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1 Walker, Physics, 3 rd Edition Chapter 22 ** Can skip to Problems: 6, 7, 9, 15, 19, 28, 29, 35, 36, 39, 42 ** Conceptual Questions (Answers to odd-numbered Conceptual Questions can be found in the back of the book, beginning on page ANSxxx.) 1. Two charged particles move at right angles to a magnetic field and deflect in opposite directions. Can one conclude that the particles have opposite charges? 2. An electron moves with constant velocity through a region of space that is free of magnetic fields. Can one conclude that the electric field is zero in this region? Explain. 3. An electron moves with constant velocity through a region of space that is free of electric fields. Can one conclude that the magnetic field is zero in this region? Explain. 4. Describe how the motion of a charged particle can be used to distinguish between an electric and a magnetic field. 5. Explain how a charged particle moving in a circle of small radius can take the same amount of time to complete an orbit as an identical particle orbiting in a circle of large radius. 6. A current-carrying wire is placed in a region with a uniform magnetic field. The wire experiences zero magnetic force. Explain. 1

2 Conceptual Exercises (Answers to odd-numbered Conceptual Exercises can be found in the back of the book, beginning on page ANSxxx.) 1. Proton 1 moves with a speed v from the east coast to the west coast in the continental United States; proton 2 moves with the same speed from the southern United States toward Canada. Is the magnitude of the magnetic force experienced by proton 2 greater than, less than, or equal to the force experienced by proton 1? Explain. 2. An electron moves west to east in the continental United States. Does the magnetic force experienced by the electron point in a direction that is generally north, south, east, west, upward, or downward? Explain. 3. Suppose particles A, B, and C in Figure have identical masses and charges of the same magnitude. Rank the particles in order of increasing speed. Indicate ties where appropriate. 4. At a point near the equator, the Earth s magnetic field is horizontal and points to the north. If an electron is moving vertically upward at this point, does the magnetic force acting on it point north, south, east, west, upward, or downward? Explain. 5. A proton is to orbit the Earth at the equator using the Earth s magnetic field to supply part of the necessary centripetal force. Should the proton move eastward or westward? Explain. 6. Referring to Figure 22 31, what is the sign of the charge for each of the three particles? Explain. 7. Suppose the three particles in Figure have the same mass and speed. Rank the particles in order of increasing magnitude of their charge. Indicate ties where appropriate. 8. A velocity selector is to be constructed using a magnetic field in the positive y direction. If positively charged particles move through the selector in the positive z direction, (a) what must be the direction of the electric field? (b) Repeat part (a) for the case of negatively charged particles. 9. An electron moving in the positive x direction, at right angles to a magnetic field, experiences a magnetic force in the positive y direction. What is the direction of the magnetic field? 2

3 10. A positively charged particle moves through a region with a uniform electric field pointing toward the top of the page and a uniform magnetic field pointing into the page. The particle can have one of the four velocities shown in Figure (a) Rank the four possibilities in order of increasing magnitude of the net force ( F1, F2, F3, and F4 ) the particle experiences. Indicate ties where appropriate. (b) Which of the four velocities could potentially result in zero net force? 11. Suppose the fields in Figure are interchanged, with the magnetic field pointing toward the top of the page and the electric field pointing into the page. (a) Rank the four possibilities in order of increasing magnitude of the net force ( F1, F2, F3, and F4 ) the particle experiences. Indicate ties where appropriate. (b) Which of the four velocities could potentially result in zero net force? 12. The accompanying photograph shows an electron beam whose initial direction of motion is horizontal, from right to left. A magnetic field deflects the beam downward. What is the direction of the magnetic field? 13. A proton follows the path shown in Figure as it moves through three regions with different uniform magnetic fields, B, B, and B 3. In each region the proton completes a half-circle. (a) Rank the three fields in 1 2 order of increasing magnitude. Indicate ties where appropriate. (b) Give the direction of each of the fields. 14. Suppose the speed of the proton in Figure is increased. (a) Does the radius of each half-circular path segment increase, decrease, or stay the same? Explain. (b) Does the time spent in each of the field regions increase, decrease, or stay the same? Explain. 15. The three wires shown in Figure are long and straight, and they each carry a current of the same magnitude, I. The currents in wires 1 and 3 are out of the page; the current in wire 2 is into the page. What is the direction of the magnetic force experienced by wire 3? 16. Each of the current-carrying wires in Figure is long and straight, and carries the current I either into or out of the page, as shown. What is the direction of the net magnetic field produced by these three wires at the center of the triangle? 17. The four wires shown in Figure are long and straight, and they each carry a current of the same magnitude, I. The currents in wires 1, 2, and 3 are out of the page; the current in wire 4 is into the page. What is the direction of the magnetic force experienced by wire 2? 3

4 18. Each of the current-carrying wires in Figure is long and straight, and carries the current I either into or out of the page, as shown. What is the direction of the net magnetic field produced by these four wires at the center of the square? 19. For each of the square current loops shown in Figure 22 36, indicate whether there will be a tendency for the loop to rotate clockwise, counterclockwise, or not at all, when viewed from above along the indicated axis. 20. A loop of wire is connected to the terminals of a battery, as indicated in Figure If the loop is to attract the bar magnet, which of the terminals, A or B, should be the positive terminal of the battery? Explain. 21. The number of turns in a solenoid is doubled, and at the same time its length is doubled. Does the magnetic field within the solenoid increase, decrease, or stay the same? Explain. 4

5 Problems: 6, 7, 9, 15, 19, 28, 29, 35, 36, 39, 42 Note: IP denotes an integrated conceptual/quantitative problem. BIO identifies problems of biological or medical interest. Red bullets (,, ) are used to indicate the level of difficulty of each problem. Section 22 2 The Magnetic Force on Moving Charges 1. What is the acceleration of a proton moving with a speed of 9.5 m/s at right angles to a magnetic field of 1.6 T? 2. An electron moves at right angles to a magnetic field of 0.12 T. What is its speed if the force exerted on it is N? 3. A negatively charged ion moves due north with a speed of ms at the Earth s equator. What is the magnetic force exerted on this ion? 4. A proton high above the equator approaches the Earth moving straight downward with a speed of 355 m/s. Find the acceleration of the proton, given that the magnetic field at its altitude is T. 5. A μ C particle moves with a speed of 16 m/s through a region where the magnetic field has a strength of 0.95 T. At what angle to the field is the particle moving if the force exerted on it is (a) N, or (c) N? 6. A particle with a charge of 14 μc experiences a force of N, (b) N when it moves at right angles to a magnetic field with a speed of 27 m/s. What force does this particle experience when it moves with a speed of 6.3 m/s at an angle of 25 relative to the magnetic field? 7. An ion experiences a magnetic force of N when moving in the positive x direction but no magnetic force when moving in the positive y direction. What is the magnitude of the magnetic force exerted on the ion when it moves in the x-y plane along the line x cases. = y? Assume that the ion s speed is the same in all 8. An electron moving with a speed of ms in the positive x direction experiences zero magnetic force. When it moves in the positive y direction, it experiences a force of direction. What is the direction and magnitude of the magnetic field? N that points in the negative z 9. IP Two charged particles with different speeds move one at a time through a region of uniform magnetic 5

6 field. The particles move in the same direction and experience equal magnetic forces. (a) If particle 1 has four times the charge of particle 2, which particle has the greater speed? Explain. (b) Find the ratio of the speeds, v1 v A μ C particle moves through a region of space where an electric field of magnitude 1250 N/C points in the positive x direction, and a magnetic field of magnitude 1.02 T points in the positive z direction. If the net force acting on the particle is N in the positive x direction, find the magnitude and direction of the particle s velocity. Assume the particle s velocity is in the x-y plane. 11. When at rest, a proton experiences a net electromagnetic force of magnitude N pointing in the positive x direction. When the proton moves with a speed of ms in the positive y direction, the net electromagnetic force on it decreases in magnitude to N, still pointing in the positive x direction. Find the magnitude and direction of (a) the electric field and (b) the magnetic field. Section 22 3 The Motion of Charged Particles in a Magnetic Field 12. Find the radius of an electron s orbit when it moves perpendicular to a magnetic field of 0.45 T with a speed of ms. 13. Find the radius of a proton s orbit when it moves perpendicular to a magnetic field of 0.45 T with a speed of ms. 14. Charged particles pass through a velocity selector with electric and magnetic fields at right angles to each other, as shown in Figure If the electric field has a magnitude of 450 N/C and the magnetic field has a magnitude of 0.18 T, what speed must the particles have to pass through the selector undeflected? 15. The velocity selector in Figure is designed to allow charged particles with a speed of ms to pass through undeflected. Find the direction and magnitude of the required electric field, given that the magnetic field has a magnitude of 0.96 T. 6

7 16. IP BIO The artery in Figure has an inside diameter of 2.75 mm and passes through a region where the magnetic field is T. (a) If the voltage difference between the electrodes is 195 μ V, what is the speed of the blood? (b) Which electrode is at the higher potential? Does your answer depend on the sign of the ions in the blood? Explain. 17. An electron accelerated from rest through a voltage of 410 V enters a region of constant magnetic field. If the electron follows a circular path with a radius of 17 cm, what is the magnitude of the magnetic field? 18. A μ Cparticle with a mass of kg moves perpendicular to a 1.01-T magnetic field in a circular path of radius 26.8 m. (a) How fast is the particle moving? (b) How long will it take the particle to complete one orbit? 19. IP When a charged particle enters a region of uniform magnetic field, it follows a circular path, as indicated in Figure (a) Is this particle positively or negatively charged? Explain. (b) Suppose that the magnetic field has a magnitude of T, the particle s speed is ms, and the radius of its path is 52.0 cm. Find the mass of the particle, given that its charge has a magnitude of C. Give your result in atomic mass units, u, where 1u = kg. 20. A proton with a kinetic energy of the radius of its circular path? J moves perpendicular to a magnetic field of 0.26 T. What is 21. IP An alpha particle (the nucleus of a helium atom) consists of two protons and two neutrons, and has a mass of kg. A horizontal beam of alpha particles is injected with a speed of ms into a region with a vertical magnetic field of magnitude T. (a) How long does it take for an alpha particle to move halfway through a complete circle? (b) If the speed of the alpha particle is doubled, does the time found in part (a) increase, decrease, or stay the same? Explain. (c) Repeat part (a) for alpha particles with a speed of ms. 22. An electron and a proton move in circular orbits in a plane perpendicular to a uniform magnetic field r B. Find the ratio of the radii of their circular orbits when the electron and the proton have (a) the same momentum and (b) the same kinetic energy. 7

8 Section 22 4 The Magnetic Force Exerted on a Current-Carrying Wire 23. What is the magnetic force exerted on a 2.15-m length of wire carrying a current of A perpendicular to a magnetic field of T? 24. A wire with a current of 2.8 A is at an angle of 36.0 relative to a magnetic field of 0.88 T. Find the force exerted on a 2.25-m length of the wire. 25. The magnetic force exerted on a 1.2-m segment of straight wire is 1.6 N. The wire carries a current of 3.0 A in a region with a constant magnetic field of 0.50 T. What is the angle between the wire and the magnetic field? 26. Consider a current loop immersed in a magnetic field, as in Figure (a) (Conceptual Exercise 19). It is given that B = 034. T and I = 95. A. In addition, the loop is a square 0.46 m on a side. Find the magnitude of the magnetic force exerted on each side of the loop. 27. A 0.45-m copper rod with a mass of 0.17 kg carries a current of 11 A in the positive x direction. What are the magnitude and direction of the minimum magnetic field needed to levitate the rod? 28. The long, thin wire shown in Figure is in a region of constant magnetic field r B. The wire carries a current of 6.2 A and is oriented at an angle of 7.5 to the direction of the magnetic field. (a) If the magnetic force exerted on this wire per meter is N, what is the magnitude of the magnetic field? (b) At what angle will the force exerted on the wire per meter be equal to N? 29. A wire with a length of 2.7 m and a mass of 0.75 kg is in a region of space with a magnetic field of 0.84 T. What is the minimum current needed to levitate the wire? 30. A high-voltage power line carries a current of 110 A at a location where the Earth s magnetic field has a magnitude of 0.59 G and points to the north, 72 below the horizontal. Find the direction and magnitude of the magnetic force exerted on a 250-m length of wire if the current in the wire flows (a) horizontally toward the east or (b) horizontally toward the south. 31. A metal bar of mass m and length L is suspended from two conducting wires, as shown in Figure A uniform magnetic field of magnitude B points vertically downward. Find the angle θ the suspending wires make with the vertical when the bar carries a current I. 8

9 Section 22 5 Loops of Current and Magnetic Torque 32. A rectangular loop of 260 turns is 22 cm wide and 16 cm high. What is the current in this loop if the maximum torque in a field of 0.48 T is 23 N m? 33. A single circular loop of radius 0.23 m carries a current of 2.6 A in a magnetic field of 0.95 T. What is the maximum torque exerted on this loop? 34. In the previous problem, find the angle the plane of the loop must make with the field if the torque is to be half its maximum value. 35. Consider a current loop in a region of uniform magnetic field, as shown in Figure (a) (Conceptual Exercise 19). Find the magnitude of the torque exerted on the loop about the vertical axis of rotation, using the data given in Problem IP Two current loops, one square the other circular, have one turn made from wires of the same length. (a) If these loops carry the same current and are placed in magnetic fields of equal magnitude, is the maximum torque of the square loop greater than, less than, or the same as the maximum torque of the circular loop? Explain. (b) Calculate the ratio of the maximum torques, τ τ. square circle 37. IP Each of the 10 turns of wire in a vertical, rectangular loop carries a current of 0.22 A. The loop has a height of 8.0 cm and a width of 15 cm. A horizontal magnetic field of magnitude T is oriented at an angle of θ = 65º relative to the normal to the plane of the loop, as indicated in Figure Find (a) the magnetic force on each side of the loop, (b) the net magnetic force on the loop, and (c) the magnetic torque on the loop. (d) If the loop can rotate about a vertical axis with only a small amount of friction, will it end up with an orientation given by θ = 0, θ = 90º, or θ = 180º? Explain. Section 22 6 Electric Currents, Magnetic Fields, and Ampère s Law 38. Find the magnetic field 6.25 cm from a long, straight wire that carries a current of 5.81 A. 39. A long, straight wire carries a current of 7.2 A. How far from this wire is the magnetic field it produces equal to the Earth s magnetic field, which is approximately T? 9

10 40. You travel to the north magnetic pole of the Earth, where the magnetic field points vertically downward. There, you draw a circle on the ground. Applying Ampère s law to this circle, show that zero current passes through its area. 41. Two power lines, each 270 m in length, run parallel to each other with a separation of 25 cm. If the lines carry parallel currents of 110 A, what is the magnitude and direction of the magnetic force each exerts on the other? 42. BIO Pacemaker Switches Some pacemakers employ magnetic reed switches to enable doctors to change their mode of operation without surgery. A typical reed switch can be switched from one position to another with a magnetic field of T. What current must a wire carry if it is to produce a T field at a distance of 0.50 m? 43. IP Consider the long, straight, current-carrying wires shown in Figure One wire carries a current of 6.2 A in the positive y direction; the other wire carries a current of 4.5 A in the positive x direction. (a) At which of the two points, A or B, do you expect the magnitude of the net magnetic field to be greater? Explain. (b) Calculate the magnitude of the net magnetic field at points A and B. 44. IP Repeat Problem 43 for the case where the 6.2-A current is reversed in direction. 45. In Oersted s experiment, suppose that the compass was 0.25 m from the current-carrying wire. If a magnetic field of half the Earth s magnetic field of T was required to give a noticeable deflection of the compass needle, what current must the wire have carried? 46. IP Two long, straight wires are separated by a distance of 12.2 cm. One wire carries a current of 2.75 A, the other carries a current of 4.33 A. (a) Find the force per meter exerted on the 2.75-A wire. (b) Is the force per meter exerted on the 4.33-A wire greater than, less than, or the same as the force per meter exerted on the A wire? Explain. 47. Two long, straight wires are oriented perpendicular to the page, as shown in Figure The current in one wire is I 1 = 30. A, pointing into the page, and the current in the other wire is I 2 = 40. A, pointing out of the page. Find the magnitude and direction of the net magnetic field at point P. 10

11 Section 22 7 Current Loops and Solenoids 48. It is desired that a solenoid 38 cm long and with 430 turns produce a magnetic field within it equal to the Earth s magnetic field ( T). What current is required? 49. A solenoid that is 75 cm long produces a magnetic field of 1.3 T within its core when it carries a current of 8.4 A. How many turns of wire are contained in this solenoid? 50. The maximum current in a superconducting solenoid can be as large as 3.75 ka. If the number of turns per meter in such a solenoid is 3250, what is the magnitude of the magnetic field it produces? 51. To construct a solenoid, you wrap insulated wire uniformly around a plastic tube 12 cm in diameter and 55 cm in length. You would like a 2.0-A current to produce a 2.5-kG magnetic field inside your solenoid. What is the total length of wire you will need to meet these specifications? General Problems 52. A stationary proton ( q = C) is located between the poles of a horseshoe magnet, where the magnetic field is 0.15 T. What is the magnitude of the magnetic force acting on the proton? 53. BIO Brain Function and Magnetic Fields Experiments have shown that thought processes in the brain can be affected if the parietal lobe is exposed to a magnetic field with a strength of 1.0 T. How much current must a long, straight wire carry if it is to produce a 1.0-T magnetic field at a distance of 0.50 m? (For comparison, a typical lightning bolt carries a current of about 20,000 A, which would melt most wires.) 54. BIO Magnetoencephalography Magnetoencephalography (MEG) is a noninvasive method for studying electrical activity in the brain. It is based on the principle that an electric current creates a magnetic field, whether the current is in a wire or in the neurons of a brain. Thus, your very thoughts at this moment are generating measurable magnetic fields outside your head. With today s technology, an MEG can detect magnetic fields as small as T. Approximating a neuron by a straight wire, what electric current is needed to produce this magnetic field strength at a distance of 5.0 cm? 55. A mixture of two isotopes is injected into a mass spectrometer. One isotope follows a curved path of radius R 1 = cm; the other follows a curved path of radius R 2 = cm. Find the mass ratio, m1 m2, assuming that the two isotopes have the same charge and speed. 11

12 56. High above the surface of the Earth, charged particles (such as electrons and protons) can become trapped in the Earth s magnetic field in regions known as Van Allen belts. A typical electron in a Van Allen belt has an energy of 45 kev and travels in a roughly circular orbit with an average radius of 220 m. What is the magnitude of the Earth s magnetic field where such an electron orbits? 57. Credit-card Magnetic Strips Experiments carried out on the television show Mythbusters determined that a magnetic field of 1000 gauss is needed to corrupt the information on a credit card s magnetic strip. (They also busted the myth that a credit card can be demagnetized by an electric eel or an eelskin wallet.) Suppose a long, straight wire carries a current of 2.5 A. How close can a credit card be held to this wire without damaging its magnetic strip? 58. Superconducting Solenoid Cryomagnetics, Inc. advertises a high-field, superconducting solenoid that produces a magnetic field of 17 T with a current of 105 A. What is the number of turns per meter in this solenoid? 59. BIO Magnetic Resonance Imaging An MRI (magnetic resonance imaging) solenoid produces a magnetic field of 1.5 T. The solenoid is 2.5 m long, 1.0 m in diameter, and wound with insulated wires 2.2 mm in diameter. Find the current that flows in the solenoid. (Your answer should be rather large. A typical MRI solenoid uses niobium titanium wire kept at liquid helium temperatures, where it is superconducting.) 60. IP A long, straight wire carries a current of 14 A. Next to the wire is a square loop with sides 1.0 m in length, as shown in Figure The loop carries a current of 2.5 A in the direction indicated. (a) What is the direction of the net force exerted on the loop? Explain. (b) Calculate the magnitude of the net force acting on the loop. 61. Suppose the 14-A current in the straight wire in Figure is reversed in direction, but the current in the loop is unchanged. (a) Calculate the magnitude and direction of the net force acting on the loop. (b) If the loop is extended in the horizontal direction, so that it is 1.0 m high and 2.0 m wide, does the net force exerted on the loop increase or decrease? By what factor? Explain. (c) If, instead, the loop is extended in the vertical direction, so it is 2.0 m high and 1.0 m wide, does the net force exerted on the loop increase or decrease? Explain. 12

13 62. A charged particle moves through a region of space containing both electric and magnetic fields. The velocity of the particle is V r = ( ms)$ x + ( msy )$ and the magnetic field is B r = ( 073T). z$. Find the electric field vector r E necessary to yield zero net force on the particle. (Note: You may wish to use cross products in this problem. They are discussed in Appendix A.) 63. IP Medical X-rays An electron in a medical X-ray machine is accelerated from rest through a voltage of 10.0 kv. (a) Find the maximum force a magnetic field of T can exert on this electron. (b) If the voltage of the X-ray machine is increased, does the maximum force found in part (a) increase, decrease, or stay the same? Explain. (c) Repeat part (a) for an electron accelerated through a potential of 25.0 kv. 64. A particle with a charge of 34 μc moves with a speed of 62 m/s in the positive x direction. The magnetic field in this region of space has a component of 0.40 T in the positive y direction, and a component of 0.85 T in the positive z direction. What are the magnitude and direction of the magnetic force on the particle? 65. IP A beam of protons with various speeds is directed in the positive x direction. The beam enters a region with a uniform magnetic field of magnitude 0.52 T pointing in the negative z direction, as indicated in Figure It is desired to use a uniform electric field (in addition to the magnetic field) to select from this beam only those protons with a speed of ms that is, only these protons should be undeflected by the two fields. (a) Determine the magnitude and direction of the electric field that yields the desired result. (b) Suppose the electric field is to be produced by a parallel-plate capacitor with a plate separation of 2.5 cm. What potential difference is required between the plates? (c) Which plate in Figure (top or bottom) should be positively charged? Explain. 66. IP A charged particle moves in a horizontal plane with a speed of ms. When this particle encounters a uniform magnetic field in the vertical direction it begins to move on a circular path of radius 15.9 cm. (a) If the magnitude of the magnetic field is 1.21 T, what is the charge-to-mass ratio (q/m) of this particle? (b) If the radius of the circular path were greater than 15.9 cm, would the corresponding charge to mass ratio be greater than, less than, or the same as that found in part (a)? Explain. (Assume that the magnetic field remains the same.) 13

14 67. Two parallel wires, each carrying a current of 2.2 A in the same direction, are shown in Figure Find the direction and magnitude of the net magnetic field at points A, B, and C. 68. Repeat Problem 67 for the case where the current in wire 1 is reversed in direction. 69. Lightning Bolts A powerful bolt of lightning can carry a current of 225 ka. (a) Treating a lightning bolt as a long, thin wire, calculate the magnitude of the magnetic field produced by such a bolt of lightning at a distance of 35 m. (b) If two such bolts strike simultaneously at a distance of 35 m from each other, what is the magnetic force per meter exerted by one bolt on the other? 70. IP Consider the two current-carrying wires shown in Figure The current in wire 1 is 3.7 A; the current in wire 2 is adjusted to make the net magnetic field at point A equal to zero. (a) Is the magnitude of the current in wire 2 greater than, less than, or the same as that in wire 1? Explain. (b) Find the magnitude and direction of the current in wire IP Consider the physical system shown in Figure 22 49, which consists of two current-carrying wires each with a length of 71 cm. (a) If the net magnetic field at the point A is out of the page, is the force between the wires attractive or repulsive? Explain. (b) Calculate the magnitude of the force exerted by each wire on the other wire, given that the magnetic field at point A is out of the page with a magnitude of 21. T 10 T. 72. Magnetars The astronomical object 4 U has the distinction of creating the most powerful magnetic field ever observed. This object is referred to as a magnetar (a subclass of pulsars), and its magnetic field is times greater than the Earth s magnetic field. (a) Suppose a 2.5-m straight wire carrying a current of 1.1 A is placed in this magnetic field at an angle of 65 to the field lines. What force does this wire experience? (b) A field this strong can significantly change the behavior of an atom. To see this, consider an electron moving with a speed of ms. Compare the maximum magnetic force exerted on the electron to the electric force a proton exerts on an electron in a hydrogen atom. The radius of the hydrogen atom is m. 14

15 73. Consider a system consisting of two concentric solenoids, as illustrated in Figure The current in the outer solenoid is I 1 = 125. A, and the current in the inner solenoid is I 2 = 217. A. Given that the number of turns per centimeter is 105 for the outer solenoid and 125 for the inner solenoid, find the magnitude and direction of magnetic field (a) between the solenoids and (b) inside the inner solenoid. 74. IP A long, straight wire on the x axis carries a current of 3.12 A in the positive x direction. The magnetic field produced by the wire combines with a uniform magnetic field of T that points in the positive z direction. (a) Is the net magnetic field of this system equal to zero at a point on the positive y axis or at a point on the negative y axis? Explain. (b) Find the distance from the wire to the point where the field vanishes. 75. Find the angle between the plane of a loop and the magnetic field for which the magnetic torque acting on the loop is equal to x times its maximum value, where 0 x Solenoids produce magnetic fields that are relatively intense for the amount of current they carry. To make a direct comparison, consider a solenoid with 55 turns per centimeter, a radius of 1.05 cm, and a current of A. (a) Find the magnetic field at the center of the solenoid. (b) What current must a long, straight wire carry to have the same magnetic field as that found in part (a)? Let the distance from the wire be the same as the radius of the solenoid, 1.05 cm. 77. The current in a solenoid with 22 turns per centimeter is 0.50 A. The solenoid has a radius of 1.5 cm. A long, straight wire runs along the axis of the solenoid, carrying a current of 16 A. Find the magnitude of the net magnetic field a radial distance of 0.75 cm from the straight wire. 78. IP BIO Transcranial Magnetic Stimulation A recently developed method to study brain function is to produce a rapidly changing magnetic field within the brain. When this technique, known as transcranial magnetic stimulation (TMS), is applied to the prefrontal cortex, for example, it can reduce a person s ability to conjugate verbs, though other thought processes are unaffected. The rapidly varying magnetic field is produced with a circular coil of 21 turns and a radius of 6.0 cm placed directly on the head. The current in this loop increases at the rate of As (by discharging a capacitor). (a) At what rate does the magnetic field at the center of the coil increase? (b) Suppose a second coil with half the area of the first coil is used instead. Would your answer to part (a) increase, decrease, or stay the same? By what factor? 15

16 79. An electron with a velocity given by v r = ( ms)$ x + ( msy )$ moves through a region of space with a magnetic field B r = ( 025. T) x$ ( 011. T)z $ and an electric field E r = ( 220 N C) x$. Using cross products, find the magnitude of the net force acting on the electron. (Cross products are discussed in Appendix A.) 80. A thin ring of radius R and charge per length λ rotates with an angular speed ω about an axis perpendicular to its plane and passing through its center. Find the magnitude of the magnetic field at the center of the ring. 81. A solenoid is made from a 25-m length of wire of resistivity Ω m. The wire, whose radius is 2.1 mm, is wrapped uniformly onto a plastic tube 4.5 cm in diameter and 1.65 m long. Find the emf to which the ends of the wire must be connected to produce a magnetic field of T within the solenoid. 82. IP A single current-carrying circular loop of radius R is placed next to a long, straight wire, as shown in Figure The current in the wire points to the right and is of magnitude I. (a) In which direction must current flow in the loop to produce zero magnetic field at its center? Explain. (b) Calculate the magnitude of the current in part (a). 83. Magnetic Fields in the Bohr Model In the Bohr model of the hydrogen atom, the electron moves in a circular orbit of radius m about the nucleus. Given that the charge on the electron is C, and that its speed is ms, find the magnitude of the magnetic field the electron produces at the nucleus of the atom. 84. A single-turn square loop carries a current of 18 A. The loop is 15 cm on a side and has a mass of kg. Initially the loop lies flat on a horizontal tabletop. When a horizontal magnetic field is turned on, it is found that only one side of the loop experiences an upward force. Find the minimum magnetic field, B necessary to start tipping the loop up from the table. 85. Consider the physical system shown in Figure (a) Find the net magnetic field (direction and magnitude) at an arbitrary point on the bottom side of the square, a distance 0 < x < 50. cm to the right of wire 1. (b) Find the magnitude of the net magnetic field at an arbitrary point on the left side of the square, a distance min, 0< y < 50. cm above wire 1. 16

17 Interactive Problems 86. IP Referring to Example 22 3 Suppose the speed of the isotopes is doubled. (a) Does the separation distance, d, increase, decrease, or stay the same? Explain. (b) Find the separation distance for this case. 87. IP Referring to Example 22 3 Suppose we change the initial speed of 238 U, leaving everything else the same. (a) If we want the separation distance to be zero, should the initial speed of 238 U be increased or decreased? Explain. (b) Find the required initial speed. 88. Referring to Active Example 22 2 The current I 1 is adjusted until the magnetic field halfway between the wires has a magnitude of T and points into the page. Everything else in the system remains the same as in Active Example Find the magnitude and direction of I Referring to Active Example 22 2 The current I 2 is adjusted until the magnetic field 5.5 cm below wire 2 has a magnitude of T and points out of page. Everything else in the system remains the same as in Active Example Find the magnitude and direction of I 2. 17

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