Birmingham Public Schools / Homepage



PROBLEMS

1, 2, 3 = straightforward, intermediate, challenging [pic] = full solution available in Student Solutions Manual/Study Guide [pic] = biomedical application

Section 19.3 Magnetic Fields

1. An electron gun fires electrons into a magnetic field that is directed straight downward. Find the direction of the force exerted by the field on an electron for each of the following directions of the electron’s velocity: (a) horizontal and due north; (b) horizontal and 30° west of north; (c) due north, but at 30° below the horizontal; (d) straight upward. (Remember that an electron has a negative charge.)

2. (a) Find the direction of the force on a proton (a positively charged particle) moving through the magnetic fields in Figure P19.2, as shown. (b) Repeat part (a), assuming the moving particle is an electron.

[pic]

Figure P19.2 (Problems 2 and 13) For Problem 13, replace the velocity vector with a current in that direction.

3. Find the direction of the magnetic field acting on the positively charged particle moving in the various situations shown in Figure P19.3, if the direction of the magnetic force acting on it is as indicated.

[pic]

Figure P19.3 (Problems 3 and 12) For Problem 12, replace the velocity vector with a current in that direction.

4. Determine the initial direction of the deflection of charged particles as they enter the magnetic fields shown in Figure P19.4.

[pic]

Figure P19.4

5. At the Equator near Earth’s surface, the magnetic field is approximately 50.0 μT northward and the electric field is about 100 N/C downward in fair weather. Find the gravitational, electric, and magnetic forces on an electron with an instantaneous velocity of 6.00 × 106 m/s directed to the east in this environment.

6. A proton travels with a speed of 3.0 × 106 m/s at an angle of 37° with the direction of a magnetic field of 0.30 T in the +y direction. What are (a) the magnitude of the magnetic force on the proton and (b) the proton’s acceleration?

7. What velocity would a proton need to circle Earth 1 000 km above the magnetic equator, where Earth’s magnetic field is directed horizontally north and has a magnitude of 4.00 × 10–8 T?

8. An electron is accelerated through 2 400 V from rest and then enters a region where there is a uniform 1.70-T magnetic field. What are the (a) maximum and (b) minimum magnitudes of the magnetic force this charge can experience?

9. A proton moves perpendicularly to a uniform magnetic field B at 1.0 × 107 m/s and experiences an acceleration of 2.0 × 1013 m/s2 in the +x direction when its velocity is in the +z direction. Determine the magnitude and direction of the field.

[pic]10. Sodium ions (Na+) move at 0.851 m/s through a bloodstream in the arm of a person standing near a large magnet. The magnetic field has a strength of 0.254 T and makes an angle of 51.0° with the motion of the sodium ions. The arm contains 100 cm3 of blood with 3.00 × 1020 Na+ ions per cubic centimeter. If no other ions were present in the arm, what would be the magnetic force on the arm?

Section 19.4 Magnetic Force on a Current-Carrying Conductor

11. A current I = 15 A is directed along the positive x axis and perpendicularly to a magnetic field. The conductor experiences a magnetic force per unit length of 0.12 N/m in the negative y direction. Calculate the magnitude and direction of the magnetic field in the region through which the current passes.

12. In Figure P19.3, assume that in each case the velocity vector shown is replaced with a wire carrying a current in the direction of the velocity vector. For each case, find the direction of the magnetic field that will produce the magnetic force shown.

13. In Figure P19.2, assume that in each case the velocity vector shown is replaced with a wire carrying a current in the direction of the velocity vector. For each case, find the direction of the magnetic force acting on the wire.

14. A wire carries a steady current of 2.40 A. A straight section of the wire is 0.750 m long and lies along the x axis within a uniform magnetic field of magnitude 1.60 T in the positive z direction. If the current is in the + x direction, what is the magnetic force on the section of wire?

15. A wire carries a current of 10.0 A in a direction that makes an angle of 30.0° with the direction of a magnetic field of strength 0.300 T. Find the magnetic force on a 5.00-m length of the wire.

16. At a certain location, Earth has a magnetic field of 0.60 × 10–4 T pointing 75° below the horizontal in a north-south plane. A 10.0-m-long straight wire carries a 15-A current. (a) If the current is directed horizontally toward the east, what are the magnitude and direction of the magnetic force on the wire? (b) What are the magnitude and direction of the force if the current is directed vertically upward?

17. A wire with a mass per unit length of 1.00 g/cm is placed on a horizontal surface with a coefficient of friction of 0.200. The wire carries a current of 1.50 A eastward and moves horizontally to the north. What are the magnitude and the direction of the smallest vertical magnetic field that enables the wire to move in this fashion?

18. A conductor suspended by two flexible wires as shown in Figure P19.18 has a mass per unit length of 0.040 0 kg/m. What current must exist in the conductor for the tension in the supporting wires to be zero when the magnetic field is 3.60 T into the page? What is the required direction for the current?

[pic]

Figure P19.18

19. An unusual message delivery system is pictured in Figure P19.19. A 15-cm length of conductor that is free to move is held in place between two thin conductors. When a 5.0-A current is directed as shown in the figure, the wire segment moves upward at a constant velocity. If the mass of the wire is 15 g, find the magnitude and direction of the minimum magnetic field that is required to move the wire. (The wire slides without friction on the two vertical conductors.)

[pic]

Figure P19.19

20. A thin, horizontal copper rod is 1.00 m long and has a mass of 50.0 g. What is the minimum current in the rod that can cause it to float in a horizontal magnetic field of 2.00 T?

21. In Figure P19.21, the cube is 40.0 cm on each edge. Four straight segments of wire—ab, bc, cd, and da—form a closed loop that carries a current I = 5.00 A, in the direction shown. A uniform magnetic field of magnitude B = 0.020 0 T is in the positive y direction. Determine the magnitude and direction of the magnetic force on each segment.

[pic]

Figure P19.21

Section 19.5 Torque on a Current Loop and Electric Motors

22. A current of 17.0 mA is maintained in a single circular loop with a circumference of 2.00 m. A magnetic field of 0.800 T is directed parallel to the plane of the loop. What is the magnitude of the torque exerted by the magnetic field on the loop?

23. An 8-turn coil encloses an elliptical area having a major axis of 40.0 cm and a minor axis of 30.0 cm (Fig. P19.23). The coil lies in the plane of the page and has a 6.00-A current flowing clockwise around it. If the coil is in a uniform magnetic field of 2.00 × 10–4 T, directed toward the left of the page, what is the magnitude of the torque on the coil? (Hint: The area of an ellipse is A = πab, where a and b are the semi-major and semi-minor axes of the ellipse.)

[pic]

Figure P19.23

24. A rectangular loop consists of 100 closely wrapped turns and has dimensions 0.40 m by 0.30 m. The loop is hinged along the y axis, and the plane of the coil makes an angle of 30.0° with the x axis (Fig. P19.24). What is the magnitude of the torque exerted on the loop by a uniform magnetic field of 0.80 T directed along the x axis, when the current in the windings has a value of 1.2 A in the direction shown? What is the expected direction of rotation of the loop?

[pic]

Figure P19.24

25. A long piece of wire with a mass of 0.100 kg and a total length of 4.00 m is used to make a square coil with a side of 0.100 m. The coil is hinged along a horizontal side, carries a 3.40-A current, and is placed in a vertical magnetic field with a magnitude of 0.010 0 T. (a) Determine the angle that the plane of the coil makes with the vertical when the coil is in equilibrium. (b) Find the torque acting on the coil due to the magnetic force at equilibrium.

26. A copper wire is 8.00 m long, and has a cross-sectional area of 1.00 × 10–4 m2. This wire forms a 1-turn loop in the shape of a square and is then connected to a battery that applies a potential difference of 0.100 V. If the loop is placed in a uniform magnetic field of magnitude 0.400 T, what is the maximum torque that can act on it? The resistivity of copper is 1.70 × 10–8 Ω · m.

Section 19.6 Motion of a Charged Particle in a Magnetic Field

27. A particle with a +2.0 μC charge and a kinetic energy of 0.090 J is fired into a uniform magnetic field of magnitude 0.10 T. If the particle moves in a circular path of radius 3.0 m, determine its mass.

28. A cosmic-ray proton in interstellar space has an energy of 10.0 MeV and executes a circular orbit having a radius equal to that of Mercury’s orbit around the Sun (5.80 × 1010 m). What is the magnetic field in that region of space?

29. Figure P19.29a is a diagram of a device called a velocity selector, in which particles of a specific velocity pass through undeflected but those with greater or lesser velocities are deflected either upward or downward. An electric field is directed perpendicularly to a magnetic field. This produces on the charged particle an electric force and a magnetic force that can be equal in magnitude and opposite in direction (Fig. P19.29b), and hence cancel. Show that particles with a speed of v = E/B will pass through undeflected.

[pic]

Figure P19.29

30. Consider the mass spectrometer shown schematically in Figure P19.30. The electric field between the plates of the velocity selector is 950 V/m, and the magnetic fields in both the velocity selector and the deflection chamber have magnitudes of 0.930 T. Calculate the radius of the path in the system for a singly charged ion with mass m = 2.18 × 10–26 kg. (Hint: See Problem 29.)

[pic]

Figure P19.30 A mass spectrometer. Charged particles are first sent through a velocity selector. They then enter a region where a magnetic field B0 (inward) causes positive ions to move in a semicircular path and strike a photographic film at P.

31. A singly charged positive ion has a mass of 2.50 × 10–26 kg. After being accelerated through a potential difference of 250 V, the ion enters a magnetic field of 0.500 T, in a direction perpendicular to the field. Calculate the radius of the ion’s path in the field.

32. A mass spectrometer is used to examine the isotopes of uranium. Ions in the beam emerge from the velocity selector at a speed of 3.00 × 105 m/s and enter a uniform magnetic field of 0.600 T directed perpendicularly to the velocity of the ions. What is the distance between the impact points formed on the photographic plate by singly charged ions of 235U and 238U?

33. An electron moves in a circular path perpendicular to a constant magnetic field with a magnitude of 1.00 mT. If the angular momentum of the electron about the center of the circle is 4.00 × 10–25 J · s, determine (a) the radius of the circular path and (b) the speed of the electron.

Section 19.7 Magnetic Field of a Long, Straight Wire and Ampère’s Law

34. Find the direction of the current in the wire in Figure P19.34 that would produce a magnetic field directed as shown, in each case.

[pic]

Figure P19.34

35. A lightning bolt may carry a current of 1.00 × 104 A for a short period of time. What is the resulting magnetic field 100 m from the bolt? Suppose that the bolt extends far above and below the point of observation.

36. In 1962, measurements of the magnetic field of a large tornado were made at the Geophysical Observatory in Tulsa, Oklahoma. If the tornado’s field was B = 1.50 × 10–8 T pointing north when the tornado was 9.00 km east of the observatory, what current was carried up or down the funnel of the tornado? Model the vortex as a long straight wire carrying a current.

37. At what distance from a long, straight wire carrying a current of 5.0 A is the magnetic field due to the wire equal to the strength of Earth’s field, approximately 5.0 × 10–5 T?

38. The two wires shown in Figure P19.38 carry currents of 5.00 A in opposite directions and are separated by 10.0 cm. Find the direction and magnitude of the net magnetic field (a) at a point midway between the wires, (b) at point P1 (10.0 cm to the right of the wire on the right), and (c) at point P2 (20.0 cm to the left of the wire on the left).

[pic]

Figure P19.38

39. Four long, parallel conductors carry equal currents of I = 5.00 A. Figure P19.39 is an end view of the conductors. The current direction is into the page at points A and B (indicated by the crosses) and out of the page at C and D (indicated by the dots). Calculate the magnitude and direction of the magnetic field at point P, located at the center of the square of edge length 0.200 m.

[pic]

Figure P19.39

40. The two wires in Figure P19.40 carry currents of 3.00 A and 5.00 A in the direction indicated. (a) Find the direction and magnitude of the magnetic field at a point midway between the wires. (b) Find the magnitude and direction of the magnetic field at point P, located 20.0 cm above the wire carrying the 5.00-A current.

[pic]

Figure P19.40

41. A wire carries a 7.00-A current along the x axis and another wire carries a 6.00-A current along the y axis as shown in Figure P19.41. What is the magnetic field at point P located at x = 4.00 m, y = 3.00 m?

[pic]

Figure P19.41

42. A long, straight wire lies on a horizontal table and carries a current of 1.20 μA. In a vacuum, a proton moves parallel to the wire (opposite the current) with a constant velocity of 2.30 × 104 m/s at a constant distance d above the wire. Determine the value of d. You may ignore the magnetic field due to Earth.

43. The magnetic field 40.0 cm away from a long, straight wire carrying current 2.00 A is 1.00 μT. (a) At what distance is it 0.100 μT? (b) At one instant, the two conductors in a long household extension cord carry equal 2.00-A currents in opposite directions. The two wires are 3.00 mm apart. Find the magnetic field 40.0 cm away from the middle of the straight cord, in the plane of the two wires. (c) At what distance is it one tenth as large? (d) The center wire in a coaxial cable carries current 2.00 A in one direction, and the sheath around it carries current 2.00 A in the opposite direction. What magnetic field does the cable create at points outside?

Section 19.8 Magnetic Force Between Two Parallel Conductors

44. Two parallel wires are 10.0 cm apart, and each carries a current of 10.0 A. (a) If the currents are in the same direction, find the force per unit length exerted by one of the wires on the other. Are the wires attracted or repelled? (b) Repeat the problem with the currents in opposite directions.

45. A wire with a weight per unit length of 0.080 N/m is suspended directly above a second wire. The top wire carries a current of 30.0 A, and the bottom wire carries a current of 60.0 A. Find the distance of separation between the wires so that the top wire will be held in place by magnetic repulsion.

46. In Figure P19.46, the current in the long, straight wire is I1 = 5.00 A, and the wire lies in the plane of the rectangular loop, which carries 10.0 A. The dimensions are c = 0.100 m, a = 0.150 m, and [pic] = 0.450 m. Find the magnitude and direction of the net force exerted by the magnetic field due to the straight wire on the loop.

[pic]

Figure P19.46

Section 19.10 Magnetic Field of a Solenoid

47. What current is required in the windings of a long solenoid that has 1 000 turns uniformly distributed over a length of 0.400 m in order to produce a magnetic field of magnitude 1.00 × 10–4 T at the center of the solenoid?

48. It is desired to construct a solenoid that has a resistance of 5.00 Ω (at 20°C) and that produces a magnetic field at its center of 4.00 × 10–2 T when it carries a current of 4.00 A. The solenoid is to be constructed from copper wire having a diameter of 0.500 mm. If the radius of the solenoid is to be 1.00 cm, determine (a) the number of turns of wire needed and (b) the length the solenoid should have.

49. A single-turn square loop of wire, 2.00 cm on a side, carries a counterclockwise current of 0.200 A. The loop is inside a solenoid, with the plane of the loop perpendicular to the magnetic field of the solenoid. The solenoid has 30 turns per centimeter and carries a counterclockwise current of 15.0 A. Find the force on each side of the loop and the torque acting on it.

50. An electron moves at a speed of 1.0 × 104 m/s in a circular path of radius of 2.0 cm inside a solenoid. The magnetic field of the solenoid is perpendicular to the plane of the electron’s path. Find (a) the strength of the magnetic field inside the solenoid and (b) the current in the solenoid if it has 25 turns per centimeter.

ADDITIONAL PROBLEMS

51. A circular coil consisting of a single loop of wire has a radius of 30.0 cm and carries a current of 25 A. It is placed in an external magnetic field of 0.30 T. Find the torque on the wire when the plane of the coil makes an angle of 35° with the direction of the field.

52. An electron enters a region of magnetic field of magnitude 0.010 0 T, traveling perpendicular to the linear boundary of the region. The direction of the field is perpendicular to the velocity of the electron. (a) Determine the time it takes for the electron to leave the “field-filled” region, noting that its path is a semicircle. (b) Find the kinetic energy of the electron if the radius of its semicircular path is 2.00 cm.

53. Two long, straight wires cross each other at right angles, as shown in Figure P19.53. (a) Find the direction and magnitude of the magnetic field at point P, which is in the same plane as the two wires. (b) Find the magnetic field at a point 30.0 cm above the point of intersection (30.0 cm out of the page, toward you).

[pic]

Figure P19.53

54. A 0.200-kg metal rod carrying a current of 10.0 A glides on two horizontal rails 0.500 m apart. What vertical magnetic field is required to keep the rod moving at a constant speed if the coefficient of kinetic friction between the rod and rails is 0.100?

55. Two species of singly charged positive ions of masses 20.0 × 10–27 kg and 23.4 × 10–27 kg enter a magnetic field at the same location with a speed of 1.00 × 105 m/s. If the strength of the field is 0.200 T, and the ions move perpendicularly to the field, find their distance of separation after they complete one half of their circular path.

56. Two parallel conductors carry currents in opposite directions, as shown in Figure P19.56. One conductor carries a current of 10.0 A. Point A is the midpoint between the wires, and point C is 5.00 cm to the right of the 10.0-A current. I is adjusted so that the magnetic field at C is zero. Find (a) the value of the current I and (b) the value of the magnetic field at A.

[pic]

Figure P19.56

[pic]57. A heart surgeon monitors the flow rate of blood through an artery using an electromagnetic flowmeter (shown schematically in Fig. P19.57). Electrodes A and B make contact with the outer surface of the blood vessel, which has interior diameter 3.00 mm. (a) For a magnetic field magnitude of 0.040 0 T, a potential difference of 160 μV appears between the electrodes. Calculate the speed of the blood. (b) Verify that electrode A is positive, as shown. Does the sign of the emf depend on whether the mobile ions in the blood are predominantly positively or negatively charged? Explain.

[pic]

Figure P19.57

58. Two circular loops are parallel, coaxial, and almost in contact, 1.00 mm apart (Fig. P19.58). Each loop is 10.0 cm in radius. The top loop carries a clockwise current of 140 A. The bottom loop carries a counterclockwise current of 140 A. (a) Calculate the magnetic force that the bottom loop exerts on the top loop. (b) The upper loop has a mass of 0.021 0 kg. Calculate its acceleration, assuming that the only forces acting on it are the force in part (a) and its weight. (Hint: The distance between the loops is small in comparison to the radius of curvature, so the loops may be treated as long, straight, parallel wires.)

[pic]

Figure P19.58

59. A 1.00-kg ball having net charge Q = 5.00 μC is thrown out of a window horizontally at a speed v = 20.0 m/s. The window is at a height h = 20.0 m above the ground. A uniform horizontal magnetic field of magnitude B = 0.010 0 T is perpendicular to the plane of the ball’s trajectory. Find the magnitude of the magnetic force acting on the ball just before it hits the ground. (Hint: Ignore magnetic forces in finding the ball’s final velocity.)

60. At the Fermilab accelerator in Batavia, Illinois, protons having momentum 4.80 × 10–16 kg · m/s are held in a circular orbit of radius 1.00 km by an upward magnetic field. What is the magnitude of this field?

61. Two long, parallel conductors carry currents I1 = 3.00 A and I2 = 3.00 A, both directed into the page in Figure P19.61. Determine the magnitude and direction of the resultant magnetic field at P.

[pic]

Figure P19.61

62. A uniform horizontal wire with a linear mass density of 0.50 g/m carries a 2.0-A current. It is placed in a constant magnetic field, with a strength of 4.0 × 10–3 T, that is horizontal and perpendicular to the wire. As the wire moves upward starting from rest, (a) what is its acceleration and (b) how long does it take to rise 50 cm? Neglect the magnetic field of Earth.

63. Three long, parallel conductors carry currents of I = 2.0 A. Figure P19.63 is an end view of the conductors, with each current coming out of the page. Given that a = 1.0 cm, determine the magnitude and direction of the magnetic field at points (a) A, (b) B, and (c) C.

[pic]

Figure P19.63

64. Two long, parallel wires, each with a mass per unit length of 40 g/m, are supported in a horizontal plane by 6.0-cm long strings, as shown in Figure P19.64. Each wire carries the same current I, causing the wires to repel each other so that the angle θ between the supporting strings is 16°. (a) Are the currents in the same or opposite directions? (b) Determine the magnitude of each current.

[pic]

Figure P19.64

65. Protons having a kinetic energy of 5.00 MeV are moving in the positive x direction and enter a magnetic field of 0.050 0 T in the z direction, out of the plane of the page, and extending from x = 0 to x = 1.00 m as in Figure P19.65. (a) Calculate the y component of the protons’ momentum as they leave the magnetic field. (b) Find the angle α between the initial velocity vector of the proton beam and the velocity vector after the beam emerges from the field. (Hint: Neglect relativistic effects and note that 1 eV = 1.60 × 10–19 J.)

[pic]

Figure P19.65

66. A straight wire of mass 10.0 g and length 5.0 cm is suspended from two identical springs that, in turn, form a closed circuit (Fig. P19.66). The springs stretch a distance of 0.50 cm under the weight of the wire. The circuit has a total resistance of 12 Ω. When a magnetic field is turned on, directed out of the page (indicated by the dots in Fig. P19.66), the springs are observed to stretch an additional 0.30 cm. What is the strength of the magnetic field? (The upper portion of the circuit is fixed.)

[pic]

Figure P19.66

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download