Review. A rod of mass 0.720 kg and radius 6.00 cm rests on

A spatially uniform magnetic field cannot exert a magnetic force on a particle in which of the following circumstances? There may be more than one correct statement. (a) The particle is charged. (b) The particle moves perpendicular to the magnetic field. (c) The particle moves parallel to the magnetic field. (d) The magnitude of the magnetic field changes with time. (e) The particle is at rest.

Rank the magnitudes of the forces exerted on the following particles from largest to smallest. In your ranking, display any cases of equality. (a) an electron moving at 1 Mm/s perpendicular to a 1-mT magnetic field (b) an electron moving at 1 Mm/s parallel to a 1-mT magnetic field (c) an electron moving at 2 Mm/s perpendicular to a 1-mT magnetic field (d) a proton moving at 1 Mm/s perpendicular to a 1-mT magnetic field (e) a proton moving at 1 Mm/s at a 458 angle to a 1-mT magnetic field

A particle with electric charge is fired into a region of space where the electric field is zero. It moves in a straight line. Can you conclude that the magnetic field in that region is zero? (a) Yes, you can. (b) No; the field might be perpendicular to the particles velocity. (c) No; the field might be parallel to the particles velocity. (d) No; the particle might need to have charge of the opposite sign to have a force exerted on it. (e) No; an observation of an object with electric charge gives no information about a magnetic field.

A proton moving horizontally enters a region where a uniform magnetic field is directed perpendicular to the protons velocity as shown in Figure OQ29.4. After the proton enters the field, does it (a) deflect downward, with its speed remaining constant; (b) deflect upward, moving in a semicircular path with constant speed, and exit the field moving to the left; (c) continue to move in the horizontal direction with constant velocity; (d) move in a circular orbit and become trapped by the field; or (e) deflect out of the plane of the paper?

At a certain instant, a proton is moving in the positive x direction through a magnetic field in the negative z direction. What is the direction of the magnetic force exerted on the proton? (a) positive z direction (b) negative z direction (c) positive y direction (d) negative y direction (e) The force is zero.

A thin copper rod 1.00 m long has a mass of 50.0 g. What is the minimum current in the rod that would allow it to levitate above the ground in a magnetic field of magnitude 0.100 T? (a) 1.20 A (b) 2.40 A (c) 4.90 A (d) 9.80 A (e) none of those answers

Electron A is fired horizontally with speed 1.00 Mm/s into a region where a vertical magnetic field exists. Electron B is fired along the same path with speed 2.00 Mm/s. (i) Which electron has a larger magnetic force exerted on it? (a) A does. (b) B does. (c) The forces have the same nonzero magnitude. (d) The forces are both zero. (ii) Which electron has a path that curves more sharply? (a) A does. (b) B does. (c) The particles follow the same curved path. (d) The particles continue to go straight.

Classify each of the following statements as a characteristic (a) of electric forces only, (b) of magnetic forces only, (c) of both electric and magnetic forces, or (d) of neither electric nor magnetic forces. (i) The force is proportional to the magnitude of the field exerting it. (ii) The force is proportional to the magnitude of the charge of the object on which the force is exerted. (iii) The force exerted on a negatively charged object is opposite in direction to the force on a positive charge. (iv) The force exerted on a stationary charged object is nonzero. (v) The force exerted on a moving charged object is zero. (vi) The force exerted on a charged object is proportional to its speed. (vii) The force exerted on a charged object cannot alter the objects speed. (viii) The magnitude of the force depends on the charged objects direction of motion.

An electron moves horizontally across the Earths equator at a speed of 2.50 3 106 m/s and in a direction 35.08 N of E. At this point, the Earths magnetic field has a direction due north, is parallel to the surface, and has a value of 3.00 3 1025 T. What is the force acting on the electron due to its interaction with the Earths magnetic field? (a) 6.88 3 10218 N due west (b) 6.88 3 10218 N toward the Earths surface (c) 9.83 3 10218 N toward the Earths surface (d) 9.83 3 10218 N away from the Earths surface (e) 4.00 3 10218 N away from the Earths surface

A charged particle is traveling through a uniform magnetic field. Which of the following statements are true of the magnetic field? There may be more than one correct statement. (a) It exerts a force on the particle parallel to the field. (b) It exerts a force on the particle along the direction of its motion. (c) It increases the kinetic energy of the particle. (d) It exerts a force that is perpendicular to the direction of motion. (e) It does not change the magnitude of the momentum of the particle.

In the velocity selector shown in Figure 29.13, electrons with speed v 5 E/B follow a straight path. Electrons moving significantly faster than this speed through the same selector will move along what kind of path? (a) a circle (b) a parabola (c) a straight line (d) a more complicated trajectory

Answer each question yes or no. Assume the motions and currents mentioned are along the x axis and fields are in the y direction. (a) Does an electric field exert a force on a stationary charged object? (b) Does a magnetic field do so? (c) Does an electric field exert a force on a moving charged object? (d) Does a magnetic field do so? (e) Does an electric field exert a force on a straight current-carrying wire? (f) Does a magnetic field do so? (g) Does an electric field exert a force on a beam of moving electrons? (h) Does a magnetic field do so?

A magnetic field exerts a torque on each of the currentcarrying single loops of wire shown in Figure OQ29.13. The loops lie in the xy plane, each carrying the same magnitude current, and the uniform magnetic field points in the positive x direction. Rank the loops by the magnitude of the torque exerted on them by the field from largest to smallest.

An accelerating voltage of 2.50 3 103 V is applied to an electron gun, producing a beam of electrons originally traveling horizontally north in vacuum toward the center of a viewing screen 35.0 cm away. What are (a) the magnitude and (b) the direction of the deflection on the screen caused by the Earths gravitational field? What are (c) the magnitude and (d) the direction of the deflection on the screen caused by the vertical component of the Earths 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 perpendicular to a constant northward component of velocity? (f) Is it a good approximation to assume it has this projectile motion? Explain.

A proton (charge 1e, mass mp), a deuteron (charge 1e, mass 2mp), and an alpha particle (charge 12e, mass 4mp) are accelerated from rest through a common potential difference DV. Each of the particles enters a uniform magnetic field B S, with its velocity in a direction perpendicular to B S. The proton moves in a circular path of radius rp. In terms of rp, determine (a) the radius rd of the circular orbit for the deuteron and (b) the radius ra for the alpha particle.

A particle with charge q and kinetic energy K travels in a uniform magnetic field of magnitude B. If the particle moves in a circular path of radius R, find expressions for (a) its speed and (b) its mass.

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 perpendicular to a uniform magnetic field of magnitude 0.044 0 T. Determine the energy (in keV) of the incident electron.

Review. One electron collides elastically with a second electron initially at rest. After the collision, the radii of their trajectories are r1 and r2. The trajectories are perpendicular to a uniform magnetic field of magnitude B. Determine the energy of the incident electron.

Review. An electron moves in a circular path perpendicular to a constant magnetic field of magnitude 1.00 mT. The angular momentum of the electron about the center of the circle is 4.00 3 10225 kg ? m2/s. Determine (a) the radius of the circular path and (b) the speed of the electron.

Review. A 30.0-g metal ball having net charge Q 5 5.00 mC is thrown out of a window horizontally north at a speed v 5 20.0 m/s. The window is at a height h 5 20.0 m above the ground. A uniform, horizontal magnetic field of magnitude B 5 0.010 0 T is perpendicular to the plane of the balls trajectory and directed toward the west. (a) Assuming the ball follows the same trajectory as it would in the absence of the magnetic field, find the magnetic force acting on the ball just before it hits the ground. (b) Based on the result of part (a), is it justified for three-significant-digit precision to assume the trajectory is unaffected by the magnetic field? Explain.

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 Mercurys orbit around the Sun (5.80 3 1010 m). What is the magnetic field in that region of space?

Assume the region to the right of a certain plane contains a uniform magnetic field of magnitude 1.00 mT and the field is zero in the region to the left of the plane as shown in Figure P29.22. An electron, originally traveling perpendicular to the boundary plane, passes into the region of the field. (a) Determine the time interval required for the electron to leave the field-filled region, noting that the electrons path is a semicircle. (b) Assuming the maximum depth of penetration into the field is 2.00 cm, find the kinetic energy of the electron.

A singly charged ion of mass m is accelerated from rest by a potential difference DV. It is then deflected by a uniform magnetic field (perpendicular to the ions velocity) into a semicircle of radius R. Now a doubly charged ion of mass m9 is accelerated through the same potential difference and deflected by the same magnetic field into a semicircle of radius R9 5 2R. What is the ratio of the masses of the ions?

A cyclotron designed to accelerate protons has a magnetic field of magnitude 0.450 T over a region of radius 1.20 m. What are (a) the cyclotron frequency and (b) the maximum speed acquired by the protons?

Consider the mass spectrometer shown schematically in Figure 29.14. The magnitude of the electric field between the plates of the velocity selector is 2.50 3 103 V/m, and the magnetic field in both the velocity selector and the deflection chamber has a magnitude of 0.035 0 T. Calculate the radius of the path for a singly charged ion having a mass m 5 2.18 3 10226 kg.

Singly charged uranium-238 ions are accelerated through a potential difference of 2.00 kV and enter a uniform magnetic field of magnitude 1.20 T directed perpendicular to their velocities. (a) Determine the radius of their circular path. (b) Repeat this calculation for uranium-235 ions. (c) What If? How does the ratio of these path radii depend on the accelerating voltage? (d) On the magnitude of the magnetic field?

A cyclotron (Fig. 29.16) designed to accelerate protons has an outer radius of 0.350 m. The protons are emitted nearly at rest from a source at the center and are accelerated through 600 V each time they cross the gap between the dees. The dees are between the poles of an electromagnet where the field is 0.800 T. (a) Find the cyclotron frequency for the protons in this cyclotron. Find (b) the speed at which protons exit the cyclotron and (c) their maximum kinetic energy. (d) How many revolutions does a proton make in the cyclotron? (e) For what time interval does the proton accelerate?

A particle in the cyclotron shown in Figure 29.16a gains energy q DV from the alternating power supply each time it passes from one dee to the other. The time interval for each full orbit is T 5 2p v 5 2pm qB so the particles average rate of increase in energy is 2q DV T 5 q 2 B DV pm Notice that this power input is constant in time. On the other hand, the rate of increase in the radius r of its path is not constant. (a) Show that the rate of increase in the radius r of the particles path is given by dr dt 5 1 r DV pB (b) Describe how the path of the particles in Figure 29.16a is consistent with the result 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? Assume the cyclotron has an outer radius of 0.350 m, an accelerating voltage of DV 5 600 V, and a magnetic field of magnitude 0.800 T. (d) By how much does the radius of the protons path increase during their last full revolution?

A velocity selector consists of electric and magnetic fields described by the expressions E S 5 E k^ and B S 5 B j ^, with B 5 15.0 mT. Find the value of E such that a 750-eV electron moving in the negative x direction is undeflected.

In his experiments on cathode rays during which he discovered the electron, J. J. Thomson showed that the same beam deflections resulted with tubes having cathodes made of different materials and containing various gases before evacuation. (a) Are these observations important? Explain your answer. (b) When he applied various potential differences to the deflection plates and turned on the magnetic coils, alone or in combination with the deflection plates, Thomson observed that the fluorescent screen continued to show a single small glowing patch. Argue whether his observation is important. (c) Do calculations to show that the charge-tomass ratio Thomson obtained was huge compared with that of any macroscopic object or of any ionized atom or molecule. How can one make sense of this comparison? (d) Could Thomson observe any deflection of the beam due to gravitation? Do a calculation to argue for your answer. Note: To obtain a visibly glowing patch on the fluorescent screen, the potential difference between the slits and the cathode must be 100 V or more.

The picture tube in an old black-and-white television uses magnetic deflection coils rather than electric deflection plates. Suppose an electron beam is accelerated through a 50.0-kV potential difference and then through a region of uniform 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 turned off, the electron beam hits the center of the screen. Ignoring relativistic corrections, what field magnitude is necessary to deflect the beam to the side of the screen?

A straight wire carrying a 3.00-A current is placed in a uniform magnetic field of magnitude 0.280 T directed perpendicular to the wire. (a) Find the magnitude of the magnetic force on a section of the wire having a length of 14.0 cm. (b) Explain why you cant determine the direction of the magnetic force from the information given in the problem.

A conductor carrying a current I 5 15.0 A is directed along the positive x axis and perpendicular to a uniform magnetic field. A magnetic force per unit length of 0.120 N/m acts on the conductor in the negative y direction. Determine (a) the magnitude and (b) the direction of the magnetic field in the region through which the current passes.

A wire 2.80 m in length carries a current of 5.00 A in a region where a uniform magnetic field has a magnitude of 0.390 T. Calculate the magnitude of the magnetic force on the wire assuming the angle between the magnetic field and the current is (a) 60.08, (b) 90.08, and (c) 1208.

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, B S 5 1.60k^ T. If the current is in the positive x direction, what is the magnetic force on the section of wire?

Why is the following situation impossible? Imagine a copper wire with radius 1.00 mm encircling the Earth at its magnetic equator, where the field direction is horizontal. A power supply delivers 100 MW to the wire to maintain a current in it, in a direction such that the magnetic force from the Earths magnetic field is upward. Due to this force, the wire is levitated immediately above the ground.

Review. A rod of mass 0.720 kg and radius 6.00 cm rests on two parallel rails (Fig. P29.37) that are d 5 12.0 cm apart and L 5 45.0 cm long. The rod carries a current of I 5 48.0A in the direction shown and rolls along the rails without slipping. A uniform magnetic field of magnitude 0.240T is directed perpendicular to the rod and the rails. If it starts from rest, what is the speed of the rod as it leaves the rails?

Review. A rod of mass m and radius R rests on two parallel rails (Fig. P29.37) that are a distance d apart and have a length L. The rod carries a current I in the direction shown and rolls along the rails without slipping. A uniform magnetic field B is directed perpendicular to the rod and the rails. If it starts from rest, what is the speed of the rod as it leaves the rails?

A wire having a mass per unit length of 0.500 g/cm carries a 2.00-A current horizontally to the south. What are (a) the direction and (b) the magnitude of the minimum magnetic field needed to lift this wire vertically upward?

Consider the system pictured in Figure P29.40. A 15.0-cm horizontal wire of mass 15.0 g is placed between two thin, vertical conductors, and a uniform magnetic field acts perpendicular to the page. The wire is free to move vertically without friction on the two vertical conductors. When a 5.00-A current is directed as shown in the figure, the horizontal wire moves upward at constant velocity in the presence of gravity. (a) What forces act on the horizontal wire, and (b) under what condition is the wire able to move upward at constant velocity? (c) Find the magnitude and direction of the minimum magnetic field required to move the wire at constant speed. (d) What happens if the magnetic field exceeds this minimum value?

A horizontal power line of length 58.0 m carries a current of 2.20 kA northward as shown in Figure P29.41. The Earths magnetic field at this location has a magnitude of 5.00 3 1025 T. The field at this location is directed toward the north at an angle 65.08 below the power line. Find (a)the magnitude and (b) the direction of the magnetic force on the power line.

A strong magnet is placed under a horizontal conducting ring of radius r that carries current I as shown in Figure P29.42. If the magnetic field B S makes an angle u with the vertical at the rings location, what are (a) the magnitude and (b) the direction of the resultant magnetic force on the ring?

Assume the Earths magnetic field is 52.0 mT northward at 60.08 below the horizontal in Atlanta, Georgia. A tube in a neon sign stretches between two diagonally opposite corners of a shop windowwhich lies in a northsouth vertical planeand carries current 35.0 mA. The current enters the tube at the bottom south corner of the shops window. It exits at the opposite corner, which is 1.40 m farther north and 0.850 m higher up. Between these two points, the glowing tube spells out DONUTS. Determine the total vector magnetic force on the tube. Hint: You may use the first important general statement presented in the Finalize section of Example 29.4.

In Figure P29.44, the cube is 40.0 cm on each edge. Four straight segments of wireab, bc, cd, and da form a closed loop that carries a current I 5 5.00 A in the direction shown. A uniform magnetic field of magnitude B 5 0.020 0 T is in the positive y direction. Determine the magnetic force vector on (a) ab, (b) bc, (c) cd, and (d) da. (e) Explain how you could find the force exerted on the fourth of these segments from the forces on the other three, without further calculation involving the magnetic field.

A typical magnitude of the external magnetic field in a cardiac catheter ablation procedure using remote magnetic navigation is B 5 0.080 T. Suppose that the permanent magnet in the catheter used in the procedure is inside the left atrium of the heart and subject to this external magnetic field. The permanent magnet has a magnetic moment of 0.10 A ? m2. The orientation of the permanent magnet is 30 from the direction of the external magnetic field lines. (a) What is the magnitude of the torque on the tip of the catheter containing this permanent magnet? (b) What is the potential energy of the system consisting of the permanent magnet in the catheter and the magnetic field provided by the external magnets?

A 50.0-turn circular coil of radius 5.00 cm can be oriented in any direction in a uniform magnetic field having a magnitude of 0.500 T. If the coil carries a current of 25.0 mA, find the magnitude of the maximum possible torque exerted on the coil.

A magnetized sewing needle has a magnetic moment of 9.70 mA ? m2. At its location, the Earths magnetic field is 55.0 mT northward at 48.08 below the horizontal. Identify the orientations of the needle that represent (a) the minimum potential energy and (b) the maximum potential energy of the needlefield system. (c) How much work must be done on the system to move the needle from the minimum to the maximum potential energy orientation?

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

An eight-turn coil encloses an elliptical area having a major axis of 40.0 cm and a minor axis of 30.0 cm (Fig. P29.49). 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 3 1024 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 5 pab, where a and b are, respectively, the semimajor and semiminor axes of the ellipse.

The rotor in a certain electric motor is a flat, rectangular coil with 80 turns of wire and dimensions 2.50 cm by 4.00 cm. The rotor rotates in a uniform magnetic field of 0.800 T. When the plane of the rotor is perpendicular to the direction of the magnetic field, the rotor carries a current of 10.0 mA. In this orientation, the magnetic moment of the rotor is directed opposite the magnetic field. The rotor then turns through onehalf revolution. This process is repeated to cause the rotor to turn steadily at an angular speed of 3.60 3 103rev/min. (a) Find the maximum torque acting on the rotor. (b) Find the peak power output of the motor. (c)Determine the amount of work performed by the magnetic field on the rotor in every full revolution. (d)What is the average power of the motor?

A rectangular coil consists of N 5 100 closely wrapped turns and has dimensions a 5 0.400 m and b 5 0.300 m. The coil is hinged along the y axis, and its plane makes an angle u 5 30.08 with the x axis (Fig. P29.51). (a) What is the magnitude of the torque exerted on the coil by a uniform magnetic field B 5 0.800 T directed in the positive x direction when the current is I 5 1.20 A in the direction shown? (b) What is the expected direction of rotation of the coil?

A rectangular loop of wire has dimensions 0.500 m by 0.300 m. The loop is pivoted at the x axis and lies in the xy plane as shown in Figure P29.52. A uniform magnetic field of magnitude 1.50 T is directed at an angle of 40.08 with respect to the y axis with field lines parallel to the yz plane. The loop carries a current of 0.900 A in the direction shown. (Ignore gravitation.) We wish to evaluate the torque on the current loop. (a) What is the direction of the magnetic force exerted on wire segment ab? (b) What is the direction of the torque associated with this force about an axis through the origin? (c) What is the direction of the magnetic force exerted on segment cd? (d) What is the direction of the torque associated with this force about an axis through the origin? (e) Can the forces examined in parts (a) and (c) combine to cause the loop to rotate around the x axis? (f) Can they affect the motion of the loop in any way? Explain. (g) What is the direction of the magnetic force exerted on segment bc? (h) What is the direction of the torque associated with this force about an axis through the origin? (i) What is the torque on segment ad about an axis through the origin? ( j) From the point of view of Figure P29.52, once the loop is released from rest at the position shown, will it rotate clockwise or counterclockwise around the x axis? (k)Compute the magnitude of the magnetic moment of the loop. (l) What is the angle between the magnetic moment vector and the magnetic field? (m) Compute the torque on the loop using the results to parts (k) and(l).

A wire is formed into a circle having a diameter of 10.0 cm and is placed in a uniform magnetic field of 3.00 mT. The wire carries a current of 5.00 A. Find (a) the maximum torque on the wire and (b) the range of potential energies of the wirefield system for different orientations of the circle.

A Hall-effect probe operates with a 120-mA current. When the probe is placed in a uniform magnetic field of magnitude 0.080 0 T, it produces a Hall voltage of 0.700 mV. (a) When it is used to measure an unknown magnetic field, the Hall voltage is 0.330 mV. What is the magnitude of the unknown field? (b) The thickness of the probe in the direction of B S is 2.00 mm. Find the density of the charge carriers, each of which has charge of magnitude e.

In an experiment designed to measure the Earths magnetic field using the Hall effect, a copper bar 0.500 cm thick is positioned along an eastwest direction. Assume n 5 8.46 3 1028 electrons/m3 and the plane of the bar is rotated to be perpendicular to the direction of B S. If a current of 8.00 A in the conductor results in a Hall voltage of 5.10 3 10212 V, what is the magnitude of the Earths magnetic field at this location?

Carbon-14 and carbon-12 ions (each with charge of magnitude e) are accelerated in a cyclotron. If the cyclotron has a magnetic field of magnitude 2.40 T, what is the difference in cyclotron frequencies for the two ions?

In Niels Bohrs 1913 model of the hydrogen atom, the single electron is in a circular orbit of radius 5.29 3 10211 m and its speed is 2.19 3 106 m/s. (a) What is the magnitude of the magnetic moment due to the electrons motion? (b) If the electron moves in a horizontal circle, counterclockwise as seen from above, what is the direction of this magnetic moment vector?

Heartlung machines and artificial kidney machines employ electromagnetic blood pumps. The blood is confined to an electrically insulating tube, cylindrical in practice but represented here for simplicity as a rectangle of interior width w and height h. Figure P29.58 shows a rectangular section of blood within the tube. Two electrodes fit into the top and the bottom of the tube. The potential difference between them establishes an electric current through the blood, with current density J over the section of length L shown in Figure P29.58. A perpendicular magnetic field exists in the same region. (a) Explain why this arrangement produces on the liquid a force that is directed along the length of the pipe. (b) Show that the section of liquid in the magnetic field experiences a pressure increase JLB. (c) After the blood leaves the pump, is it charged? (d) Is it carrying current? (e) Is it magnetized? (The same electromagnetic pump can be used for any fluid that conducts electricity, such as liquid sodium in a nuclear reactor.)

A particle with positive charge q 5 3.20 3 10219 C moves with a velocity v S 5 12i ^ 1 3j ^ 2 k^ 2 m/s through a region where both a uniform magnetic field and a uniform electric field exist. (a) Calculate the total force on the moving particle (in unit-vector notation), taking B S 5 12 i ^ 1 4j ^ 1 k^ 2 T and E S 5 14i ^ 2 j ^ 2 2k^ 2 V/m. (b) What angle does the force vector make with the positive x axis?

Figure 29.11 shows a charged particle traveling in a nonuniform magnetic field forming a magnetic bottle. (a) Explain why the positively charged particle in the figure must be moving clockwise when viewed from the right of the figure. The particle travels along a helix whose radius decreases and whose pitch decreases as the particle moves into a stronger magnetic field. If the particle is moving to the right along the x axis, its velocity in this direction will be reduced to zero and it will be reflected from the right-hand side of the bottle, acting as a magnetic mirror. The particle ends up bouncing back and forth between the ends of the bottle. (b) Explain qualitatively why the axial velocity is reduced to zero as the particle moves into the region of strong magnetic field at the end of the bottle. (c) Explain why the tangential velocity increases as the particle approaches the end of the bottle. (d) Explain why the orbiting particle has a magnetic dipole moment.

Review. The upper portion of the circuit in Figure P29.61 is fixed. 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 upper portion of the circuit. The springs stretch 0.500 cm under the weight of the wire, and the circuit has a total resistance of 12.0 V. When a magnetic field is turned on, directed out of the page, the springs stretch an additional 0.300 cm. Only the horizontal wire at the bottom of the circuit is in the magnetic field. What is the magnitude of the magnetic field?

Within a cylindrical region of space of radius 100 Mm, a magnetic field is uniform with a magnitude 25.0 mT and oriented parallel to the axis of the cylinder. The magnetic field is zero outside this cylinder. A cosmicray proton traveling at one-tenth the speed of light is heading directly toward the center of the cylinder, moving perpendicular to the cylinders axis. (a) Find the radius of curvature of the path the proton follows when it enters the region of the field. (b) Explain whether the proton will arrive at the center of the cylinder.

Review. A proton is at rest at the plane boundary of a region containing a uniform magnetic field B (Fig. P29.63). An alpha particle moving horizontally makes a head-on elastic collision with the proton. Immediately after the collision, both particles enter the magnetic field, moving perpendicular to the direction of the field. The radius of the protons trajectory is R. The mass of the alpha particle is four times that of the proton, and its charge is twice that of the proton. Find the radius of the alpha particles trajectory.

(a) A proton moving with velocity v S 5 vi i ^ experiences a magnetic force F S 5 Fi j ^. Explain what you can and cannot infer about B S from this information. (b) What If? In terms of Fi, what would be the force on a proton in the same field moving with velocity v S 5 2vi i ^? (c) What would be the force on an electron in the same field moving with velocity v S 5 2vi i ^?

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

Review. A metal rod of mass m carrying a current I glides on two horizontal rails a distance d apart. If the coefficient of kinetic friction between the rod and rails is m, what vertical magnetic field is required to keep the rod moving at a constant speed?

A proton having an initial velocity of 20.0i ^ Mm/s enters a uniform magnetic field of magnitude 0.300 T with a direction perpendicular to the protons velocity. It leaves the field-filled region with velocity 220.0 j S Mm/s. Determine (a) the direction of the magnetic field, (b) the radius of curvature of the protons path while in the field, (c) the distance the proton traveled in the field, and (d) the time interval during which the proton is in the field.

Model the electric motor in a handheld electric mixer as a single flat, compact, circular coil carrying electric current in a region where a magnetic field is produced by an external permanent magnet. You need consider only one instant in the operation of the motor. (We will consider motors again in Chapter 31.) Make order-ofmagnitude estimates of (a) the magnetic field, (b) the torque on the coil, (c) the current in the coil, (d) the coils area, and (e) the number of turns in the coil. The input power to the motor is electric, given by P 5 I DV, and the useful output power is mechanical, P 5 tv.

A nonconducting sphere has mass 80.0 g and radius 20.0 cm. A flat, compact coil of wire with five turns is wrapped tightly around it, with each turn concentric with the sphere. The sphere is placed on an inclined plane that slopes downward to the left (Fig. P29.69), making an angle u with the horizontal so that the coil is parallel to the inclined plane. A uniform magnetic field of 0.350 T vertically upward exists in the region of the sphere. (a) What current in the coil will enable the sphere to rest in equilibrium on the inclined plane? (b) Show that the result does not depend on the value of u.

Why is the following situation impossible? Figure P29.70 shows an experimental technique for altering the direction of travel for a charged particle. A particle of charge q 5 1.00 mC and mass m 5 2.00 3 10213 kg enters the bottom of the region of uniform magnetic field at speed v 5 2.00 3 105 m/s, with a velocity vector perpendicular to the field lines. The magnetic force on the particle causes 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 magnitude B 5 0.400 T and is directed out of the page. The length h of the magnetic field region is 0.110m. An experimenter performs the technique and measures the angle u at which the particles exit the top of the field. She finds that the angles of deviation are exactly as predicted.

Figure P29.71 shows a schematic representation of an apparatus that can be used to measure magnetic fields. A rectangular coil of wire contains N turns and has a width w. The coil is attached to one arm of a balance and is suspended between the poles of a magnet. The magnetic field is uniform and perpendicular to the plane of the coil. The system is first balanced when the current in the coil is zero. When the switch is closed and the coil carries a current I, a mass m must be added to the right side to balance the system. (a) Find an expression for the magnitude of the magnetic field. (b) Why is the result independent of the vertical dimensions of the coil? (c) Suppose the coil has 50 turns and a width of 5.00 cm. When the switch is closed, the coil carries a current of 0.300 A, and a mass of 20.0 g must be added to the right side to balance the system. What is the magnitude of the magnetic field?

A heart surgeon monitors the flow rate of blood through an artery using an electromagnetic flowmeter (Fig. P29.72). Electrodes A and B make contact with the outer surface of the blood vessel, which has a diameter of 3.00 mm. (a) For a magnetic field magnitude of 0.040 0 T, an emf of 160 mV appears between the electrodes. Calculate the speed of the blood. (b) Explain 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? Explain.

A uniform magnetic field of magnitude 0.150 T is directed along the positive x axis. A positron moving at a speed of 5.00 3 106 m/s enters the field along a direction that makes an angle of u 5 85.08 with the x axis (Fig. P29.73). The motion of the particle is expected to be a helix as described in Section 29.2. Calculate (a) the pitch p and (b) the radius r of the trajectory as defined in Figure P29.73.

Review. (a) Show that a magnetic dipole in a uniform magnetic field, displaced from its equilibrium orientation and released, can oscillate as a torsional pendulum (Section 15.5) in simple harmonic motion. (b) Is this statement true for all angular displacements, for all displacements less than 1808, or only for small angular displacements? Explain. (c) Assume the dipole is a compass needlea light bar magnetwith a magnetic moment of magnitude m. It has moment of inertia I about its center, where it is mounted on a frictionless, vertical axle, and it is placed in a horizontal magnetic field of magnitude B. Determine its frequency of oscillation. (d) Explain how the compass needle can be conveniently used as an indicator of the magnitude of the external magnetic field. (e) If its frequency is 0.680 Hz in the Earths local field, with a horizontal component of 39.2 mT, what is the magnitude of a field parallel to the needle in which its frequency of oscillation is 4.90 Hz?

The accompanying table shows measurements of the Hall voltage and corresponding magnetic field for a probe used to measure magnetic fields. (a) Plot these data and deduce a relationship between the two variables. (b) If the measurements were taken with a current of 0.200 A and the sample is made from a material having a charge-carrier density of 1.00 3 1026 carriers/m3, what is the thickness of the sample?

A metal rod having a mass per unit length l carries a current I. The rod hangs from two wires in a uniform vertical magnetic field as shown in Figure P29.76. The wires make an angle u with the vertical when in equilibrium. Determine the magnitude of the magnetic field.

Consider an electron orbiting a proton and maintained in a fixed circular path of radius R 5 5.29 3 10211 m by the Coulomb force. Treat the orbiting particle as a current loop. Calculate the resulting torque when the electronproton system is placed in a magnetic field of 0.400 T directed perpendicular to the magnetic moment of the loop.

Protons having a kinetic energy of 5.00 MeV (1 eV 5 1.60 3 10219 J) are moving in the positive x direction and enter a magnetic field B S 5 0.050 0k^ T directed out of the plane of the page and extending from x 5 0 to x 5 1.00 m as shown in Figure P29.78. (a) Ignoring relativistic effects, find the angle a between the initial velocity vector of the proton beam and the velocity vector after the beam emerges from the field. (b) Calculate the y component of the protons momenta as they leave the magnetic field.

Review. A wire having a linear mass density of 1.00 g/cm is placed on a horizontal surface that has a coefficient of kinetic friction of 0.200. The wire carries a current of 1.50 A toward the east and slides horizontally to the north at constant velocity. What are (a) the magnitude and (b) the direction of the smallest magnetic field that enables the wire to move in this fashion?

A proton moving in the plane of the page has a kinetic energy of 6.00 MeV. A magnetic field of magnitude B 5 1.00 T is directed into the page. The proton enters the magnetic field with its velocity vector at an angle u 5 45.08 to the linear boundary of the field as shown in Figure P29.80. (a) Find x, the distance from the point of entry to where the proton will leave the field. (b) Determine u, the angle between the boundary and the protons velocity vector as it leaves the field.