? (Ans: 2.8 µt) 41. A long, straight, hollow conductor, that has an inner radius of 1.0 mm and an outer radius of 3.0 mm, carries a current of 15 A uniformly distributed over its cross section. What is the magnitude of the magnetic field at a distance of 4.0 mm from the axis of the wire? (Ans: 0.75 mt)

4 42. A long solenoid has 1000 turns/m and carries current I. An electron moves within the solenoid in a circle of radius 2.30 cm perpendicular to the solenoid axis. The speed of the electron is m/s. What is the current in the solenoid? (Ans: 2.66 A) 43. Five long, straight, insulated wires are closely bound together to form a small cable of diameter 1.0 cm. Currents carried by the wires are I1 = 20 A, I 2 = -6 A, I 3 = 12 A, I 4 = -7 A, and I 5 = 18 A (negative currents are opposite in direction to the positive). Find the magnitude of the magnetic field at a distance 10 cm from the cable. (Ans: 74 µt) 44. A copper wire is of total length 1.0 m. You want to make N-turn circular current loop, using the entire wire, that generates a 1.0-mT magnetic field at the center of the coil when the current is 1.0 A. What will be the diameter of your coil? (Ans: m) 45. Two long straight current-carrying parallel wires cross the x-axis and carry currents I and 3I in the same direction, as shown in figure 43. At what value of x is the net magnetic field zero? (Ans: 1.0 cm) 46. Two long straight current-carrying parallel wires cross the x axis and carry currents I and 3I in the directions shown in figure 44. At what value of x is the net magnetic field zero? (Ans: 0) 47. Figure 45 shows two long, parallel current-carrying wires. These wires carry equal currents I 1 = I 2 = 20 A, and are separated by a distance d = 0.50 m. Calculate the magnetic field at point P. (Ans: +8.0 ĵ µt) 48. A cylindrical infinitely long wire has a radius of 4.0 mm. What current will produce a maximum magnetic field of 0.50 mt? (Ans: 10 A) 49. The radius of a coil of wire with N turns is r = 0.22 m. A clockwise current of I coil = 2.0 A flows in the coil, as shown in figure 47. A long, straight wire carrying a current I wire = 31 A toward the left is located at a distance d = 0.05 m from the edge of the coil. The magnetic field at the center of the coil is zero. Determine N, the number of turns. (Ans: 4) 50. Figure 48 shows a wire consisting of concentric circular arcs of radii r, 2r and 3r, where r = 1.0 cm. The wire carries a current i = 1.0 A. Calculate the magnetic field at point C. (Ans: 13 µt, out of the page) 51. Two long straight wires enter a room through a door. One carries a current of 3.0 A into the room while the other carries a current of 5.0 A out of the room. What is the magnitude of the path integral B s. d around the door frame? (Ans: T m) 52. In figure 49, a closed loop carries current i = 0.20 A. The loop consists of two straight wires and two concentric circular arcs of radii R1 = 2.0 m and R 2 = 4.0 m. What is the magnetic field at the center P? (Ans: T into the page) 53. Figure 50 shows three long parallel wires at the vertices of an equilateral triangle of side a = 10 cm, and carrying the same current I = 5.0 A. The currents in wires 1 and 2 are out of the page, while in wire 3 the current is into the page. Calculate the net magnetic force per unit length on wire 3 due to wires 1 and 2. (Ans: ĵ N/m) 54. Two infinitely long wires carry currents in opposite directions, as shown in figure 51. The current in each wire is 5.0 A, and the distance d = 1.0 m. What is the net magnetic field at point P? (Ans: 0.50 µt, to the left) 55. A long, solid, cylindrical wire carries a uniformly distributed current. If the radius of the wire is 3.5 mm, and the magnitude of the current density is 1.5 A/cm 2, what is the magnitude of the magnetic field at a distance of 2.5 mm from the axis of the wire? (Ans: 24 µt) Dr. M. F. Al-Kuhaili PHYS 102 Chapter 29 Page 4

5 56. Figure 55 shows two wires each carrying a current in the direction indicated in the figure. Wire 1, which consists of a circular arc of radius R and two radial lengths, carries a current I 1 = 1.5 A. Wire 2 is long and straight; carries a current I 2 = 0.4 A and is at a distance R/2 from the center of the arc. For what value of arc angle φ the net magnetic field B at point P due to the two currents is zero? (Ans: 61 o ) A B C D E Conceptual Problems 1. Four long straight wires, carrying equal currents out of the page, are placed at the corners of a square, as shown in figure 2. What is the net magnetic force on wire P? A. East B. North C. South D. West E. South west 2. Figure 6 shows three wires that are perpendicular to the page. The currents are all equal, two being out of page and one being into the page. Rank the wires according to the magnitudes of the magnetic forces on them, greatest first. A. 1 and 3 tie, then 2 B. 2, then 1 and 3 tie C. 1, then 2, then 3 D. 2 and 3 tie, then 1 E. 3, then 2, then 1 3. Two long straight, parallel wires are 3.0 cm apart. They carry currents I 1 = 3.0 A and I 2 = 5.0 A in opposite directions, as shown in the figure 7. At what point, beside infinity, could the magnetic field be zero? A. A B. B C. C D. D E. E 4. Figure 10 shows four wires carrying equal current and four Amperian loops. Rank the loops according to the magnitude of B s. d along each, greatest first A. c, b and d tie, then a B. a, b and a tie, then c C. c, a, b, d D. c, d, a, b E. a, b, c, d 5. An infinitely long wire has a charge density λ per unit length. The wire moves along its axis with velocity v. The ratio of the magnetic field to the eclectic field at a point r from the wire is: A. independent of r B. directly proportional to λ C. proportional to r D. inversely proportional to r E. inversely proportional to r 2 Dr. M. F. Al-Kuhaili PHYS 102 Chapter 29 Page 5

6 6. Four long straight wires carry equal currents into the page, as shown in the figure 11. What is the direction of the net magnetic force exerted on wire A by the other three wires? A. East B. North C. South D. West E. zero 7. Figure 14 shows a small positive charge q moving toward a long current-carrying wire. Which of the arrows labeled A to D correctly represents the direction of the magnetic force on the charge? A. C B. A C. B D. D E. The force points in a direction perpendicular to the plane of the figure. 8. Figure 20 shows three arrangements of circular loops, centered on vertical axes and carrying identical currents in the directions indicated. Rank the arrangements according to the magnitudes of the magnetic fields at the midpoints between the loops on the central axes, maximum to minimum. A. 1, 3, 2 B. 3, 2, 1 C. 3, 1, 2 D. 2, 3, 1 E. 2, 1, 3 9. Two long parallel straight wires carry equal currents in opposite directions. At a point midway between the wires, the magnetic field they produce is A. non-zero and perpendicular to the plane of the two wires. B. zero. C. none of the answers. D. non-zero and parallel to the wires. E. non-zero and along a line connecting the wires. 10. An electron is moving along the axis of a solenoid carrying a current. Which of the following is a correct statement about the magnetic force acting on the electron? A. No force acts B. The force acts radially inward. C. The force acts radially outward. D. The force acts in the direction of motion E. The force acts opposite to the direction of motion 11. Figure 25 shows four circular loops that are concentric with a wire whose current is directed out of the page. The current is uniform across the cross section of the wire. Rank the loops according to the magnitude of the enclosed current, greatest first. [Loops a and b are inside the wires. Loops c and d are outside] A. d = c > b > a. B. d > c > b > a. C. a > c > b > d. D. a > c > b > d. E. a = b > c > d. Dr. M. F. Al-Kuhaili PHYS 102 Chapter 29 Page 6

7 12. A proton is moving along the axis of a solenoid carrying a current. Which of the following statement is correct about the magnetic force acting on the proton? A. No force acts. B. The force acts radially inwards. C. The force acts radially outwards. D. The force acts in the direction of motion. E. The force acts in the opposite direction of motion. 13. Figure 27 shows two wires carrying anti-parallel currents. If i 2 is greater than i 1, the point at which the resultant magnetic field of the two wires will be zero is located in the region (regions): A. I B. II and III B. I C. I and III D. II E. III 14. An infinitely long wire carrying current I is bent at a right angle as shown in figure 29. What is the magnetic field at point P that is a distance y from the corner of the wire? 15. Wires A, B, and C carry equal currents in the directions shown in Figure 31. What is the direction of the net magnetic force on wire C due to wires A and B? A. Negative y direction B. Positive x direction C. Negative x direction D. Positive y direction E. Positive z direction 16. Three long straight wires are perpendicular to the page, and are placed at three corners of a square of edge length a = 20 cm, as shown in figure 33. Wires number 1, 2 and 3 each carries a 1.0-A current out of the page. What is the direction of the net force on wire number 1 due to wires number 2 and 3? A. Towards point P B. Away from point P C. Towards wire number 2 D. Away from wire number 2 E. Towards wire number 3 Dr. M. F. Al-Kuhaili PHYS 102 Chapter 29 Page 7

8 17. Three wires carrying currents of magnitudes i 1, i 2 and i 3 in the directions shown in figure 37. Which one of the following integrals is correct? 18. In figure 39, two long straight wires are perpendicular to the page. Each carries a current of 25.0 A directed out of the page. In unit vector notation, what is the net magnetic force per unit length on the wire at the origin? A ( î + ĵ ) N/m B ( î - ĵ ) N/m C (- î - ĵ ) N/m D (- î - ĵ ) N/m E (- î + ĵ ) N/m 19. A cylindrical wire of radius R carries current I that is uniformly distributed across its cross-section. Find the magnitude of the magnetic field inside the wire at r < R from the axis. A. µ 0Ir/(2πR 2 ) B. µ I/(2πR) 0 C. µ I/(2πr) 0 D. µ 0IR 2 /(2πr 2 ) E. µ 0I/(4πr) 20. Two long parallel wires, a distance d apart, carry currents of I and 5I in the same direction. Locate the point r, from I, at which their magnetic fields cancel each other. A. r=d/6 B. r=d/2 C. r=2d D. r=3d/2 E. r=d/4 21. Two parallel wires, carrying equal currents of 10 A, attract each other with a force F. If both currents are doubled, and the distance between them is reduced by 50%, what will be the new force? A. 8F B. 16F C. 4F D. F E. F/4 Dr. M. F. Al-Kuhaili PHYS 102 Chapter 29 Page 8

9 22. Figure 40 shows the cross section of a long solid wire carrying a uniform current i. The radius of the wire is R. What is the value of the integral B. d s over the circular closed path of radius r shown in the figure? A. µ oi(r/r) B. µ oir/r C. µ ir/r o 2 D. µ oir 2 /r E. µ i o Two long wires are parallel to the z-axis as shown in figure 42. Find the resultant magnetic field at the origin, given that the wires carry equal current I and moves in the same direction. [Take I = 1.0 A, and a =0.5 m] A. Zero. B T, In the negative x-direction. C D E T, In the positive z-direction. T, In the positive z-direction. T, In the positive x-direction. 24. Three current-carrying wires A, B and C are placed as shown in figure 46. All currents have the same value. Wire C has a current that flows into the page. The three wires are at equal distances from the origin. If the net magnetic force on wire C due to wires A and B is in the positive y direction, then A. Wires A and B must have currents flowing into the page. B. Wires A and B must have currents flowing out of the page. C. Wire A has a current flowing out of the page, and wire B has a current flowing into the page. D. Wire A has a current flowing into the page, and wire B has a current flowing out of the page. E. The net magnetic force can never be along the positive y axis. 25. Figure 52 shows a snapshot of the velocity vectors of five electrons near a wire carrying current i. The five velocities have the same magnitude. Which electron experiences the largest magnetic force? A. 1 B. 2 C. 3 D. 4 E Two long wires are placed in the xy plane, as shown in figure 53. Each wire carries a current of 1.5 A, directed out of the page. If the distance d = 3.0 m, what is the net magnetic field due to these wires at the origin? A. (+0.10 î 0.10 ĵ ) (µt) B. (+0.10 î ĵ ) (µt) C. ( 0.10 î 0.10 ĵ ) (µt) D. ( 0.10 î ĵ ) (µt) E. zero Dr. M. F. Al-Kuhaili PHYS 102 Chapter 29 Page 9

10 27. In figure 54, two infinitely long wires carry currents i. Each follows a 90 o arc on the circumference of the same circle of radius R. What is the magnitude of the net magnetic field at the center of the circle (point C)? A. µ i o 2 πr B. µ i o πr C. µ i o 4 πr D. µ i o µ oi + 2πR 16R E. µ i o µ oi + πr 16R 28. Figure 56 shows three circuits consisting of straight radial lengths and concentric circular arcs (either halfor quarter-circles of radii r or 2r). The circuits carry the same current in the indicated direction. Rank the circuits according to the magnitude of the magnetic field produced at the center of curvature (the dot), greatest first. A) B2, B1, B3 B) B1, B2, B3 C) B3, B2, B1 D) B2, B3, B1 E) B1, B3, B2 Dr. M. F. Al-Kuhaili PHYS 102 Chapter 29 Page 10

11 Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 Dr. M. F. Al-Kuhaili PHYS 102 Chapter 29 Page 11

12 Figure 17 Figure 18 Figure 19 Figure 20 Figure 21 Figure 22 Figure 23 Figure 24 Figure 25 Figure 26 Figure 27 Figure 28 Figure 29 Figure 30 Figure 31 Dr. M. F. Al-Kuhaili PHYS 102 Chapter 29 Page 12

13 Figure 32 Figure 33 Figure 34 Figure 35 Figure 36 Figure 37 Figure 38 Figure 39 Figure 40 Figure 41 Figure 42 Figure 43 Figure 44 Figure 45 Dr. M. F. Al-Kuhaili PHYS 102 Chapter 29 Page 13

14 Figure 46 Figure 47 Figure 48 Figure 49 Figure 50 Figure 51 Figure 52 Figure 53 Figure 54 Figure 55 Figure 56 Dr. M. F. Al-Kuhaili PHYS 102 Chapter 29 Page 14