Three particles, a, b, and c, enter a magnetic field as shown in the figure. What can you say about the charge on each particle?

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1 1 Three particles, a, b, and c, enter a magnetic field as shown in the figure. What can you say about the charge on each particle? 6 Determine the magnitude and direction of the force on an electron traveling 8.75 x 10 5 m/s horizontally to the east in a vertically upward magnetic field of strength 0.75 T x N North 7 Find the direction of the force on a negative charge for each diagram shown in the figure, where v (green) is the velocity of the charge and B (blue) is the direction of the magnetic field. ( means the vector points inward e means it points outward, toward you.) Charge A upward force and must be positive Charge B has no force and no charge Charge C downward force and negative charge. 2 A positively charged particle in a nonuniform magnetic field follows the trajectory shown in the figure. Indicate the direction of the magnetic field everywhere in space, assuming the path is always in the plane of the page, and indicate the relative magnitudes of the field in each region. (a) Left (b) Left (c) Upward (d) Into the Page (e) No Force (f) Downward 3 Explain why a strong magnet held near a CRT television screen causes the picture to become distorted. Also, explain why the picture sometimes goes completely black where the field is the strongest. [But don t risk damage to your TV by trying this.] 8 Determine the direction of B for each case in the figure, where F represents the maximum magnetic force on a positively charged particle moving with velocity v. 4 If a moving charged particle is deflected sideways in some region of space, can we conclude, for certain, that B 0 in that region? Explain. (a) Downward (b) inward into the paper (c) right 5 Alpha particles of charge q = +2e and mass m = 6.6 x kg are emitted from a radioactive source at a speed of 1.6 x 10 7 m/s. What magnetic field strength would be required to bend them into a circular path of radius r = 0.25 m? 1.3 T

2 9 An electron is projected vertically upward with a speed of 1.70 x 10 6 m/s into a uniform magnetic field of T that is directed horizontally away from the observer. Describe the electron s path in this field. 15 Suppose the Earth s magnetic field at the equator has magnitude 0.40 x 10-4 T and a northerly direction at all points. Estimate the speed a singly ionized uranium ion (m = 238 u, q = e) would need to circle the Earth 5.0 km above the equator. Can you ignore gravity? 1.0 x 10 8 m/s F B /F g = 1.7 x10 8 can ignore gravity x 10-5 m A 5.0 MeV (kinetic energy) proton enters a 0.20 T field, in a plane perpendicular to the field. What is the radius of its path? 16 the following questions: (a) What value of magnetic field would make a beam of electrons, traveling to the right at a speed of 4.8 x 10 6 m/s, go undeflected through a region where there is a uniform electric field of 10, 000 V/m pointing vertically up? (b) What is the direction of the magnetic field if it is known to be perpendicular to the electric field? (c) What is the frequency of the circular orbit of the electrons if the electric field is turned off? 1.6 m (a) x 10-3 T (b) out (c) 5.8 x 10 7 Hz 11 An electron experiences the greatest force as it travels 2.9 x 10 6 m/s in a magnetic field when it is moving northward. The force is upward and of magnitude 7.2 x N. What are the magnitude and direction of the magnetic field? 17 A proton moving horizontally enters a uniform magnetic field perpendicular to the proton's velocity, as shown in the figure. (a) Describe the subsequent motion of the proton. (b) How would an electron behave under the same circumstances? 1.6 T East 12 A proton (mass m p ), a deuteron (m = 2m p, Q = e), and an alpha particle (m = 4m p, Q = 2e) are accelerated by the same potential difference V and then enter a uniform magnetic field B, where they move in circular paths perpendicular to B. Determine the radius of the paths for the deuteron and alpha particle in terms of that for the proton. r d = 2r p r a = 2r p 13 Show that the time T required for a particle of charge q moving with constant speed v to make one circular revolution in a uniform magnetic field B ( v ) is T = (2πm)/(qB). 18 Consider an electron near the Earth s equator. (a) In which direction does it tend to deflect if its velocity is directed downward? (b) In which direction does it tend to deflect if its velocity is directed northward? (c) In which direction does it tend to deflect if its velocity is directed westward? (d) In which direction does it tend to deflect if its velocity is directed southeastward? 14 A 3.40 g bullet moves with a speed of 160 m/s perpendicular to the Earth s magnetic field of 5.00 x 10-5 T. If the bullet possesses a net charge of 13.5 x 10-9 C by what distance will it be deflected from its path due to the Earth s magnetic field after it has traveled 1.00 km? (a) Down North = East, so the force is directed West. (b) North North = sin 0 = 0 : Zero deflection. (c) West North = Down, so the force is directed Up. (d) Southeast North = Up, so the force is Down. 6.2 x 10-7 m

3 19 A proton moves perpendicular to a uniform magnetic field B at m/s and experiences an acceleration of m/s 2 in the +x direction when its velocity is in the +z direction. Determine the magnitude and direction of the field. 26 Consider the mass spectrometer shown schematically in the figure. The magnitude 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 magnitude of T. Calculate the radius of the path for a singly charged ion having a mass m = kg x 10-2 T 20 An electron is accelerated through V from rest and then enters a uniform 1.70-T magnetic field. (a) What is the maximum value of the magnetic force this charge can experience? (b) What is the minimum value of the magnetic force this charge can experience? (a) 7.90 pn; (b) 0 21 A proton moves with a velocity of v = (2 i 4j + k) m/s in a region in which the magnetic field is B = (i + 2j -3k ) T. What is the magnitude of the magnetic force this charge experiences? m At the Fermilab accelerator in Batavia, Illinois, protons having momentum 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? 2.34 x N 22 The magnetic field of the Earth at a certain location is directed vertically downward and has a magnitude of 50.0 µt. A proton is moving horizontally toward the west in this field with a speed of m/s. (a) What are the direction and magnitude of the magnetic force the field exerts on this charge? (b) What is the radius of the circular arc followed by this proton? (a) 4.96 x N (b) 1.29 km T The picture tube in a 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. What field magnitude is necessary to deflect the beam to the side of the screen? Ignore relativistic corrections. 23 A singly charged positive ion has a mass of kg. After being accelerated from rest through a potential difference of 833 V, the ion enters a magnetic field of T along a direction perpendicular to the direction of the field. Calculate the radius of the path of the ion in the field mt A singly charged ion completes five revolutions in a uniform magnetic field of magnitude T in 1.50 ms. Calculate the mass of the ion in kilograms cm 3.82 x kg 24 A proton moving freely in a circular path perpendicular to a constant magnetic field takes 1.00 µs to complete one revolution. Determine the magnitude of the magnetic field x 10-2 T 25 A velocity selector consists of electric and magnetic fields described by the expressions E = E k and B = B j, with B = 15.0 mt. Find the value of E such that a 750 ev electron moving along the positive x axis is undeflected. 244 kv/m

4 30 The figure here shows the circular paths of two particles that travel at the same speed in a uniform magnetic field B, which is directed into the page. One particle is a proton; the other is an electron (which is less massive). (a) Which particle follows the smaller circle? (b) Does that particle travel clockwise or counterclockwise? 34 A proton moves through a uniform magnetic field given by B = (10i - 20j + 30jk) mt, At time t 1, the proton has a velocity given by v = v x i + v y j + (2.0 km/s)k and the magnetic force on the proton is F B = (4.0 x N)i + (2.0 x N)j. (a) At that instant, what is v x? (b) At that instant, what is v y? (a) -3.5 km/s; (b) 7.0 km/s (a) electron; (b) clockwise 35 An electron moves through a uniform magnetic field given by B = B x i + (3.0B x )j. At a particular instant, the electron has velocity v = (2.0i + 4.0j) m/s and the magnetic force acting on it is (6.4 x N)k. Find B. 31 A proton traveling at with respect to the direction of a magnetic field of strength 2.60 mt experiences a magnetic force of 6.50 x N. (a) Calculate the proton's speed. (b) Calculate its kinetic energy in electron-volts. -2.0T (a) 400 km/s; (b) 835 ev 36 What uniform magnetic field, applied perpendicular to a beam of electrons moving at 1.30 x 10 6 m/s, is required to make the electrons travel in a circular arc of radius m? 32 An alpha particle travels at a velocity v of magnitude 550 m/s through a uniform magnetic field B of magnitude T. (An alpha particle has a charge of x C and a mass of 6.6 x kg.) The angle between v and B is (a) What is the magnitude of the force F B acting on the particle due to the field? (b) What is the magnitude of the acceleration of the particle due to F B? (c) Does the speed of the particle increase, decrease, or remain the same? µt A source injects an electron of speed v = 1.5 x 10 7 m/s into a uniform magnetic field of magnitude B = 1.0 x 10-3 T. The velocity of the electron makes an angle θ = 10 0 with the direction of the magnetic field. Find the distance d from the point of injection at which the electron next crosses the field line that passes through the injection point. 33 (a) 6.2 x N; (b) 9.5 x 10 8 m/s 2 ; (c) same An electron that has velocity v = (2.0 x 10 6 m/s)i + (3.0 x 10 6 m/s)j moves through the uniform magnetic field B = (0.030 T)i - (0.15 T)j. (a) Find the force on the electron. (b) Repeat your calculation for a proton having the same velocity m An electron is accelerated from rest through potential difference V and then enters a region of uniform magnetic field, where it undergoes uniform circular motion. The figure gives the radius r of that motion versus V 1/2. What is the magnitude of the magnetic field? (a) (6.2 x N) k; (b) (-6.2 x N) k 6.7 x 10-2 T

5 39 Physicist S. A. Goudsmit devised a method for measuring the mass of heavy ions by timing their period of revolutionin a known magnetic field. A singly charged ion of iodine makes 7.00 rev in a 45.0 mt field in 1.29 ms. Calculate its mass in atomic mass units. 127 u 43 A 5.0 μc particle moves through a region containing a magnetic field -20i mt and the electric field 300i V/m. At one instant the velocity of the particle is (17i - 11j + 7.0k) km/s. At that instant and in unit-vector notation, what is the net electromagnetic force (the sum of the electric and magnetic forces) on the particle? (0.80 mn) k 40 A certain particle is sent into a uniform magnetic field, with the particle's velocity vector perpendicular to the direction of the field. The figure gives the period T of the particle's motion versus the inverse of the field magnitude B. What is the ratio m/q of the particle's mass to the magnitude of its charge? 44 In the figure, a particle moves along a circle in a region of uniform magnetic field of magnitude B = 4.00 mt. The particle is either a proton or an electron (you must decide which). It experiences a magnetic force of magnitude 3.20 x N. (a) What is the particle's speed? (b) What is the radius of the circle? (c) What is the period of the motion? (a) 4.99 x 10 6 m/s; (b) 7.10mm; (c) 8.93 ns x 10-9 kg/c In the figure, an electron moves at speed v = 100 m/s along an x axis through uniform electric and magnetic fields. The magnetic field B is directed into the page and has magnitude 5.00 T. In unit-vector notation, what is the electric field? 45 An electron that is moving through a uniform magnetic field has a velocitl v = (40 km/s)i + (35 km/s)j when it experiences a force F = -(4.2 f N)i + (4.8 1 N)j due to the magnetic field. If B x = 0, calculate the magnetic field B. (0.75 T) k (-500 V/m) j 46 the following questions. (a) What is the velocity of a beam of electrons that go undeflected when passing through perpendicular electric and magnetic fields of magnitude 8.8 x 10 3 V/m and 3.5 x 10-3 T respectively? (b) What is the radius of the electron orbit if the electric field is turned off? 42 A proton travels through uniform magnetic and electric fields. The magnetic field is B = -2.50i mt. At one instant the velocity of the proton is v = 2000j m/s. (a) At that instant and in unit-vector notation, what is the net force acting on the proton if the electric field is 4.00k V/m? (b) At that instant and in unit-vector notation, what is the net force acting on the proton if the electric field is -4.00k V/m? (c) At that instant and in unit-vector notation, what is the net force acting on the proton if the electric field is 4.00i V/m? 47 (a) 2.5 x 10 6 m/s (b) 4.1 x 10-3 m A doubly charged helium atom whose mass is 6.6 x kg is accelerated by a voltage of 2100 V. (a) What will be its radius of curvature if it moves in a plane perpendicular to a uniform T field? (b) What is its period of revolution? (a) (1.44 x N) k; (b) (1.60 x N) k; (c) (6.41 x N) i + (8.01 x N) k (a) 2.7 x 10-2 m (b) 3.8 x 10-7 s

6 48 A particle of mass m and charge q moves in a circular path in a magnetic field B. Show that its kinetic energy is proportional to r 2, the square of the radius of curvature of its path. 53 Two charged particles are projected into a magnetic field perpendicular to their velocities. If the charges are deflected in opposite directions, what can you say about them? 49 Protons move in a circle of radius 5.10 cm in a T magnetic field. (a) What value of electric field could make their paths straight? (b) In what direction must it point? 54 The electron beam in the figure is projected to the right. The beam deflects downward in the presence of a magnetic field produced by a pair of current-carrying coils, (a) What is the direction of the magnetic field? (b) What would happen to the beam if the magnetic field were reversed in direction? 1.57 x 10 6 V/m 50 A doubly charged helium atom, whose mass is 7.67 x T is accelerated by a voltage of 2400 V. (a) What will be its radius of curvature in a uniform T field? (b) What is its period of revolution? (a) x 10-2 m (b) 5.5 x 10-7 s Estimate the approximate maximum deflection of the electron beam near the center of a TV screen due to the Earth s 5.0 x 10-5 T field. The CRT screen is 22 cm from the electron gun. (a) Assume the electrons are accelerated by 2.0 kv. (b) Assume the electrons are accelerated by 30 kv. Note that in color TV sets, the CRT beam must be directed accurately to within less than 1 mm in order to strike the correct phosphor. Because the Earth s field is significant here, mu metal shields are used to reduce the Earth s field in the CRT. (a) 8.02 x 10-3 mm (b) 2.07 x 10-3 mm A proton follows a spiral path through a gas in a magnetic field of T, perpendicular to the plane of the spiral, as shown in the figure. In two successive loops, at points P and Q, the radii are 10.0 mm and 8.5 mm, respectively. Calculate the change in the kinetic energy of the proton as it travels from P to Q Charged particles from outer space, called cosmic rays, strike the Earth more frequently near the poles than near the equator. Why? An electron has a velocity of m/s (in the positive x direction), and an acceleration of m/s 2 (in the positive z direction) in a uniform electric and magnetic field. The the electric field has a magnitude of 20.0 N/C (in the positive z direction). (a) What can you determine about the magnetic field in the region? (b) What can you not determine? The magnetic field may have any x-component (a) B z = 0 (b) B y = 2.62 mt 57 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 ( m). What is the magnetic field in that region of space? 7.88 x T -2.1 x J ev

7 58 Singly charged uranium 238 ions are accelerated through a potential difference of 2.00 kv and enter a uniform magnetic field of 1.20 T directed perpendicular to their velocities. (a) Determine the radius of their circular path. (b) Repeat for uranium- 235 ions. (c) How does the ratio of these path radii depend on the accelerating voltage and on the magnitude of the magnetic field? 62 An electron of kinetic energy 1.20 kev circles in a plane perpendicular to a uniform magnetic field. The orbit radius is 25.0 cm. (a) Find the electron's speed. (b) Find the magnetic field magnitude. (c) Find the circling frequency. (d) Find the period of the motion. (a) 8.28 cm (b) 8.23 cm (c) (a) 2.05 x 10 7 m/s; (b) 467 µt; (c) 13.1 MHz; (d) 76.3 ns 59 Assume that the region to the right of a certain vertical plane contains a vertical magnetic field of magnitude 1.00 mt, and the field is zero in the region to the left of the plane. 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 its path is a semicircle. (b) Find the kinetic energy of the electron if the maximum depth of penetration into the field is 2.00 cm. 63 the following questions. (a) Find the frequency of revolution of an electron with an energy of 100 ev in a uniform magnetic field of magnitude 35.0 μt. (b) Calculate the radius of the path of this electron if its velocity is perpendicular to the magnetic field. (a) 1.79 x 10-8 s (b) 35.1 ev (a) MHz; (b) 96.4 cm 60 The figure shows three situations in which a charged particle with velocity V travels through a uniform magnetic field B. In each situation, what is the direction of the magnetic force F B on the particle? 64 In a nuclear experiment a proton with kinetic energy 1.0 MeV moves in a circular path in a uniform magnetic field. (a) What energy must an alpha particle (q = +2e, m = 4.0 u) have if it circulates in the same circular path? (b) What energy must a deuteron (q = +e, m = 2.0 u) have if it circulates in the same circular path? (a) 1.0 MeV; (b) 0.5 MeV a, +z; b, -x; c,f B = 0 65 A particle undergoes uniform circular motion of radius 26.1 μm in a uniform magnetic field. The magnetic force on the particle has a magnitude of 1.60 x N. What is the kinetic energy of the particle? 2.09 x J 61 An electron is accelerated from rest by a potential difference of 350 V. It then enters a uniform magnetic field of magnitude 200 mt with its velocity perpendicular to the field. (a) Calculate the speed of the electron. (b) Calculate the radius of its path in the magnetic field. 66 A positron with kinetic energy 2.00 kev is projected into a uniform magnetic field B of magnitude T, with its velocity vector making an angle of with B. (a) Find the period of its helical path. (b) Find the pitch p, of its helical path. (c) Find the radius r of its helical path. (a) 1.11 x 10 7 m/s; (b) mm (a) ns; (b) mm; (c) 1.51 mm

8 67 In the figure, a charged particle moves into a region of uniform magnetic field B, goes through half a circle, and then exits that region. The particle is either a proton or an electron (you must decide which) It spends 130 ns in the region. (a) What is the magnitude of B? (b) If the particle is sent back through the magnetic field (along the same initial path) but with 2.00 times its previous kinetic energy, how much time does it spend in the field during this trip? 70 A proton, a deuteron (q = +e, m = 2.0 u), and an alpha particle (q = +2e, m = 4.0 u) are accelerated through the same potential difference and then enter the same region of uniform magnetic field B, moving perpendicular to B. (a) What is the ratio of the proton's kinetic energy K P to the alpha particle's kinetic energy K α? (b) What is the ratio of the deuteron's kinetic energy K d to K α? (c) If the radius of the proton's circular path is 10 cm, what is the radius of the deuteron's path? (d) If the radius of the proton's circular path is 10 cm, what is the radius of the alpha particle's path? (a) T; (b) 130 ns (a) 0.50; (b) 0.50; (c) 14 cm; (d) 14 cm 68 In the figure, an electron with an initial kinetic energy of 4.0 kev enters region 1 at time t = 0. That region contains a uniform magnetic field directed into the page, with magnitude T. The electron goes through a half-circle and then exits region 1, headed toward region 2 across a gap of 25.0 cm. There is an electric potential difference V = 2000V across the gap, with a polarity such that the electron's speed increases uniformly as it traverses the gap. Region 2 contains a uniform magnetic field directed out of the page, with magnitude T. The electron goes through a half-circle and then leaves region 2. At what time t does it leave? 71 An electron with kinetic energy 2.5 kev moving along the positive direction of an x axis enters a region in which a uniform electric field of magnitude 10 kv/m is in the negative direction of the y axis. A uniform magnetic field B is to be set up to keep the electron moving along the x axis, and the direction of B is to be chosen to minimize the required magnitude of B. In unit-vector notation, what B should be set up? (-0.34 mt) k 72 A beam of electrons whose kinetic energy is K emerges from a thin-foil "window" at the end of an accelerator tube. A metal plate at distance d from this window is perpendicular to the direction of the emerging beam (see the figure). (a) Show that we can prevent the beam from hitting the plate if we apply a uniform magnetic field B such that B is defined in the equation below the diagram in which m and e are the electron mass and charge. (b) How should B be oriented? 8.7 ns 69 A particle of mass 10 g and charge 80 μc moves through a uniform magnetic field, in a region where the free-fall acceleration is m/s 2. The velocity of the particle is a constant 20i km/s, which is perpendicular to the magnetic field. What, then, is the magnetic field? (-61 mt) k (a) (b) out of the plane of the page

9 73 In the figure shows three situations in which a positively charged particle moves at velocity v through a uniform magnetic field B and experiences a magnetic force F B. In each situation, determine whether the orientations of the vectors are physically reasonable. 76 The figure shows the path of a particle through six regions of uniform magnetic field, where the path is eith, half-circle or a quarter-circle. Upon leaving the last region the particle travels between two charged, parallel plates and is deflected toward the plate of higher potential. What is the direction of the magnetic field in each of the six regions? (a) no because v and FB must be perpendicular; (b) yes; (c) no because B and F B must be perpendicular 74 The figure shows crossed uniform electric and magnetic fields E and B and, at a certain instant, the velocity vectors of the 10 charged particles listed in the table. (The vectors are not drawn to scale.) The speeds given in the table are either less than or greater than E/B. Which particles will move out of the page toward you after the instant shown in the figure? into page: a, d, e; out of page: b, c, f (the particle is negatively charged) 77 The figure shows the path of an electron that passes through two regions containing uniform magnetic fields of magnitudes B 1 and B 2, Its path in each region is a half-circle. (a) Which field is stronger? (b) What is the direction of each field? (c) Is the time spent by the electron in the B 1 region greater than, less than, or the same as the time spent in the B 2 region? 2, 5, 6, 9, In the figure, a charged particle enters a uniform magnetic field d with speed v 0, moves through a half-circle in time T 0, and then leaves the field. (a) Is the charge positive or negative? (b) Is the final speed of the particle greater than, less than, or equal to v 0? (c) If the initial speed had been 0.5v 0, would the time spent in field B have been greater than, less than, or equal to T 0? (d) Would the path have been a half-circle, more than a halfcircle, or less than a half-circle? (a) B 1 ; (b) B 1 into page, B 2 out of page; (c) less (a) negative; (b) equal; (c) equal; (d) half circle

10 78 The figure shows 11 paths through a region of uniform magnetic field. One path is a straight line; the rest are halfcircles. The table gives the masses, charges, and speeds of 11 particles that take these paths through the field in the directions shown. Which path in the figure corresponds to which particle in the table? 81 If a negatively charged particle enters a region of uniform magnetic field which is perpendicular to the particle s velocity, will the kinetic energy of the particle increase, decrease, or stay the same? Explain your answer. (Neglect gravity and assume there is no electric field.) 82 A particle of charge q moves in a circular path of radius r in a uniform magnetic field B. Show that its momentum is p = qbr. 83 A particle of mass m and charge q moves in a circular path in a magnetic field B, Show that the angular momentum of the particle is L = qbr 2 about the center of the circle. 1i, 2e, 3c, 4a, 5g, 6j, 7d, 8b, 9h, 10f, 11k 79 The figure shows the path of an electron in a region of uniform magnetic field. The path consists of two straight sections, each between a pair of uniformly charged plates, and two halfcircles. (a) Which plate is at the higher electric potential in the top pair of plates? (b) Which plate is at the higher electric potential in the bottom pair? (c) What is the direction of the magnetic field? 84 Show that the magnetic dipole moment M of an electron orbiting the proton nucleus of a hydrogen atom is related to the orbital angular momentum L of the electron by M = (e/2m)l. M = el/2m 85 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 T. Determine the energy (in kev) of the incident electron. 80 (a) upper plate; (b) lower plate; (c) out of page A uniform magnetic field B, with magnitude 1.2 mt, is directed vertically upward throughout the volume of a laboratory chamber. A proton with kinetic energy 5.3 MeV enters the chamber, moving horizontally from south to north. What magnetic deflecting force acts on the proton as it enters the chamber? The proton mass is 1.67 x kg. (Neglect Earth's magnetic field.) x10 5 ev x V x J 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 J s. (a) Determine the radius of the circular path. (b) Determine the speed of the electron. (a) 5.00 cm (b) 8.78 x 10 6 m/s 87 A proton is at rest at the plane vertical boundary of a region containing a uniform vertical magnetic field B. 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 proton s trajectory is R. Find the radius of the alpha particle s trajectory. The mass of the alpha particle is four times that of the proton, and its charge is twice that of the proton. 6.1 x N 3/4 R

11 88 A proton moving with speed 2.0 x 10 5 m/s in a field free region abruptly enters an essentially uniform magnetic field B = T (B v ). (a) If the proton enters the magnetic field region at a 45 angle as shown in the figure at what angle does it leave? (b) If the proton enters the magnetic field region at a 45 angle as shown in the figure at what distance x does it exit the field? 93 Protons having a kinetic energy of 5.00 MeV are moving in the positive x direction and enter a magnetic field B = T directed out of the plane of the page and extending from x = 0 to x = 1.00 m, as shown in the figure. (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. Ignore relativistic effects and note that 1 ev = J. (a) θ = 45 0 (b) 3.5 x 10-3 m (a) x kg.m/s (b) A straight stream of protons passes a given point in space at a rate of 1.5 x 10 9 protons/s. What magnetic field do they produce 2.0 m from the beam? 2.4 x T 94 the following questions. (a) A proton moving in the +x direction with velocity v = v i î experiences a magnetic force F = F i ĵ in the +y direction. Explain what you can and cannot infer about B from this information. (b) What If? In terms of F i, what would be the force on a proton in the same field moving with velocity v = v i î? (c) What would be the force on an electron in the same field moving with velocity v = v i î? 90 An electron moving along the positive x-axis perpendicular to a magnetic field experiences a magnetic deflection in the negative y direction. What is the direction of the magnetic field? negative z direction (a) B x could have any value B y = 0 B z = -F i /ev i (b) -F i j (c) -F i j 91 A proton travels with a speed of m/s at an angle of 37.0 with the direction of a magnetic field of T in the +y direction. What are (a) What is the magnitude of the magnetic force on the proton? (b) What is its acceleration? 95 A positive charge q = C moves with a velocity v = (2 i + 3 j k) 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 charge (in unitvector notation), taking B = (2i + 4j + k )T and E = (4 i j 2 k)v/m. (b) What angle does the force vector make with the positive x axis? N m/s 2 A proton moving at m/s through a magnetic field of 1.70 T experiences a magnetic force of magnitude N. What is the angle between the proton s velocity and the field? (a) F = (3.52 i j )x N (b) or 131

12 96 A proton moving in the plane of the page has a kinetic energy of 6.00 MeV. A magnetic field of magnitude B = 1.00 T is directed into the page. The proton enters the magnetic field with its velocity vector at an angle θ = 45.0 to the linear boundary of the field as shown in the figure. (a) Find x, the distance from the point of entry to where the proton will leave the field. (b) Determine θ, the angle between the boundary and the proton s velocity vector as it leaves the field. 98 The figure shows a metallic, rectangular solid that is to move at a certain speed v through the uniform magnetic field B. The dimensions of the solid are multiples of d, as shown. You have six choices for the direction of the velocity: parallel to x, y, or z in either the positiveor negative direction. (a) Rank the six choices according to the potential difference set up across the solid, greatest first. (b) For which choice is the front face at lower potential? (a) +z and -z tie, then +y and -y tie, then +x and -x tie (zero); (b) +y (a) m (b) An electric field of 1.50 kv/m and a magnetic field of T act on a moving electron to produce no net force. If the fields are perpendicular to each other, what is the electron's speed? 97 The figure shows four directions for the velocity vector v of a positively charged particle moving through a uniform electric field E (directed out of the page and represented with an encircled dot) and a uniform magnetic field B. (a) Rank directions 1, 2, and 3 according to the magnitude of the net force on the particle, greatest first. (b) Of all four directions, which might result in a net force of zero? km/s In the figure, an electron accelerated from rest through potential difference V 1 = 1.00 kv enters the gap between two parallel plates having separation d = 20.0 min and potential difference V 2 = 100 V. The lower plate is at the lower potential. Neglect fringing and assume that the electron's velocity vector is perpendicular to the electric field vector between the plates. In unit-vector notation, what uniform magnetic field allows the electron to travel in a straight line in the gap? (a) 2, then 1 and 3 tie (zero); (b) mt 101 An electron has an initial velocity of (12.0j k) km/s and a constant acceleration of (2.00 x m/s 2 )i in a region in which uniform electric and magnetic fields are present. If B = (400 μt)i, find the electric field E. (-11.4 V/m) i - (6.00 V/m) j + (4.80 V/m) k

13 102 At time t 1, an electron is sent along the positive direction of an x axis, through both an electric field E and a magnetic field B, with E directed parallel to the y axis. The figure gives the y component F net,y of the net force on the electron due to the two fields, as a function of the electron's speed v at time t 1 The x and z components of the net force are zero at t 1. (a) Assuming B x = 0, find the magnitude E in unit-vector notation. (b) Assuming B x = 0, find B in unit-vector notation. 106 At one instant, v = (-2.00i j k) m/s is the velocity of a proton in a uniform magnetic field B = (2.00i j k) mt. (a) At that instant, what is the magnetic force F on the proton, in unit-vector notation? (b) At that instant, what is the angle between v and F,? (c) At that instant, what is the angle between v and B? (a) (12.8 N) i N) j; (b) 90 0 ; (c) An electron has velocity v = (32i + 40j) km/s as it enters a uniform magnetic fieldd = 60i /-tt. (a) What is the radius of the helical path taken by the electron? (b) What is the pitch of that path? (c) To an observer looking into the magnetic field region from the entrance point of the electron, does the electron spiral clockwise or counterclockwise as it moves? (a) 1.25 V/m; (b) (25.0 mt) k (a) 3.8 mm; (b) 19 mm; (c) clockwise 103 A particle of mass 6.0 g moves at 4.0 km/s in an xy plane, in a region with a uniform magnetic field given by 5.0i mt. At one instant, when the particle's velocity is directed 37 0 counterclockwise from the positive direction of the x axis, the magnetic force on the particle is 0.48k N. What is the particle's charge? -40 mc 108 An electron in an old-fashioned TV camera tube is moving at 7.20 x 10 6 M/S in a magnetic field of strength 83.0 mt. (a) What is the maximum magnitude of the force acting on the electron due to the field? (b) What is the minimum magnitude of the force acting on the electron due to the field? (c) At one point the electron has an acceleration of magnitude 4.90 x m/s 2. What is the angle between the electron's velocity and the magnetic field? 104 A particle with charge 2.0 C moves through a uniform magnetic field. At one instant the velocity of the particle is (2.0i + 4.0j + 6.0k) m/s and the magnetic force on the particle is (4.0i - 20j +. 12k) N. The x and y components of the magnetic field are equal. What is B? (a) 9.56 x N; (b) 0; (c) (3.0 T) i - (3.0 T) j - (4.0 T) ˆk 109 A proton of charge +e and mass m enters a uniform magnetic field Bi with an initial velocity v = v 0x i + v oy j. Find an expression in unit-vector notation for its velocity v at any later time t. 105 An electron follows a helical path in a uniform magnetic field given by B = (20i - 50j - 30k) mt. At time t = 0, the electron's velocity is given by v = (20i - 30j + 50k) m/s. (a) What is the angle φ between v and B? The electron's velocity changes with time. (b) Does its speed change with time? (c) Does the angle φ change with time? (d) What is the radius of the helical path? v = v 0x i + v 0y cos(ωt) j - v 0y sin(ωt) k, where ω = eb/m (a) 84 0 ; (b) no; (c) no; (d) 5.7 nm

14 110 At time t = 0, an electron with kinetic energy 12 kev moves through x = 0 in the positive direction of an x axis that is parallel to the horizontal component of Earth's magnetic field A The field's vertical component is downward and has magnitude 55.0 μt. (a) What is the magnitude of the electron's acceleration due to B? (b) What is the electron's distance from the x axis when the electron reaches coordinate x = 20 cm? 113 *62. (II) Suppose the electric field between the electric plates in the mass spectrometer of the figure is 2.48 x 10 4 V/m and the magnetic fields B = B = 0.68 T. The source contains carbon isotopes of mass numbers 12, 13, and 14 from a long dead piece of a tree. (To estimate atomic masses, multiply by 1.67 x kg ) (a) How far apart are the lines formed by the singly charged ions of each type on the photographic film? (b) What if the ions were doubly charged? (a) 6.3 x m/s 2 ; (b) (c) 3.0 mm 111 Two ions have the same mass, but one is singly ionized and the other is doubly ionized. How will their positions on the film of a mass spectrometer (see figure) differ? (a) 1.1 x 10-3 m 1.1 x 10-3 m (b) 5.6 x 10-4 m 5.6 x 10-4 m 114 One form of mass spectrometer accelerates ions by a voltage V before they enter a magnetic field B. The ions are assumed to start from rest. Show that the mass of an ion is m = (qb 2 R 2 )/ (2V) where R is the radius of the ions path in the magnetic field and q is their charge. 112 In a mass spectrometer, germanium atoms have radii of curvature equal to 21.0, 21.6, 21.9, 22.2, and 22.8 cm. The largest radius corresponds to an atomic mass of 76 u. What are the atomic masses of the other isotopes? m 21.6 = 72 u m 22.7 = 74 u 115 m = (qr 2 B 2 )/2V The figure shows a particle of mass m having positive charge q and initially traveling with velocity v j. At the origin of coordinates it enters a region between y = 0 and y = h containing a uniform magnetic field B directed perpendicularly out of the page. (a) What is the critical value of v such that the particle just reaches y = h? Describe the path of the particle under this condition, and predict its final velocity. (b) Specify the path the particle takes and its final velocity, if v is less than the critical value. (c) Specify the path the particle takes and its final velocity, if v is greater than the critical value.

15 116 A certain commercial mass spectrometer is used to separate uranium ions of mass 3.92 x kg and charge 3.20 x C from related species. The ions are accelerated through a potential difference of 100 kv and then pass into a uniform magnetic field, where they are bent in a path of radius 1.00 m. After traveling through and passing through a slit of width 1.00 mm turn and height 1.00 cm, they are collected in a cup. (a) What is the magnitude of the (perpendicular) magnetic field in the separator? (b) If the machine is used to separate out 100 mg of material per hour, calculate the current of the desired ions in the machine, (c) If the machine is used to separate out 100 mg of material per hour, calculate the thermal energy produced in the cup in 1.00 h. 118 The cyclotron shown in the figure is a device used to accelerate elementary particles such as protons to high speeds. Particles starting at point A with some initial velocity travel in circular orbits in the magnetic field B. The particles are accelerated to higher speeds each time they pass through the gap between the metal dees, where there is an electric field E. (There is no electric field inside the hollow metal dees.) The electric field changes direction each half cycle, owing to an ac voltage V = V 0 sin 2πft, so that the particles are increased in speed at each passage through the gap. (a) Show that the frequency f of the voltage must be f = Bq/2πm, where q is the charge on the particles and m their mass. (b) Show that the kinetic energy of the particles increases by 2qV 0 each revolution, assuming that the gap is small. (c) If the radius of the cyclotron is 2.0 m and the magnetic field strength is 0.50 T, what will be the maximum kinetic energy of accelerated protons in MeV? 117 (a) 495 mt; (b) 22.7 ma; (c) 8.17 MJ Atom I of mass 35 u and atom 2 of mass 37 u are both singly ionized with a charge of +e. After being introduced into a mass spectrometer (see figure) and accelerated from rest through a potential difference V = 7.3 kv, each ion follows a circular path in a uniform magnetic field of magnitude B = 0.50 T. What is the distance x between the points where the ions strike the detector? 119 (a) (Bq)/(2πm) (b) 2qV 0 (c) 48 MeV Magnetic fields are very useful in particle accelerators for beam steering ; that is, the magnetic fields can be used to change the beam s direction without altering its speed as shown in the figure. (a) Show how this works with a beam of protons. (b) What happens to protons that are not moving with the speed that the magnetic field is designed for? (c) If the field extends over a region 5.0 cm wide and has a magnitude of 0.33 T, by approximately what angle will a beam of protons traveling at 1.0 x 10 7 m/s be bent? 8.2 mm Calculate the cyclotron frequency of a proton in a magnetic field of magnitude 5.20 T. 4.8 x 10 8 rad/s

16 121 A cyclotron designed to accelerate protons has a magnetic field of magnitude T over a region of radius 1.20 m. (a) What is the cyclotron frequency? (b) What is the maximum speed acquired by the protons? (a) 4.31 x 10 7 rad/s (b) 5.17 x 10 7 m/s 126 A cyclotron with dee radius 53.0 cm is operated at an oscillator frequency of 12.0 MHz to accelerate protons. (a) What magnitude B of magnetic field is required to achieve resonance? (b) At that field magnitude, what is the kinetic energy of a proton emerging from the cyclotron? Suppose, instead, that B = 1.57 T. (c) What oscillator frequency is required to achieve resonance now? (d) At that frequency, what is the kinetic energy of an emerging proton? 122 What is the required radius of a cyclotron designed to accelerate protons to energies of 34.0 MeV using a magnetic field of 5.20 T? 0.162m (a) T; (b) 8.34 MeV; (c) l3.9 MHz; (d) 33.2 MeV 123 A cyclotron designed to accelerate protons has an outer radius of 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 T. (a) Find the cyclotron frequency. (b) Find the speed at which protons exit the cyclotron. (c) Find their maximum kinetic energy. (d) How many revolutions does a proton make in the cyclotron? (e) For what time interval does one proton accelerate? 127 Suppose a cyclotron is operated at an oscillator frequency of 12 MHz and has a dee radius R = 53 cm. (a) What is the magnitude of the magnetic field needed for Jeuterons to be accelerated in the cyclotron? The deuteron mass is m = 3.34 x kg (b) What is the resulting kinetic energy of the deuterons? (a) 1.57 (b) 2.7 x10-12 J or about 17 MeV (a) 7.66 x 10 7 rad/s (b) 2.68 x 10 7 m/s (c) 3.76 x 10 6 ev (d) 3.13 x 10 3 revolutions (e) 2.57 x 10-4 s 128 An electron enters a uniform magnetic field B = 0.23 T.at a 45 angle to B. Determine the radius r and pitch p (distance between loops) of the electron s helical path assuming its speed is m/s as shown in the figure. 124 A cyclotron is sometimes used for carbon dating. Carbon-14 and carbon-12 ions are obtained from a sample of the material to be dated, and accelerated in the cyclotron. If the cyclotron has a magnetic field of magnitude 2.40 T, what is the difference in cyclotron frequencies for the two ions? khz 4.38 x 10 5 s -1 In a certain cyclotron a proton moves in a circle of radius m. The magnitude of the magnetic field is 1.20 T. (a) What is the oscillator frequency? (b) What is the kinetic energy of the proton, in electron-volts? 5.3 x 10-5 m 3.3 x 10-4 m (a) 18.3 MHz; (b) 17.2 MeV

17 129 A uniform magnetic field of magnitude T is directed along the positive x axis. A positron moving at m/s enters the field along a direction that makes an angle of 85.0 with the x axis in the figure. The motion of the particle is expected to be a helix. (a) Calculate the pitch p of the trajectory (b) Calculate the radius r of the trajectory. 132 In the figure, a conducting rectangular solid of dimensions d x = 5.00 m, d y = 3.00 m, and d z = 2.00 m moves at constant velocity v = (20.0 m/s)i through a uniform magneticfield B = (30.0 mt)j. (a) What is the resulting electric field within the solid, in unitvector notation? (b) What is the resulting potential difference across the solid? (a) (-600 mv/m) k; (b) 1.20 V 130 (a) 1.04 x 10-4 m (b) 1.89 x 10-4 m Consider an electron orbiting a proton and maintained in a fixed circular path of radius R = m by the Coulomb force. Treating the orbiting charge as a current loop, calculate the resulting torque when the system is in a magnetic field of T directed perpendicular to the magnetic moment of the electron. Note: Need to show relationship between current and moving charge: derive the formula IlXB from qvxb. 133 An electron follows a helical path in a uniform magnetic field of magnitude T. The pitch of the path is 6.00 μm, and the magnitude of the magnetic force on the electron is 2.00 x N. What is the electron's speed? 65.3 m/s x N.m The figure shows four orientations, at angle θ, of a magnetic dipole moment μ in a magnetic field. (a) Rank the orientations according to the magnitude of the torque on the dipole, greatest first. (b) Rank the orientations according to the potential energy of the dipole, greatest first. 134 An electron moves in a circle of radius r = 5.29 x m with speed 2.19 x 10 6 m/s. Treat the circular path as a current loop with a constant current equal to the ratio of the electron's charge magnitude to the period of the motion. If the circle lies in a uniform magnetic field of magnitude B = 7.10 mt, what is the maximum possible magnitude of the torque produced on the loop by the field? 6.58 x N.m 135 The magnetic dipole moment of Earth has magnitude 8.00 x J/T. Assume that this is produced by charges flowing in Earth's molten outer core. If the radius of their circular path is 3500 km, calculate the current they produce. (a) all tie; (b) 1 and 4 tie, then 2 and 3 tie 2.08 GA

18 136 A magnetic dipole with a dipole moment of magnitude J/T is released from rest in a uniform magnetic field of magnitude 52 mt. The rotation of the dipole due to the magnetic force on it is unimpeded. When the dipole rotates through the orientation where its dipole moment is aligned with the magnetic field, its kinetic energy is 0.80 mj. (a) What is the initial angle between the dipole moment and the magnetic field? (b) What is the angle when the dipole is next (momentarily) at rest? (a) 77 0 ; (b) The figure 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. 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. 137 The figure gives the potential energy U of a magnetic dipole in an external magnetic field B, as a function of angle φ between the directions of B and the,dipole moment. The dipole can be rotated about an axle with negligible friction so as to change φ. Counterclockwise rotation from φ = 0 yields positive values of and clockwise rotations yield negative values.the dipole is to be released at angle φ = 0 with a rotational kinetic energy of 6.7 x 10-4 J, so that it rotates counterclock-wise. To what maximum value of φ will it rotate? (In the language of energy, what value φ is the turning point in the potential well of the figure?) The figure shows a wire that carries current to the right through a uniform magnetic field. It also shows four choices for the direction of that field. (a) Rank the choices according to the magnitude of the electric potential difference that would be set up across the width of the wire, greatest first. (b) For which choice is the top side of the wire at higher potential than the bottom side of the wire? 138 An electron with a kinetic energy of 22.5 ev moves into a region of uniform magnetic field B of magnitude 4.55 x 10-4 T. The angle between the directions of B and the electron's velocity v is What is the pitch of the helical path taken by the electron? (a) 3 and 4 tie, then 1 and 2 tie (zero); (b) 4 (assuming that the rightward current is due to leftward motion of electrons in the wire) 139 At the equator, near the surface of the Earth, the magnetic field is approximately 50.0 µt northward, and the electric field is about 100 N/C downward in fair weather. Assume the electron has an instantaneous velocity of m/s directed to the east. (a) Find the gravitational force on an electron in this environment. (a) Find the electric force on an electron in this environment. (a) Find the magnetic force on an electron in this environment. (a) N down (b) N up (c) N up

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