Conceptual Questions. Problems. 852 CHAPTER 29 Magnetic Fields

Transcription

1 852 CHAPTER 29 Magnetic Fields magnitde crrent, and the niform magnetic field points in the positive direction. Rank the loops by the magnitde of the torqe eerted on them by the field from largest to smallest. y (m) A C (m) Figre OQ Conceptal Qestions 1. Two charged particles are projected in the same direction into a magnetic field perpendiclar to their velocities. f the particles are deflected in opposite directions, what can yo say abot them? 2. How can the motion of a moving charged particle be sed to distingish between a magnetic field and an electric field? Give a specific eample to jstify yor argment. 3. s it possible to orient a crrent loop in a niform magnetic field sch that the loop does not tend to rotate? Eplain. 4. Eplain why it is not possible to determine the charge and the mass of a charged particle separately by measring denotes answer available in tdent oltions Manal/tdy Gide accelerations prodced by electric and magnetic forces on the particle. 5. How can a crrent loop be sed to determine the presence of a magnetic field in a given region of space? 6. Charged particles from oter space, called cosmic rays, strike the Earth more freqently near the poles than near the eqator. Why? 7. Can a constant magnetic field set into motion an electron initially at rest? Eplain yor answer. Problems The problems fond in this chapter may be assigned online in Enhanced WebAssign 1. denotes straightforward problem; 2. denotes intermediate problem; 3. denotes challenging problem 1. fll soltion available in the tdent oltions Manal/tdy Gide 1. denotes problems most often assigned in Enhanced WebAssign; these provide stdents with targeted feedback and either a Master t ttorial or a Watch t soltion video. shaded denotes asking for qantitative and conceptal reasoning denotes symbolic reasoning problem denotes Master t ttorial available in Enhanced WebAssign denotes gided problem denotes paired problems that develop reasoning with symbols and nmerical vales ection 29.1 Magnetic Fields and Forces in p Problems 1 throgh 4 and 6 throgh 8 in Chapter 11 can be assigned with this section as review for the vector prodct. 1. A proton is projected into a magnetic field that is directed along the positive ais. Find the direction of the magnetic force eerted on the proton for each of the following directions of the proton s velocity: (a) the positive y direction, (b) the negative y direction, (c) the positive direction. a right b at Determine the initial direction of the deflection of charged particles as they enter the magnetic fields shown in Figre P Find the direction of the magnetic field acting on a positively charged particle moving in the varios sitations c Figre P29.2 d

2 Problems 853 shown in Figre P29.3 if the direction of the magnetic force acting on it is as indicated. F v F (ot) a b c v Figre P29.3 v (in) 4. Consider an electron near the Earth s eqator. n which direction does it tend to deflect if its velocity is (a) directed downward? (b) Directed northward? (c) Directed westward? (d) Directed sotheastward? 5. A proton travels with a speed of m/s in a direction that makes an angle of with the direction of a magnetic field of magnitde T in the positive direction. What are the magnitdes of (a) the magnetic force on the proton and (b) the proton s acceleration? 6. A proton moving at m/s throgh a magnetic field of magnitde 1.70 T eperiences a magnetic force of magnitde N. What is the angle between the proton s velocity and the field? 7. An electron is accelerated throgh V from rest and then enters a niform 1.70-T magnetic field. What are (a) the maimm and (b) the minimm vales of the magnetic force this particle eperiences? 8. A proton moves with a velocity of v 5 12 i^2 4 j^1 k^2 m/s in a region in which the magnetic field is 5 1i^1 2j^2 k^2t. What is the magnitde of the magnetic force this particle eperiences? 9. A proton moves perpendiclar to a niform magnetic field at a speed of m/s and eperiences an acceleration of m/s 2 in the positive direction when its velocity is in the positive z direction. Determine the magnitde and direction of the field. 10. A laboratory electromagnet prodces a magnetic field of magnitde 1.50 T. A proton moves throgh this field with a speed of m/s. (a) Find the magnitde of the maimm magnetic force that cold be eerted on the proton. (b) What is the magnitde of the maimm acceleration of the proton? (c) Wold the field eert the same magnetic force on an electron moving throgh the field with the same speed? (d) Wold the electron eperience the same acceleration? Eplain. 11. Review. A charged particle of mass 1.50 g is moving at a speed of m/s. ddenly, a niform magnetic field of magnitde mt in a direction perpendiclar to the particle s velocity is trned on and then trned off in a time interval of 1.00 s. Dring this time interval, the magnitde and direction of the velocity of the particle ndergo F a negligible change, bt the particle moves by a distance of m in a direction perpendiclar to the velocity. Find the charge on the particle. ection 29.2 Motion of a Charged Particle in a Uniform Magnetic Field 12. An electron moves in a circlar path perpendiclar to a niform magnetic field with a magnitde of 2.00 mt. f the speed of the electron is m/s, determine (a) the radis of the circlar path and (b) the time interval reqired to complete one revoltion. 13. A proton (charge 1e, mass m p ), a deteron (charge 1e, mass 2m p ), and an alpha particle (charge 12e, mass 4m p ) are accelerated from rest throgh a common potential difference DV. Each of the particles enters a niform magnetic field, with its velocity in a direction perpendiclar to. The proton moves in a circlar path of radis r p. n terms of r p, determine (a) the radis r d of the circlar orbit for the deteron and (b) the radis r a for the alpha particle. 14. An accelerating voltage of V is applied to an electron gn, prodcing a beam of electrons originally traveling horizontally north in vacm toward the center of a viewing screen 35.0 cm away. What are (a) the magnitde and (b) the direction of the deflection on the screen cased by the Earth s gravitational field? What are (c) the magnitde and (d) the direction of the deflection on the screen cased by the vertical component of the Earth s magnetic field, taken as 20.0 mt down? (e) Does an electron in this vertical magnetic field move as a projectile, with constant vector acceleration perpendiclar to a constant northward component of velocity? (f) s it a good approimation to assme it has this projectile motion? Eplain. 15. Review. One electron collides elastically with a second electron initially at rest. After the collision, the radii of their trajectories are 1.00 cm and 2.40 cm. The trajectories are perpendiclar to a niform magnetic field of magnitde T. Determine the energy (in kev) of the incident electron. 16. Review. One electron collides elastically with a second electron initially at rest. After the collision, the radii of their trajectories are r 1 and r 2. The trajectories are perpendiclar to a niform magnetic field of magnitde. Determine the energy of the incident electron. 17. Review. An electron moves in a circlar path perpendiclar to a constant magnetic field of magnitde 1.00 mt. The anglar momentm of the electron abot the center of the circle is kg? m 2 /s. Determine (a) the radis of the circlar path and (b) the speed of the electron. 18. A particle with charge q and kinetic energy K travels in a niform magnetic field of magnitde. f the particle moves in a circlar path of radis R, find epressions for (a) its speed and (b) its mass. 19. A cosmic-ray proton in interstellar space has an energy of 10.0 MeV and eectes a circlar orbit having a radis

3 854 CHAPTER 29 Magnetic Fields eqal to that of Mercry s orbit arond the n ( m). What is the magnetic field in that region of space? 20. Review. A 30.0-g metal ball having net charge Q mc is thrown ot of a window horizontally north at a speed v m/s. The window is at a height h m above the grond. A niform, horizontal magnetic field of magnitde T is perpendiclar to the plane of the ball s trajectory and directed toward the west. (a) Assming the ball follows the same trajectory as it wold in the absence of the magnetic field, find the magnetic force acting on the ball jst before it hits the grond. (b) ased on the reslt of part (a), is it jstified for threesignificant-digit precision to assme the trajectory is naffected by the magnetic field? Eplain. 21. A singly charged ion of mass m is accelerated from rest by a potential difference DV. t is then deflected by a niform magnetic field (perpendiclar to the ion s velocity) into a semicircle of radis R. Now a dobly charged ion of mass m9 is accelerated throgh the same potential difference and deflected by the same magnetic field into a semicircle of radis R9 5 2R. What is the ratio of the masses of the ions? 22. Assme the region to the right of a certain plane contains a niform magnetic field of magnitde 1.00 mt and the field is zero in the region to the left of the plane as shown in Figre P An electron, originally traveling perpendiclar to the bondary plane, passes into the region of the field. (a) Determine the time interval reqired for the electron to leave the field-filled region, noting that the electron s path is a semicircle. (b) Assming the maimm depth of penetration into the field is 2.00 cm, find the kinetic energy of the electron m. What are (a) the cyclotron freqency and (b) the maimm speed acqired by the protons? 25. A velocity selector consists of electric and magnetic fields described by the epressions E5 E k^ and 5 j^, with mt. Find the vale of E sch that a 750-eV electron moving in the negative direction is ndeflected. 26. ingly charged ranim-238 ions are accelerated throgh a potential difference of 2.00 kv and enter a niform magnetic field of magnitde 1.20 T directed perpendiclar to their velocities. (a) Determine the radis of their circlar path. (b) Repeat this calclation for ranim-235 ions. (c) What f? How does the ratio of these path radii depend on the accelerating voltage? (d) On the magnitde of the magnetic field? 27. A cyclotron (Fig ) designed to accelerate protons has an oter radis of m. The protons are emitted nearly at rest from a sorce at the center and are accelerated throgh 600 V each time they cross the gap between the dees. The dees are between the poles of an electromagnet where the field is T. (a) Find the cyclotron freqency for the protons in this cyclotron. Find (b) the speed at which protons eit the cyclotron and (c) their maimm kinetic energy. (d) How many revoltions does a proton make in the cyclotron? (e) For what time interval does the proton accelerate? 28. A particle in the cyclotron shown in Figre 29.16a gains energy q DV from the alternating power spply each time it passes from one dee to the other. The time interval for each fll orbit is T5 2p v 5 2pm q so the particle s average rate of increase in energy is e v mt Figre P29.22 ection 29.3 Applications nvolving Charged Particles Moving in a Magnetic Field 23. Consider the mass spectrometer shown schematically in Active Figre The magnitde of the electric field between the plates of the velocity selector is V/m, and the magnetic field in both the velocity selector and the deflection chamber has a magnitde of T. Calclate the radis of the path for a singly charged ion having a mass m kg. 24. A cyclotron designed to accelerate protons has a magnetic field of magnitde T over a region of radis 2q DV T 5 q 2 DV pm Notice that this power inpt is constant in time. On the other hand, the rate of increase in the radis r of its path is not constant. (a) how that the rate of increase in the radis r of the particle s path is given by dr dt 5 1 r DV p (b) Describe how the path of the particles in Figre 29.16a is consistent with the reslt of part (a). (c) At what rate is the radial position of the protons in a cyclotron increasing immediately before the protons leave the cyclotron? Assme the cyclotron has an oter radis of m, an accelerating voltage of DV V, and a magnetic field of magnitde T. (d) y how mch does the radis of the protons path increase dring their last fll revoltion? 29. The pictre tbe in an old black-and-white television ses magnetic deflection coils rather than electric deflection plates. ppose an electron beam is accelerated throgh a 50.0-kV potential difference and then throgh a region of

4 Problems 855 niform magnetic field 1.00 cm wide. The screen is located 10.0 cm from the center of the coils and is 50.0 cm wide. When the field is trned off, the electron beam hits the center of the screen. gnoring relativistic corrections, what field magnitde is necessary to deflect the beam to the side of the screen? 30. n his eperiments on cathode rays dring which he discovered the electron, J. J. Thomson showed that the same beam deflections reslted with tbes having cathodes made of different materials and containing varios gases before evacation. (a) Are these observations important? Eplain yor answer. (b) When he applied varios potential differences to the deflection plates and trned on the magnetic coils, alone or in combination with the deflection plates, Thomson observed that the florescent screen contined to show a single small glowing patch. Arge whether his observation is important. (c) Do calclations to show that the charge-to-mass ratio Thomson obtained was hge compared with that of any macroscopic object or of any ionized atom or molecle. How can one make sense of this comparison? (d) Cold Thomson observe any deflection of the beam de to gravitation? Do a calclation to arge for yor answer. Note: To obtain a visibly glowing patch on the florescent screen, the potential difference between the slits and the cathode mst be 100 V or more. ection 29.4 Magnetic Force Acting on a Crrent-Carrying Condctor 31. A condctor carrying a crrent A is directed along the positive ais and perpendiclar to a niform magnetic field. A magnetic force per nit length of N/m acts on the condctor in the negative y direction. Determine (a) the magnitde and (b) the direction of the magnetic field in the region throgh which the crrent passes. 32. A straight wire carrying a 3.00-A crrent is placed in a niform magnetic field of magnitde T directed perpendiclar to the wire. (a) Find the magnitde of the magnetic force on a section of the wire having a length of 14.0 cm. (b) Eplain why yo can t determine the direction of the magnetic force from the information given in the problem. 33. A wire carries a steady crrent of 2.40 A. A straight section of the wire is m long and lies along the ais within a niform magnetic field, k^ T. f the crrent is in the positive direction, what is the magnetic force on the section of wire? (a) the direction and (b) the magnitde of the minimm magnetic field needed to lift this wire vertically pward? 36. Why is the following sitation impossible? magine a copper wire with radis 1.00 mm encircling the Earth at its magnetic eqator, where the field direction is horizontal. A power spply delivers 100 MW to the wire to maintain a crrent in it, in a direction sch that the magnetic force from the Earth s magnetic field is pward. De to this force, the wire is levitated immediately above the grond. 37. Review. A rod of mass kg and radis 6.00 cm rests on two parallel rails (Fig. P29.37) that are d cm apart and L cm long. The rod carries a crrent of A in the direction shown and rolls along the rails withot slipping. A niform magnetic field of magnitde T is directed perpendiclar to the rod and the rails. f it starts from rest, what is the speed of the rod as it leaves the rails? d L Figre P29.37 Problems 37 and Review. A rod of mass m and radis R rests on two parallel rails (Fig. P29.37) that are a distance d apart and have a length L. The rod carries a crrent in the direction shown and rolls along the rails withot slipping. A niform magnetic field is directed perpendiclar to the rod and the rails. f it starts from rest, what is the speed of the rod as it leaves the rails? 39. A horizontal power line of length 58.0 m carries a crrent of 2.20 ka northward as shown in Figre P The Earth s magnetic field at this location has a magnitde of T. The field at this location is directed toward the north at an angle below the power line. Find (a) the magnitde and (b) the direction of the magnetic force on the power line. 34. A wire 2.80 m in length carries a crrent of 5.00 A in a region where a niform magnetic field has a magnitde of T. Calclate the magnitde of the magnetic force on the wire assming the angle between the magnetic field and the crrent is (a) 60.08, (b) 90.08, and (c) A wire having a mass per nit length of g/cm carries a 2.00-A crrent horizontally to the soth. What are Figre P29.39

5 856 CHAPTER 29 Magnetic Fields 40. Consider the system pictred in Figre P A 15.0-cm horizontal wire of mass 15.0 g is placed between two thin, vertical condctors, and a niform magnetic field acts perpendiclar to the page. The wire is free to move vertically withot friction on the two vertical condctors. When a 5.00-A crrent is directed as shown in the figre, the horizontal wire moves pward at constant velocity in the presence of gravity. (a) What forces act on the horizontal wire, and (b) nder what condition is the wire able to move pward at constant velocity? (c) Find the magnitde and direction of the minimm magnetic field reqired to move the wire at constant speed. (d) What happens if the magnetic field eceeds this minimm vale? three, withot frther calclation involving the magnetic field. z d y c b Figre P29.42 a 5.00 A 15.0 cm 5.00 A Figre P A 43. Assme the Earth s magnetic field is 52.0 mt northward at below the horizontal in Atlanta, Georgia. A tbe in a neon sign stretches between two diagonally opposite corners of a shop window which lies in a north soth vertical plane and carries crrent 35.0 ma. The crrent enters the tbe at the bottom soth corner of the shop s window. t eits at the opposite corner, which is 1.40 m farther north and m higher p. etween these two points, the glowing tbe spells ot DONUT. Determine the total vector magnetic force on the tbe. Hint: Yo may se the first important general statement presented in the Finalize section of Eample A strong magnet is placed nder a horizontal condcting ring of radis r that carries crrent as shown in Figre P f the magnetic field makes an angle with the vertical at the ring s location, what are (a) the magnitde and (b) the direction of the resltant magnetic force on the ring? N r Figre P n Figre P29.42, the cbe is 40.0 cm on each edge. For straight segments of wire ab, bc, cd, and da form a closed loop that carries a crrent A in the direction shown. A niform magnetic field of magnitde T is in the positive y direction. Determine the magnetic force vector on (a) ab, (b) bc, (c) cd, and (d) da. (e) Eplain how yo cold find the force eerted on the forth of these segments from the forces on the other ection 29.5 Torqe on a Crrent Loop in a Uniform Magnetic Field 44. A crrent of 17.0 ma is maintained in a single circlar loop of 2.00 m circmference. A magnetic field of T is directed parallel to the plane of the loop. (a) Calclate the magnetic moment of the loop. (b) What is the magnitde of the torqe eerted by the magnetic field on the loop? 45. A magnetized sewing needle has a magnetic moment of 9.70 ma? m 2. At its location, the Earth s magnetic field is 55.0 mt northward at below the horizontal. dentify the orientations of the needle that represent (a) the minimm potential energy and (b) the maimm potential energy of the needle field system. (c) How mch work mst be done on the system to move the needle from the minimm to the maimm potential energy orientation? 46. A 50.0-trn circlar coil of radis 5.00 cm can be oriented in any direction in a niform magnetic field having a magnitde of T. f the coil carries a crrent of 25.0 ma, find the magnitde of the maimm possible torqe eerted on the coil. 47. A rectanglar coil consists of N closely wrapped trns and has dimensions a m and b m. The coil is hinged along the y ais, and its plane makes an angle with the ais (Fig. P29.47). (a) What is the magnitde of the torqe eerted on the coil by a niform magnetic field T directed in the positive direction when the crrent is A in the direction shown? (b) What is the epected direction of rotation of the coil?

6 Problems The rotor in a certain electric motor is a flat, rectanglar coil with 80 trns of wire and dimensions 2.50 cm by 4.00 cm. The rotor rotates in a niform magnetic field of T. When the plane of the rotor is perpendiclar to the direction of the magnetic field, the rotor carries a crrent of 10.0 ma. n this orientation, the magnetic moment of the rotor is directed opposite the magnetic field. The rotor then trns throgh one-half revoltion. This process is repeated to case the rotor to trn steadily at an anglar speed of rev/min. (a) Find the maimm torqe acting on the rotor. (b) Find the peak power otpt of the motor. (c) Determine the amont of work performed by the magnetic field on the rotor in every fll revoltion. (d) What is the average power of the motor? 49. A wire is formed into a circle having a diameter of 10.0 cm and is placed in a niform magnetic field of 3.00 mt. The wire carries a crrent of 5.00 A. Find (a) the maimm torqe on the wire and (b) the range of potential energies of the wire field system for different orientations of the circle. 50. A rectanglar loop of wire has dimensions m by m. The loop is pivoted at the ais and lies in the y plane as shown in Figre P A niform magnetic field of magnitde 1.50 T is directed at an angle of with respect to the y ais with field lines parallel to the yz plane. The loop carries a crrent of A in the direction shown. (gnore gravitation.) We wish to evalate the torqe on the crrent loop. (a) What is the direction of the magnetic force eerted on wire segment ab? (b) What is the direction of the torqe associated with this force abot an ais throgh the origin? (c) What is the direction of the magnetic force eerted on segment cd? (d) What is the direction of the torqe associated with this force abot an ais throgh the origin? (e) Can the forces eamined in z d a y b Figre P29.47 a z c Figre P b y parts (a) and (c) combine to case the loop to rotate arond the ais? (f) Can they affect the motion of the loop in any way? Eplain. (g) What is the direction of the magnetic force eerted on segment bc? (h) What is the direction of the torqe associated with this force abot an ais throgh the origin? (i) What is the torqe on segment ad abot an ais throgh the origin? (j) From the point of view of Figre P29.50, once the loop is released from rest at the position shown, will it rotate clockwise or conterclockwise arond the ais? (k) Compte the magnitde of the magnetic moment of the loop. (l) What is the angle between the magnetic moment vector and the magnetic field? (m) Compte the torqe on the loop sing the reslts to parts (k) and (l). ection 29.6 The Hall Effect 51. n an eperiment designed to measre the Earth s magnetic field sing the Hall effect, a copper bar cm thick is positioned along an east west direction. Assme n electrons/m 3 and the plane of the bar is rotated to be perpendiclar to the direction of. f a crrent of 8.00 A in the condctor reslts in a Hall voltage of V, what is the magnitde of the Earth s magnetic field at this location? 52. A Hall-effect probe operates with a 120-mA crrent. When the probe is placed in a niform magnetic field of magnitde T, it prodces a Hall voltage of mv. (a) When it is sed to measre an nknown magnetic field, the Hall voltage is mv. What is the magnitde of the nknown field? (b) The thickness of the probe in the direction of is 2.00 mm. Find the density of the charge carriers, each of which has charge of magnitde e. Additional Problems 53. n Niels ohr s 1913 model of the hydrogen atom, the single electron is in a circlar orbit of radis m and its speed is m/s. (a) What is the magnitde of the magnetic moment de to the electron s motion? (b) f the electron moves in a horizontal circle, conterclockwise as seen from above, what is the direction of this magnetic moment vector? 54. Carbon-14 and carbon-12 ions (each with charge of magnitde e) are accelerated in a cyclotron. f the cyclotron has a magnetic field of magnitde 2.40 T, what is the difference in cyclotron freqencies for the two ions? 55. A particle with positive charge q C moves with a velocity v 5 12i^1 3j^2 k^2 m/s throgh a region where both a niform magnetic field and a niform electric field eist. (a) Calclate the total force on the moving particle (in nit-vector notation), taking 5 12i^1 4j^1 k^2 T and E 514i^2 j^2 2k^2 V/m. (b) What angle does the force vector make with the positive ais? 56. Heart lng machines and artificial kidney machines employ electromagnetic blood pmps. The blood is confined to an electrically inslating tbe, cylindrical in

7 858 CHAPTER 29 Magnetic Fields practice bt represented here for simplicity as a rectangle of interior width w and height h. Figre P29.56 shows a rectanglar section of blood within the tbe. Two electrodes fit into the top and the bottom of the tbe. The potential difference between them establishes an electric crrent throgh the blood, with crrent density J over the section of length L shown in Figre P A perpendiclar magnetic field eists in the same region. (a) Eplain why this arrangement prodces on the liqid a force that is directed along the length of the pipe. (b) how that the section of liqid in the magnetic field eperiences a pressre increase JL. (c) After the blood leaves the pmp, is it charged? (d) s it carrying crrent? (e) s it magnetized? (The same electromagnetic pmp can be sed for any flid that condcts electricity, sch as liqid sodim in a nclear reactor.) h w J L 57. Review. The pper portion of the circit in Figre P29.57 is fied. The horizontal wire at the bottom has a mass of 10.0 g and is 5.00 cm long. This wire hangs in the gravitational field of the Earth from identical light springs connected to the pper portion of the circit. The springs stretch cm nder the weight of the wire, and the circit has a total resistance of 12.0 V. When a magnetic field is trned on, directed ot of the page, the springs stretch an additional cm. Only the horizontal wire at the bottom of the circit is in the magnetic field. What is the magnitde of the magnetic field? 24.0 V Figre P29.56 of the figre. The particle travels along a heli whose radis decreases and whose pitch decreases as the particle moves into a stronger magnetic field. f the particle is moving to the right along the ais, its velocity in this direction will be redced to zero and it will be reflected from the right-hand side of the bottle, acting as a magnetic mirror. The particle ends p boncing back and forth between the ends of the bottle. (b) Eplain qalitatively why the aial velocity is redced to zero as the particle moves into the region of strong magnetic field at the end of the bottle. (c) Eplain why the tangential velocity increases as the particle approaches the end of the bottle. (d) Eplain why the orbiting particle has a magnetic dipole moment. 59. Review. A proton is at rest at the plane bondary of a region containing a niform magnetic field (Fig. P29.59). An alpha particle moving horizontally makes a head-on elastic collision with the proton. mmediately after the collision, both particles enter the magnetic field, moving perpendiclar to the direction of the field. The radis of the proton s trajectory is R. The mass of the alpha particle is for times that of the proton, and its charge is twice that of the proton. Find the radis of the alpha particle s trajectory. Proton v Alpha particle 0 Figre P Within a cylindrical region of space of radis 100 Mm, a magnetic field is niform with a magnitde 25.0 mt and oriented parallel to the ais of the cylinder. The magnetic field is zero otside this cylinder. A cosmic-ray proton traveling at one-tenth the speed of light is heading directly toward the center of the cylinder, moving perpendiclar to the cylinder s ais. (a) Find the radis of crvatre of the path the proton follows when it enters the region of the field. (b) Eplain whether the proton will arrive at the center of the cylinder cm Figre P29.57 ot 61. Review. A kg metal rod carrying a crrent of 10.0 A glides on two horizontal rails m apart. f the coefficient of kinetic friction between the rod and rails is 0.100, what vertical magnetic field is reqired to keep the rod moving at a constant speed? 58. Figre shows a charged particle traveling in a nonniform magnetic field forming a magnetic bottle. (a) Eplain why the positively charged particle in the figre mst be moving clockwise when viewed from the right 62. Review. A metal rod of mass m carrying a crrent glides on two horizontal rails a distance d apart. f the coefficient of kinetic friction between the rod and rails is m, what vertical magnetic field is reqired to keep the rod moving at a constant speed?

8 Problems A proton having an initial velvocity of 20.0i^ Mm/s enters a niform magnetic field of magnitde T with a direction perpendiclar to the proton s velocity. t leaves the field-filled region with velocity j Mm/s. Determine (a) the direction of the magnetic field, (b) the radis of crvatre of the proton s path while in the field, (c) the distance the proton traveled in the field, and (d) the time interval dring which the proton is in the field. 64. (a) A proton moving with velocity v 5 v i i^ eperiences a magnetic force F 5 F i j^. Eplain what yo can and cannot infer abot from this information. (b) What f? n terms of F i, what wold be the force on a proton in the same field moving with velocity v 52v i i^? (c) What wold be the force on an electron in the same field moving with velocity v 52v i i^? 65. A noncondcting sphere has mass 80.0 g and radis 20.0 cm. A flat, compact coil of wire with five trns is wrapped tightly arond it, with each trn concentric with the sphere. The sphere is placed on an inclined plane that slopes downward to the left (Fig. P29.65), making an angle with the horizontal so that the coil is parallel to the inclined plane. A niform magnetic field of T vertically pward eists in the region of the sphere. (a) What crrent in the coil will enable the sphere to rest in eqilibrim on the inclined plane? (b) how that the reslt does not depend on the vale of. Figre P Model the electric motor in a handheld electric mier as a single flat, compact, circlar coil carrying electric crrent in a region where a magnetic field is prodced by an eternal permanent magnet. Yo need consider only one instant in the operation of the motor. (We will consider motors again in Chapter 31.) Make order-of-magnitde estimates of (a) the magnetic field, (b) the torqe on the coil, (c) the crrent in the coil, (d) the coil s area, and (e) the nmber of trns in the coil. The inpt power to the motor is electric, given by P 5 DV, and the sefl otpt power is mechanical, P 5 tv. 67. Figre P29.67 shows a schematic representation of an apparats that can be sed to measre magnetic fields. A rectanglar coil of wire contains N trns and has a width w. The coil is attached to one arm of a balance and is sspended between the poles of a magnet. The magnetic field is niform and perpendiclar to the plane of the coil. The system is first balanced when the crrent in the coil is zero. When the switch is closed and the coil carries a crrent, a mass m mst be added to the right side to balance the system. (a) Find an epression for the magnitde of the magnetic field. (b) Why is the reslt independent of the vertical dimensions of the coil? (c) ppose the coil has 50 trns and a width of 5.00 cm. When the switch is closed, the coil carries a crrent of A, and a mass of 20.0 g mst be added to the right side to balance the system. What is the magnitde of the magnetic field? w Coil R 68. Why is the following sitation impossible? Figre P29.68 shows an eperimental techniqe for altering the direction of travel for a charged particle. A particle of charge q mc and mass m kg enters the bottom of the region of niform magnetic field at speed v m/s, with a velocity vector perpendiclar to the field lines. The magnetic force on the particle cases its direction of travel to change so that it leaves the region of the magnetic field at the top traveling at an angle from its original direction. The magnetic field has magnitde T and is directed ot of the page. The length h of the magnetic field region is m. An eperimenter performs the techniqe and measres the angle at which the particles eit the top of the field. he finds that the angles of deviation are eactly as predicted. h e Figre P29.67 v Figre P A metal rod having a mass per nit length l carries a crrent. The rod hangs from two wires in a niform vertical magnetic field as shown in Figre P The wires m

9 860 CHAPTER 29 Magnetic Fields make an angle with the vertical when in eqilibrim. Determine the magnitde of the magnetic field. 70. A heart srgeon monitors the flow rate of blood throgh an artery sing an electromagnetic flowmeter (Fig. P29.70). Electrodes A and make contact with the oter srface of the blood vessel, which has a diameter of 3.00 mm. (a) For a magnetic field magnitde of T, an emf of 160 mv appears between the electrodes. Calclate the speed of the blood. (b) Eplain why electrode A has to be positive as shown. (c) Does the sign of the emf depend on whether the mobile ions in the blood are predominantly positively or negatively charged? Eplain. Figre P29.69 g 72. Review. (a) how that a magnetic dipole in a niform magnetic field, displaced from its eqilibrim orientation and released, can oscillate as a torsional pendlm (ection 15.5) in simple harmonic motion. (b) s this statement tre for all anglar displacements, for all displacements less than 1808, or only for small anglar displacements? Eplain. (c) Assme the dipole is a compass needle a light bar magnet with a magnetic moment of magnitde m. t has moment of inertia abot its center, where it is monted on a frictionless, vertical ale, and it is placed in a horizontal magnetic field of magnitde. Determine its freqency of oscillation. (d) Eplain how the compass needle can be conveniently sed as an indicator of the magnitde of the eternal magnetic field. (e) f its freqency is Hz in the Earth s local field, with a horizontal component of 39.2 mt, what is the magnitde of a field parallel to the needle in which its freqency of oscillation is 4.90 Hz? 73. A niform magnetic field of magnitde T is directed along the positive ais. A positron moving at a speed of m/s enters the field along a direction that makes an angle of with the ais (Fig. P29.73). The motion of the particle is epected to be a heli as described in ection Calclate (a) the pitch p and (b) the radis r of the trajectory as defined in Figre P Artery y Electrodes N A lood flow To voltmeter z v r p Figre P The accompanying table shows measrements of the Hall voltage and corresponding magnetic field for a probe sed to measre magnetic fields. (a) Plot these data and dedce a relationship between the two variables. (b) f the measrements were taken with a crrent of A and the sample is made from a material having a charge- carrier density of carriers/m 3, what is the thickness of the sample? DV H (mv) (T) Challenge Problems Figre P Protons having a kinetic energy of 5.00 MeV (1 ev J) are moving in the positive direction and enter a magnetic field k^ T directed ot of the plane of the page and etending from 5 0 to m as shown in Figre P (a) gnoring relativistic effects, find the angle a between the initial velocity vector of the proton beam and the velocity vector after the beam p y Figre P29.74 e 1.00 m

10 Problems 861 emerges from the field. (b) Calclate the y component of the protons momenta as they leave the magnetic field. 75. Review. A wire having a linear mass density of 1.00 g/cm is placed on a horizontal srface that has a coefficient of kinetic friction of The wire carries a crrent of 1.50 A toward the east and slides horizontally to the north at constant velocity. What are (a) the magnitde and (b) the direction of the smallest magnetic field that enables the wire to move in this fashion? 76. A proton moving in the plane of the page has a kinetic energy of 6.00 MeV. A magnetic field of magnitde T is directed into the page. The proton enters the magnetic field with its velocity vector at an angle to the linear bondary of the field as shown in Figre P (a) Find, the distance from the point of entry to where the proton will leave the field. (b) Determine, the angle between the bondary and the proton s velocity vector as it leaves the field. p Figre P Consider an electron orbiting a proton and maintained in a fied circlar path of radis R m by the Colomb force. Treat the orbiting particle as a crrent loop. Calclate the reslting torqe when the electron proton system is placed in a magnetic field of T directed perpendiclar to the magnetic moment of the loop.