* 65. (I) A long thin solenoid has 430 loops of wire per

1. (I) (a) What is the magnitude of the force per meter of length on a straight wire earning an 8.40-A current when perpendicular to a 0.90-T uniform magnetic field? (b) What if the angle between the wire and field is 45.0?

2. (I) Calculate the magnitude of the magnetic force on a 160-m length of straight wire stretched between two towers carrying a 150-A current. The Earths magnetic field of 5.0 x 10~5T makes an angle of 65 with the wire.

3. (I) How much current is flowing in a wire 4.80 m long if the maximum force on it is 0.750 N when placed in a uniform 0.0800-T field?

4. (II) A 1.5-m length of wire carrying 4.5 A of current is oriented horizontally. At that point on the Earth's surface, the dip angle of the Earth's magnetic field makes an angle of 38 to the wire. Estimate the magnitude of the magnetic force on the wire due to the Earth's magnetic field of 5.5 x 10-5T at this point.

5. (II) The force on a wire carrying 8.75 A is a maximum of 1.28 N when placed between the pole faces of a magnet. If the pole faces are 55.5 cm in diameter, what is the approximate strength of the magnetic field ?

6. (II) The magnetic force per meter on a wire is measured to be only 35% of its maximum possible value. Sketch the relationship of the wire and the field if the force had been a maximum, and sketch the relationship as it actually is. calculating the angle between the wire and the magnetic field.

7. (II) The force on a wire is a maximum of 6.50 x 10-2N when placed between the pole faces of a magnet. The current flows horizontally to the right and the magnetic field is vertical. The wire is observed to jump" toward the observer when the current is turned on. (a) What type of magnetic pole is the top pole face? (/>) If the pole faces have a diameter of 10.0 cm. estimate the current in the wire if the field is 0.16T. (c) If the wire is tipped so that it makes an angle of 10.0 with the horizontal, w hat force will it now feel?

8. (II) Suppose a straight 1.00-mm-diamctcr copper wire could just float" horizontally in air because of the force due to the Earth's magnetic field B. which is horizontal, perpendicular to the wire, and of magnitude 5.0 x 10-5T. What current would the wire carry? Does the answer seem feasible? Explain briefly.

9. (I) Alpha particles of charge q = +2e and mass m = 6.6 x 10~27kg are emitted from a radioactive source at a speed of 1.6 X 107m/s. What magnetic field strength would be required to bend them into a circular path of radius r = 0.25 m?

10. (I) Determine the magnitude and direction of the force on an electron traveling 8.75 X 105m/s horizontally to the east in a vertically upward magnetic field of strength 0.75 T.

11. (I) Find the direction of the force on a negative charge for each diagram shown in Fig. 20-51. where V (green) is the velocity of the charge and B (blue) is the direction of the magnetic field. (<8> means the vector points inward. O means it points outward, toward you.)

12. (I) Determine the direction of B for each case in Fig. 20-52. where F represents the maximum magnetic force on a positively charged particle moving with velocity v.

13. (I) An electron is projected vertically upward with a speed of 1.70 x 106m/s into a uniform magnetic field of 0.350 T that is directed horizontally away from the observer. Describe the electron's path in this field.

14. (II) A 5.0-MeV (kinetic energy) proton enters a 0.20-T field, in a plane perpendicular to the field. What is the radius of its path?

15. (II) An electron experiences the greatest force as it travels 2.9 X 10hm/s in a magnetic field when it is moving northward. The force is upward and of magnitude 7.2 X 10-1' N. What are the magnitude and direction of the magnetic Field?

16. (II) What is the velocity of a beam of electrons that go undeflected when passing through perpendicular electric and magnetic fields of magnitude 8.8 X 103V/m and 3.5 X 10-3T. respectively? What is the radius of the electron orbit if the electric field is turned off?

17. (II) A doubly charged helium atom whose mass is 6.6 X 10_27kg is accelerated by a voltage of 21(X)V. (a) What will be its radius of curvature if it moves in a plane perpendicular to a uniform 0.340- T field? (6) What is its period of revolution ?

18. (II) A proton (mass mp), a deuteron (m = 2wp. Q = e). and an alpha particle (m = 4mp. Q = 2e) are accelerated by the same potential difference V and then enter a uniform magnetic field B. where they move in circular paths perpendicular to B. Determine the radius of the paths for the deuteron and alpha particle in terms of that for the proton.

19. (II) Show that the time T required for a particle of charge q moving with constant speed v to make one circular revolution in a uniform magnetic field B (i. v) is qB [Hint: see Example 20-5 and Chapter 5.]

20. (II) A particle of charge q moves in a circular path of radius r in a uniform magnetic field B. Show that its momentum is p = qBr.

21. (II) A particle of mass m and charge q moves in a circular path in a magnetic field B. Show that its kinetic energy is proportional to r2, the square of the radius of curvature of its path.

22. (II) Show that the angular momentum of the particle in Problem 21 is L = qBr2 about the center of the circle.

23. (Ill) A 3.40-g bullet moves with a speed of 160m/s perpendicular to the Earth's magnetic field of 5.00 x 10 5T. If the bullet possesses a net charge of 13.5 x 10_9C. by what distance will it be deflected from its path due to the Eartlvs magnetic field after it has traveled 1.00 km?

24. (Ill) Suppose the Earth's magnetic field at the equator has magnitude 0.40 x 10-4T and a northerly direction at all points. Estimate the speed a singly ionized uranium ion (m = 238 u, q = e) would need to circle the Earth 5.0 km above the equator. Can you ignore gravity?

25. (Ill) A proton moving with speed v = 2.0 X 10^ m/s in a field-free region abruptly enters an essentially uniform magnetic field B = 0.850T (Blv). If the proton enters the magnetic field region at a 45 angle as shown in Fig. 20-53. (a) at what angle does it leave, and (b) at what distance x does it exit the field?

26. (I) A jumper cable used to start a stalled vehicle carries a 65-A current. How strong is the magnetic field 6.0 cm away from it? Compare to the Earth's magnetic field.

27. (I) If an electric wire is allowed to produce a magnetic field no larger than that of the Earth (0.55 X 10 4 T) at a distance of 25 cm, what is the maximum current the wire can carry?

28. (I) In Fig. 20-54. a long straight wire carries current / out of the page toward you. Indicate, with appropriate arrows, the direction of B at each of the points C, ^ * D. and E in the plane of the page. !)

29. (I) A vertical straight wire earn ing an upward 24-A current exerts an attractive force per unit length of 8.8 X 10-4 N/m on a second parallel wire 7.0 cm away. What current (magnitude and direction) flows in the second wire?

30. (I) Determine the magnitude and direction of the force between two parallel wires 35 m long and 6.0 cm apart, each earning 25 A in the same direction.

31. (II) An experiment on the F.arths magnetic field is being carried out 1.00 m from an electric cable. What is the maximum allowable current in the cable if the experiment is to be accurate to 1.0%?

32. (II) A power line carries a current of 95 A along the tops of 8.5-m-high poles. What is the magnitude of the magnetic field produced by this wire at the ground? How does this compare with the Earths field of about 3C1?

33. (II) Two long thin parallel wires 13.0 cm apart carry 25-A currents in the same direction. Determine the magnetic field at point P. 12.0 cm from one wire and 5.0 cm from the other (Fig. 20-55).

34. (II) A horizontal compass is placed 18cm due south from a straight vertical wire carrying a 35-A current downward. In what direction does the compass needle point at this location? Assume the horizontal component of the Farths field at this point is 0.45 X 10~4T and the magnetic declination is 0.

35. (II) A long horizontal wire carries 22.0 A of current due north. What is the net magnetic field 20.0 cm due west of the wire if the Earths field there points north but downward, 37 below the horizontal, and has magnitude 5.0 x 10-5 T?

36. (II) A straight stream of protons passes a given point in space at a rate of 1.5 x 10g protons/s. What magnetic field do they produce 2.0 m from the beam?

37. (II) Determine the magnetic field midway between two long straight wires 2.0 cm apart in terms of the current / in one when the other carries 15 A. Assume these currents are (a) in the same direction, and (b) in opposite directions.

38. (II) A long pair of wires conducts 25.0 A of dc current to, and from, an instrument. If the insulated wires are of negligible diameter but are 2.8 mm apart, what is the magnetic field 10.00 cm from their midpoint, in their plane (Fig. 20-56)? Compare to the magnetic field of the Earth.

39. (II) A third wire is placed in the plane of the two wires shown in Fig. 20-56. parallel and just to the right. If it carries 25.0 A upward, what force per meter of length does it exert on each of the other two wires? Assume it is 2.8 mm from the nearest wire, center to center.

40. (II) A compass needle points 23 E of N outdoors. However, when it is placed 12.0 cm to the east of a vertical wire inside a building, it points 55 E of N. What arc the magnitude and direction of the current in the wire? The Earths field there is 0.50 x 10_4T and is horizontal.

41. (II) A rectangular loop of wire lies in the same plane as a straight wire, as shown in Fig. 20-57. There is a current of 2.5 A in both wires. Determine the magnitude and direction of the net force on the loop.

42. (II) A long horizontal wire carries a current of 48 A. A second wire, made of 2.5-mm-diameter copper wire and parallel to the first, is kept in suspension magnetically 15 cm below (Fig. 20-58). (a) Determine the magnitude and direction of the current in the lower wire. (b) Is the lower wire in stable equilibrium? (c) Repeat parts (a) and (/>) if the second wire is suspended 15 cm above the first due to the latters field.

43. (II) Two long wires are oriented so that they are perpendicular to each other. At their closest, they are 20.0 cm apart (Fig. 20-59). What is the magnitude of the magnetic field at a point midway between them if the top one carries a current of 20.0 A and the bottom one carries 5.0 A?

44. (II) Two long straight parallel wires are 15 cm apart. Wire A carries 2.0 A current. Wire B's current is 4.0 A in the same direction, (a) Determine the magnetic field magnitude due to wire A at the position of wire B. (b) Determine the magnetic field due to wire B at the position of wire A. (c) Arc these two magnetic fields equal and opposite? Why or why not? (d) Determine the force on wire A due to w'ire B. and the force on w'ire B due to wrire A. Are these two forces equal and opposite? Why or why not?

45. (II) Three long parallel wires are 3.8cm from one another. (Looking along them, they are at three corners of an equilateral triangle.) The current in each wire is 8.00 A. but its direction in wire M is opposite to that in wires N and P (Fig. 20-60). Determine the magnetic force per unit length on each wire due to the other twro.

46. (II) In Fig. 20-60. determine the magnitude and direction of the magnetic field at the midpoint of the side of the triangle between wire M and wire N.

47. (II) Let two long parallel wires, a distance d apart, carry equal currents 1 in the same direction. One wire is at x = 0. the other is at .v = d, Fig. 20-61. Determine B along the x axis between the wires as a function of .v.

48. (I) A thin 12-cm-long solenoid has a total of 420 turns of wire and carries a current of 2.0 A. Calculate the field inside near the center.

49. (I) A 30.0-cm long solenoid 1.25 cm in diameter is to produce a field of 0.385 T at its center. How- much current should the solenoid carry if it has 975 turns of the wrire?

50. (II) A 550-turn solenoid is 15cm long.'ITie current in it is 33 A. A 3.0-cm-long straight wire cuts through the center of the solenoid, along a diameter. This wire carries a 22-A current downward (and is connected by other wires that don't concern us). What is the force on this wrire assuming the solenoid's field points due east?

51. (Ill) You have 1.0 kg of copper and want to make a practical solenoid that produces the greatest possible magnetic field for a given voltage. Should you make your copper wrire long and thin, short and fat, or something else? Consider other variables, such as solenoid diameter, length, and so on.

*52. (II) A toroid is a solenoid in the shape of a circle (Fig. 20-62). Use Ampere's law along the circular path, shown dashed in Fig. 20-62a. to determine that the magnetic field (d) inside the toroid is B = fi0NI/2irR, where N is the total number of turns, and (b) outside the toroid is B = 0. (c) Is the field inside a toroid uniform like a solenoid's? If not, how does it vary?

*53. (Ill) (a) Use Amperes law to show that the magnetic field between the conductors of a coaxial cable (Fig. 20-63) is B = jlirr if r is greater than the radius of the inner wire and less than the radius of the outer cylindrical braid. (b) Show that B = 0 outside the coaxial cable.

*54. (I) A single square loop of wire 22.0cm on a side is placed with its face parallel to the magnetic field between the pole pieces of a large magnet. When 6.30 A flows in the coil, the torque on it is 0.325 m-N. What is the magnetic field strength?

* 55. (I) A galvanometer needle deflects full scale for a 53.0-/aA current. What current will give full- scale deflection if the magnetic field weakens to 0.860 of its original value?

* 56. (I) If the restoring spring of a galvanometer weakens by 25% over the years, what current will give full-scale deflection if it originally required 36 juA?

*57. (I) If the current to a motor drops by 12%, by what factor does the output torque change?

* 58. (II) Show that the magnetic dipole moment A/ of an electron orbiting the proton nucleus of a hydrogen atom is related to the orbital angular momentum L of the electron by

* 59. (II) A circular coil 16.0cm in diameter and containing nine loops lies flat on the ground. The Earth's magnetic field at this location has magnitude 5.50 X 10~5T and points into the Earth at an angle of 56.0 below a line pointing due north. If a 7.20-A clockwise current passes through the coil, (a) determine the torque on the coil, and (b) which edge of the coil rises up: north, cast, south, or west?

* 60. (I) Protons move in a circle of radius 5.10 cm in a 0.566-T magnetic field. What value of electric field could make their paths straight? In what direction must it point?

*61. (I) In a mass spectrometer, germanium atoms have radii of curvature equal to 21.0, 21.6, 21.9, 22.2. and 22.8 cm. The largest radius corresponds to an atomic mass of 76 u. What are the atomic masses of the other isotopes?

62. (II) Suppose the electric field between the electric plates in the mass spectrometer of Fig. 20-39 is 2.48 x 104 V/m and the magnetic fields B = B' = 0.68 T. The source contains carbon isotopes of mass numbers 12, 13, and 14 from a long-dead piece of a tree. (To estimate atomic masses, multiply by 1.67 x 1027kg.) How far apart arc the lines formed by the singly charged ions of each type on the photographic film ? What if the ions were doubly charged?

*63. (II) A mass spectrometer is being used to monitor air pollutants. It is difficult, however, to separate molecules with nearly equal mass such as CO (28.0106 u) and N2 (28.0134 u). How large a radius of curvature must a spectrometer have if these two molecules are to be separated on the film by 0.50 mm?

*64. (II) One form of mass spectrometer accelerates ions by a voltage V before they enter a magnetic field B. The ions are assumed to start from rest. Show that the mass of an ion is m = qB2R2/2V, where R is the radius of the ions' path in the magnetic field and q is their charge.

* 65. (I) A long thin solenoid has 430 loops of wire per meter, and a 25-A current flows through the wire. If the permeability of the iron is 3000/xu, what is the total field B inside the solenoid?

*66. (II) An iron-core solenoid is 38 cm long and 1.8cm in diameter, and has 640 turns of wire. TTie magnetic field inside the solenoid is 2.2 T when 48 A flows in the wire. What is the permeability /x at this high field strength ?

67. Protons with momentum 4.8 X 10~,6kg-m/s are magnetically steered clockwise in a circular path 2.0 km in diameter at Fermi National Accelerator Laboratory in Illinois. Determine the magnitude and direction of the field in the magnets surrounding the beam pipe.

68. A proton and an electron have the same kinetic energy upon entering a region of constant magnetic field. What is the ratio of the radii of their circular paths?

69. The power cable for an electric trolley (Fig. 20-64) carries a horizontal current of 330 A toward the east. The Earths magnetic field has a strength 5.0 X 10-5T and makes an angle of dip of 22 at this location. Calculate the magnitude and direction of the magnetic force on a 15-m length of this cable.

70. Calculate the force on an airplane which has acquired a net charge of 1550/xC and moves with a speed of 120 m/s perpendicular to the Earths magnetic field of 5.0 x 10-5 T.

71. Near the equator, the Earths magnetic field points almost horizontally to the north and has magnitude B = 0.50 X 10_4T. What should be the magnitude and direction for the velocity of an electron if its weight is to be exactly balanced by the magnetic force?

72. A doubly charged helium atom, whose mass is 6.6 X 10~27kg. is accelerated by a voltage of 2400 V. (a) What will be its radius of curvature in a uniform 0.240-T field? (/>) What is its period of revolution?

73. A sort of projectile launcher" is shown in Fig. 20-65. A large current moves in a closed loop composed of fixed rails, a power supply, and a very light, almost frictionless bar touching the rails. A magnetic field is perpendicular to the plane of the circuit. If the bar has a length L = 22 cm, a mass of 1.5 g. and is placed in a Field of 1.7 T. what constant current flow is needed to accelerate the bar from rest to 28 m/s in a distance of 1.0 m? In what direction must the magnetic field point?

74. In Fig. 20-60 the top wire is 1.00-mm-diameter copper wire and is suspended in air due to the two magnetic forces from the bottom two wires. ITie current flow through the two bottom wires is 95 A in each. Calculate the required current flow in the suspended wire.

75. Two stiff parallel wires a distance / apart in a horizontal plane act as rails to support a light metal rod of mass m (perpendicular to each rail). Fig. 20-66. A magnetic field B. directed vertically upward (outward in the diagram), acts throughout. At t = 0, wires connected to the rails are connected to a constant current source and a current / begins to flow through the system. Determine the speed of the rod. which starts from rest at t = 0. as a function of time (a) assuming no friction between the rod and the rails, and (/>) if the coefficient of friction is . (c) Does the rod move cast or west if the current through it heads north?

76. Estimate the approximate maximum deflection of the electron beam near the center of a TV screen due to the Earth's 5.0 x 10-5T field. Assume the CRT screen (Section 17-10) is 22 cm from the electron gun. where the electrons are accelerated (tf) by 2.0 kV. or (b) by 30 kV. Note that in color TV sets, the CRT beam must be directed accurately to within less than 1 mm in order to strike the correct phosphor. Because the Earths field is significant here, mu-metal shields are used to reduce the Earth's field in the CRT.

77. The cyclotron (Fig. 20-67) is a device used to accelerate elementary particles such as protons to high speeds. Particles starting at point A with some initial velocity travel in circular orbits in the magnetic field B. The particles are accelerated to higher speeds each time they pass through the gap between the metal dees," where there is an electric field F,. (There is no electric field inside the hollow metal dees.) The electric field changes direction each half-cycle, owing to an ac voltage V = VQ sin lir/t, so that the particles are increased in speed at each passage through the gap. (a) Show that the frequency / of the voltage must be f = Bq/lnnis where q is the charge on the particles and m their mass, (b) Show that the kinetic energy of the particles increases by 2qV() each revolution, assuming that the gap is small, (c) If the radius of the cyclotron is 2.0 m and the magnetic field strength is 0.50 T. what will be the maximum kinetic energy of accelerated protons in MeV?

78. Four very long straight parallel wires, located at the corners of a square of side /. carry equal currents /0 perpendicular to the page as shown in Fig. 20-68. Determine the magnitude and direction of B at the center C of the square.

79. Magnetic fields are very useful in particle accelerators for beam steering": that is, the magnetic fields can be used to change the beams direction without altering its speed (Fig. 20-69). Show how this works with a beam of protons. What happens to protons that are not moving with the speed that the magnetic field is designed for? If the field extends over a region 5.0 cm wide and has a magnitude of 0.33 T. by approximately what angle will a beam of protons traveling at 1.0 x 107m/s be bent?

80. The magnetic field B at the center of a circular coil of wire carrying a current / (as in Fig. 20-9) is fioNI B = -- 2 r where .-V is the number of loops in the coil and r is its radius. Suppose that an electromagnet uses a coil 1.2 m in diameter made from square copper wire 1.6 mm on a side. The power supply produces 120 V at a maximum power output of 4.0 kW. (a) Howr many turns are needed to run the power supply at maximum power? (b) What is the magnetic field strength at the center of the coil? (c) If you use a greater number of turns and this same power supply (so the voltage remains at 120 V), will a greater magnetic field strength result? Explain.

81. Near the Earth's poles the magnetic field is about I G (l X l(T4T). Imagine a simple model in which the Earth's field is produced by a single current loop around the equator. Roughly estimate the current this loop would carry. [Him: use the formula given in Problem 80.|

82. You want to get an idea of the magnitude of magnetic fields produced by overhead power lines. You estimate that the two wires arc each about 30 m above the ground and are about 3 m apart. The local power company tells you that the lines operate at 10 kV and provide a maximum of 40 MW to the local area. Estimate the maximum magnetic field you might experience walking under these power lines, and compare to the Earths field. [For an ac current, values are rms. and the magnetic field will be changing.)

83. (a) What value of magnetic field would make a beam of electrons, traveling to the right at a speed of 4.8 x 106m/s, go undeflected through a region where there is a uniform electric field of 10,000 V/m pointing vertically up? (/>) What is the direction of the magnetic field if it is known to be perpendicular to the electric field? (c) What is the frequency of the circular orbit of the electrons if the electric field is turned off?

84. A proton follows a spiral path through a gas in a magnetic field of 0.010 T, perpendicular to the plane of the spiral, as shown in Fig. 20-70. In two successive loops, at points P and Q, the radii are 10.0 mm and 8.5 mm, respectively. Calculate the change in the kinetic energy of the proton as it travels from P to O.

85. A 32-cm-long solenoid, 1.8 cm in diameter, is to produce a 0.30-T magnetic field at its center. If the maximum current is 5.7 A. how many turns must the solenoid have?

86. Two long straight aluminum wires, each of diameter 0.50 mm. carry the same current but in opposite directions. They are suspended by 0.50-m-long strings as shown in Fig. 20-71. If the suspension strings make an angle of 3.0 with the vertical, what is the current in the wires?

87. An electron enters a uniform magnetic field B = 0.23 T at a 45 angle to B. Determine the radius r and pitch p (distance between loops) of the electrons helical path assuming its speed is 3.0 x 106m/s. See Fig. 20-72.