Chapter 27

A particle with a charge of -1.24 \(\times\) 10\(^{-8}\) C is moving with instantaneous velocity \(\vec{v} =\) 14.19 \(\times\) 10\(^4\) m/s)\(\hat{\imath}\) + (-3.85 \(\times\) 10\(^4\) m/s)\(\hat{\jmath}\). What is the force exerted on this particle by a magnetic field (a) \(\overrightarrow{B} =\) (1.40 T)\(\hat{\imath}\) and (b) \(\overrightarrow{B} =\) (1.40 T) \(\hat{k}\) ?

A particle of mass 0.195 g carries a charge of -2.50 \(\times\) 10\(^{-8}\) C. The particle is given an initial horizontal velocity that is due north and has magnitude 4.00 \(\times\) 10\(^4\) m/s. What are the magnitude and direction of the minimum magnetic field that will keep the particle moving in the earth's gravitational field in the same horizontal, northward direction?

An electron experiences a magnetic force of magnitude 4.60 \(\times\) 10\(^{-15}\) N when moving at an angle of 60.0\(^\circ\) with respect to a magnetic field of magnitude 3.50 \(\times\) 10\(^{-3}\) T. Find the speed of the electron.

An electron moves at 1.40 \(\times\) 10\(^6\) m/s through a region in which there is a magnetic field of unspecified direction and magnitude 7.40 \(\times\) 10\(^{-2}\) T. (a) What are the largest and smallest possible magnitudes of the acceleration of the electron due to the magnetic field? (b) If the actual acceleration of the electron is one-fourth of the largest magnitude in part (a), what is the angle between the electron velocity and the magnetic field?

A particle with charge 7.80 \(\mu\)C is moving with velocity \(\vec{v} =\) - 13.80 \(\times\) 103m/s\(\hat{\jmath}\). The magnetic force on the particle is measured to be \(\overrightarrow{F} =\) (7.60 \(\times\) 10\(^{-3}\) N)\(\hat{\imath}\) - (5.20 \(\times\) 10\(^{-3}\) N)\(\hat{k}\). (a) Calculate all the components of the magnetic field you can from this information. (b) Are there components of the magnetic field that are not determined by the measurement of the force? Explain. (c) Calculate the scalar product \(\overrightarrow{B}\) \(\cdot\) \(\overrightarrow{F}\). What is the angle between \(\overrightarrow{B}\) and \(\overrightarrow{F}\)?

A particle with charge -5.60 nC is moving in a uniform magnetic field \(\overrightarrow{B} =\) -(1.25 T)\(\hat{k}\). The magnetic force on the particle is measured to be \(\overrightarrow{F} =\) -(3.40 \(\times\) 10\(^{-7}\)N)\(\hat{\imath}\) + (7.40 \(\times\) 10\(^{-7}\)N)\(\hat{\jmath}\). (a) Calculate all the components of the velocity of the particle that you can from this information. (b) Are there components of the velocity that are not determined by the measurement of the force? Explain. (c) Calculate the scalar product \(\vec{v}\) \(\cdot\) \(\overrightarrow{F}\). What is the angle between \(\vec{v}\) and \(\overrightarrow{F}\)?

A group of particles is traveling in a magnetic field of unknown magnitude and direction. You observe that a proton moving at 1.50 km/s in the \(+x\)-direction experiences a force of 2.25 \(\times\) 10\(^{-16}\) N in the \(+y\)-direction, and an electron moving at 4.75 km/s in the \(-z\)-direction experiences a force of 8.50 \(\times\) 10-16 N in the \(+y\)-direction. (a) What are the magnitude and direction of the magnetic field? (b) What are the magnitude and direction of the magnetic force on an electron moving in the \(-y\)-direction at 3.20 km/s?

A flat, square surface with side length 3.40 cm is in the xy-plane at \(z =\) 0. Calculate the magnitude of the flux through this surface produced by a magnetic field \(\overrightarrow{B} =\) (0.200 T)\(\hat{\imath}\) + (0.300 T)\(\hat{\jmath}\) - (0.500 T)\(\hat{k}\).

A circular area with a radius of 6.50 cm lies in the \(xy\)-plane. What is the magnitude of the magnetic flux through this circle due to a uniform magnetic field \(B =\) 0.230 T (a) in the \(+z\)-direction; (b) at an angle of 53.1\(^\circ\) from the \(+z\)-direction; (c) in the \(+y\)-direction?

A horizontal rectangular surface has dimensions 2.80 cm by 3.20 cm and is in a uniform magnetic field that is directed at an angle of 30.0\(^\circ\) above the horizontal. What must the magnitude of the magnetic field be to produce a flux of 3.10 \(\times\) 10\(^{-4}\) Wb through the surface?

A 150-g ball containing 4.00 \(\times\) 10\(^8\) excess electrons is dropped into a 125-m vertical shaft. At the bottom of the shaft, the ball suddenly enters a uniform horizontal magnetic field that has magnitude 0.250 T and direction from east to west. If air resistance is negligibly small, find the magnitude and direction of the force that this magnetic field exerts on the ball just as it enters the field.

An alpha particle (a He nucleus, containing two protons and two neutrons and having a mass of 6.64 \(\times\) 10\(^{-27}\) kg) traveling horizontally at 35.6 km>s enters a uniform, vertical, 1.80-T magnetic field. (a) What is the diameter of the path followed by this alpha particle? (b) What effect does the magnetic field have on the speed of the particle? (c) What are the magnitude and direction of the acceleration of the alpha particle while it is in the magnetic field? (d) Explain why the speed of the particle does not change even though an unbalanced external force acts on it.

Cyclotrons are widely used in nuclear medicine for producing short-lived radioactive isotopes. These cyclotrons typically accelerate H\(^-\) (the \(hydride\) ion, which has one proton and two electrons) to an energy of 5 MeV to 20 MeV. This ion has a mass very close to that of a proton because the electron mass is negligible-about \\(\frac{1}{2000}\\) of the proton's mass. A typical magnetic field in such cyclotrons is 1.9 T. (a) What is the speed of a 5.0-MeV H\(^-\)? (b) If the H\(^-\) has energy 5.0 MeV and \(B =\) 1.9 T, what is the radius of this ion's circular orbit?

A deuteron (the nucleus of an isotope of hydrogen) has a mass of 3.34 \(\times\) 10\(^{-27}\) kg and a charge of \(+e\). The deuteron travels in a circular path with a radius of 6.96 mm in a magnetic field with magnitude 2.50 T. (a) Find the speed of the deuteron. (b) Find the time required for it to make half a revolution. (c) Through what potential difference would the deuteron have to be accelerated to acquire this speed?

In a cyclotron, the orbital radius of protons with energy 300 keV is 16.0 cm. You are redesigning the cyclotron to be used instead for alpha particles with energy 300 keV. An alpha particle has charge \(q =\) +2\(e\) and mass \(m =\) 6.64 \(\times\) 10\(^{-27}\) kg. If the magnetic field isn't changed, what will be the orbital radius of the alpha particles?

An electron in the beam of a cathode-ray tube is accelerated by a potential difference of 2.00 kV. Then it passes through a region of transverse magnetic field, where it moves in a circular arc with radius 0.180 m. What is the magnitude of the field?

A proton (\(q\) = 1.60 \(\times\) 10\(^{-19}\) C, \(m =\) 1.67 \(\times\) 10\(^{-27}\) kg) moves in a uniform magnetic field \(\overrightarrow{B} =\) (0.500 T)\(\hat{\imath}\). At \(t =\) 0 the proton has velocity components \(\upsilon_x =\) 1.50 \(\times\) 10\(^5\) m/s, \(\upsilon_y =\) 0, and \(\upsilon_z =\) 2.00 \(\times\) 10\(^5\) m/s (see Example 27.4). (a) What are the magnitude and direction of the magnetic force acting on the proton? In addition to the magnetic field there is a uniform electric field in the +\(x\)-direction, \(\overrightarrow{E} =\) (+2.00 \(\times\) 10\(^4\) V/m)\(\hat{\imath}\). (b) Will the proton have a component of acceleration in the direction of the electric field? (c) Describe the path of the proton. Does the electric field affect the radius of the helix? Explain. (d) At \(t =\) \(T\)/2, where T is the period of the circular motion of the proton, what is the \(x\)-component of the displacement of the proton from its position at \(t =\) 0?

A singly charged ion of \(^7\)Li (an isotope of lithium) has a mass of 1.16 \(\times\) 10\(^{-26}\) kg. It is accelerated through a potential difference of 220 V and then enters a magnetic field with magnitude 0.874 T perpendicular to the path of the ion. What is the radius of the ion's path in the magnetic field?

A particle with initial velocity \(\vec{v}$$_0 =\) (5.85 \(\times\) 10\(^3\)m/s)\(\hat{\jmath}\) enters a region of uniform electric and magnetic fields. The magnetic field in the region is \(\overrightarrow{B} =\) - (1.35 T)\(\hat{k}\). Calculate the magnitude and direction of the electric field in the region if the particle is to pass through undeflected, for a particle of charge (a) +0.640 nC and (b) -0.320 nC. You can ignore the weight of the particle.

(a) What is the speed of a beam of electrons when the simultaneous influence of an electric field of 1.56 \(\times\) 10\(^4\) V/m and a magnetic field of 4.62 \(\times\) 10\(^{-3}\) T, with both fields normal to the beam and to each other, produces no deflection of the electrons? (b) In a diagram, show the relative orientation of the vectors \(\vec{v}\), \(\overrightarrow{E}\), and \(\overrightarrow{B}\). (c) When the electric field is removed, what is the radius of the electron orbit? What is the period of the orbit?

A singly ionized (one electron removed) \(^{40}\)K atom passes through a velocity selector consisting of uniform perpendicular electric and magnetic fields. The selector is adjusted to allow ions having a speed of 4.50 km/s to pass through undeflected when the magnetic field is 0.0250 T. The ions next enter a second uniform magnetic field (\(B'\)) oriented at right angles to their velocity. \(^{40}\)K contains 19 protons and 21 neutrons and has a mass of 6.64 \(\times\) 10\(^{-26}\) kg. (a) What is the magnitude of the electric field in the velocity selector? (b) What must be the magnitude of \(B'\) so that the ions will be bent into a semicircle of radius 12.5 cm?

Singly ionized (one electron removed) atoms are accelerated and then passed through a velocity selector consisting of perpendicular electric and magnetic fields. The electric field is 155 V/m and the magnetic field is 0.0315 T. The ions next enter a uniform magnetic field of magnitude 0.0175 T that is oriented perpendicular to their velocity. (a) How fast are the ions moving when they emerge from the velocity selector? (b) If the radius of the path of the ions in the second magnetic field is 17.5 cm, what is their mass?

In the Bainbridge mass spectrometer (see Fig. 27.24), the magnetic-field magnitude in the velocity selector is 0.510 T, and ions having a speed of 1.82 \(\times\) 10\(^6\) m/s pass through undeflected. (a) What is the electric-field magnitude in the velocity selector? (b) If the separation of the plates is 5.20 mm, what is the potential difference between the plates?

The amount of meat in prehistoric diets can be determined by measuring the ratio of the isotopes \(^{15}\)N to \(^{14}\)N in bone from human remains. Carnivores concentrate \(^{15}\)N, so this ratio tells archaeologists how much meat was consumed. For a mass spectrometer that has a path radius of 12.5 cm for \(^{12}\)C ions (mass 1.99 \(\times\) 10\(^{-26}\) kg), find the separation of the \(^{14}\)N 1mass 2.32 \(\times\) 10\(^{-26}\) kg2 and 15N (mass 2.49 \(\times\) 10\(^{-26}\) kg) isotopes at the detector.

A straight, 2.5-m wire carries a typical household current of 1.5 A (in one direction) at a location where the earth's magnetic field is 0.55 gauss from south to north. Find the magnitude and direction of the force that our planet's magnetic field exerts on this wire if it is oriented so that the current in it is running (a) from west to east, (b) vertically upward, (c) from north to south. (d) Is the magnetic force ever large enough to cause significant effects under normal household conditions?

An electromagnet produces a magnetic field of 0.550 T in a cylindrical region of radius 2.50 cm between its poles. A straight wire carrying a current of 10.8 A passes through the center of this region and is perpendicular to both the axis of the cylindrical region and the magnetic field. What magnitude of force does this field exert on the wire?

A straight, vertical wire carries a current of 2.60 A downward in a region between the poles of a large superconducting electromagnet, where the magnetic field has magnitude \(B =\) 0.588 T and is horizontal. What are the magnitude and direction of the magnetic force on a 1.00-cm section of the wire that is in this uniform magnetic field, if the magnetic field direction is (a) east; (b) south; (c) 30.0\(^\circ\) south of west?

The plane of a 5.0 cm \(\times\) 8.0 cm rectangular loop of wire is parallel to a 0.19-T magnetic field. The loop carries a current of 6.2 A. (a) What torque acts on the loop? (b) What is the magnetic moment of the loop? (c) What is the maximum torque that can be obtained with the same total length of wire carrying the same current in this magnetic field?

A coil with magnetic moment 1.45 A \(\cdot\) m\(^2\) is oriented initially with its magnetic moment antiparallel to a uniform 0.835-T magnetic field. What is the change in potential energy of the coil when it is rotated 180\(^\circ\) so that its magnetic moment is parallel to the field?

A dc motor with its rotor and field coils connected in series has an internal resistance of 3.2 \(\Omega\). When the motor is running at full load on a 120-V line, the emf in the rotor is 105 V. (a) What is the current drawn by the motor from the line? (b) What is the power delivered to the motor? (c) What is the mechanical power developed by the motor?

A particle with charge 7.26 \(\times\) 10\(^{-8}\) C is moving in a region where there is a uniform 0.650-T magnetic field in the +\(x\)-direction. At a particular instant, the velocity of the particle has components \(v_x =\) -1.68 \(\times\) 10\(^4\) m/s, \(v_y =\) -3.11 \(\times\) 104 m/s, and \(v_z =\) 5.85 \(\times\) 10\(^4\) m/s. What are the components of the force on the particle at this time?

If two deuterium nuclei (charge \(+e\), mass 3.34 \(\times\) 10\(^{-27}\) kg) get close enough together, the attraction of the strong nuclear force will fuse them to make an isotope of helium, releasing vast amounts of energy. The range of this force is about 10\(^{-15}\) m. This is the principle behind the fusion reactor. The deuterium nuclei are moving much too fast to be contained by physical walls, so they are confined magnetically. (a) How fast would two nuclei have to move so that in a head-on collision they would get close enough to fuse? (Assume their speeds are equal. Treat the nuclei as point charges, and assume that a separation of 1.0 \(\times\) 10\(^{-15}\) is required for fusion.) (b) What strength magnetic field is needed to make deuterium nuclei with this speed travel in a circle of diameter 2.50 m?

In the Bohr model of the hydrogen atom (see Section 39.3), in the lowest energy state the electron orbits the proton at a speed of 2.2 \(\times\) 10\(^6\) m/s in a circular orbit of radius 5.3 \(\times\) 10\(^{-11}\) m. (a) What is the orbital period of the electron? (b) If the orbiting electron is considered to be a current loop, what is the current \(I\)? (c) What is the magnetic moment of the atom due to the motion of the electron?

The magnetic poles of a small cyclotron produce a magnetic field with magnitude 0.85 T. The poles have a radius of 0.40 m, which is the maximum radius of the orbits of the accelerated particles. (a) What is the maximum energy to which protons (\(q =\) 1.60 \(\times\) 10\(^{-19}\)C, \(m =\) 1.67 \(\times\) 10\(^{-27}\) kg) can be accelerated by this cyclotron? Give your answer in electron volts and in joules. (b) What is the time for one revolution of a proton orbiting at this maximum radius? (c) What would the magnetic-field magnitude have to be for the maximum energy to which a proton can be accelerated to be twice that calculated in part (a)? (d) For \(B =\) 0.85 T, what is the maximum energy to which alpha particles (\(q =\) 3.20 \(\times\) 10\(^{-19}\) C, \(m =\) 6.64 \(\times\) 10\(^{-27}\) kg) can be accelerated by this cyclotron? How does this compare to the maximum energy for protons?

A particle with negative charge q and mass \(m =\) 2.58 \(\times\) 10\(^{-15}\) kg is traveling through a region containing a uniform magnetic field \(\overrightarrow{B} =\) -(0.120 T)\(\hat{k}\). At a particular instant of time the velocity of the particle is \(\vec{v}\) (1.05 \(\times\) 10\(^6\) m/s (-3\(\hat{\imath}\)+4\(\hat{\jmath}\)+12\(\hat{k}\)) and the force \(\overrightarrow{F}\) on the particle has a magnitude of 2.45 N. (a) Determine the charge \(q\). (b) Determine the acceleration \(\overrightarrow{a}\) of the particle. (c) Explain why the path of the particle is a helix, and determine the radius of curvature \(R\) of the circular component of the helical path. (d) Determine the cyclotron frequency of the particle. (e) Although helical motion is not periodic in the full sense of the word, the \(x\)- and \(y\)-coordinates do vary in a periodic way. If the coordinates of the particle at \(t =\) 0 are (\(x, y, z\)) = (\(R\), 0, 0), determine its coordinates at a time \(t =\) 2\(T\), where \(T\) is the period of the motion in the \(xy\)-plane.

A particle of charge \(q\) > 0 is moving at speed v in the \(+z\)-direction through a region of uniform magnetic field \(\overrightarrow{B}\). The magnetic force on the particle is \(\overrightarrow{F} =\) \(F_0\)(3\(\hat{\imath}\) + 4 \(\hat{\jmath}\)), where \(F_0\) is a positive constant. (a) Determine the components \(B_x\), \(B_y\), and \(B_z\), or at least as many of the three components as is possible from the information given. (b) If it is given in addition that the magnetic field has magnitude 6\(F_0/qv\), determine as much as you can about the remaining components of \(\overrightarrow{B}\).

A mass spectrograph is used to measure the masses of ions, or to separate ions of different masses (see Section 27.5). In one design for such an instrument, ions with mass \(m\) and charge \(q\) are accelerated through a potential difference \(V\). They then enter a uniform magnetic field that is perpendicular to their velocity, and they are deflected in a semicircular path of radius \(R\). A detector measures where the ions complete the semicircle and from this it is easy to calculate \(R\). (a) Derive the equation for calculating the mass of the ion from measurements of \(B\), \(V\), \(R\), and \(q\). (b) What potential difference \(V\) is needed so that singly ionized \(^{12}\)C atoms will have \(R =\) 50.0 cm in a 0.150-T magnetic field? (c) Suppose the beam consists of a mixture of \(^{12}\)C and \(^{14}\)C ions. If \(v\) and \(B\) have the same values as in part (b), calculate the separation of these two isotopes at the detector. Do you think that this beam separation is sufficient for the two ions to be distinguished? (Make the assumption described in Problem 27.59 for the masses of the ions.)

\(\textbf{Determining Diet.}\) One method for determining the amount of corn in early Native American diets is the \(stable\) \(isotope\) \(ratio\) \(analysis\) (SIRA) technique. As corn photosynthesizes, it concentrates the isotope carbon-13, whereas most other plants concentrate carbon-12. Overreliance on corn consumption can then be correlated with certain diseases, because corn lacks the essential amino acid lysine. Archaeologists use a mass spectrometer to separate the \(^{12}\)C and \(^{13}\)C isotopes in samples of human remains. Suppose you use a velocity selector to obtain singly ionized (missing one electron) atoms of speed 8.50 km /s, and you want to bend them within a uniform magnetic field in a semicircle of diameter 25.0 cm for the \(^{12}\)C. The measured masses of these isotopes are 1.99 \(\times\) 10\(^{-26}\) kg (\(^{12}\)C) and 2.16 \(\times\) 10\(^{-26}\) kg (\(^{13}\)C). (a) What strength of magnetic field is required? (b) What is the diameter of the \(^{13}\)C semicircle? (c) What is the separation of the \(^{12}\)C and \(^{13}\)C ions at the detector at the end of the semicircle? Is this distance large enough to be easily observed?

A plastic circular loop has radius \(R\), and a positive charge q is distributed uniformly around the circumference of the loop. The loop is then rotated around its central axis, perpendicular to the plane of the loop, with angular speed \(\omega\). If the loop is in a region where there is a uniform magnetic field \(\overrightarrow{B}\) directed parallel to the plane of the loop, calculate the magnitude of the magnetic torque on the loop.

A conducting bar with mass m and length \(L\) slides over horizontal rails that are connected to a voltage source. The voltage source maintains a constant current \(I\) in the rails and bar, and a constant, uniform, vertical magnetic field \(\overrightarrow{B}\) fills the region between the rails (\(\textbf{Fig. P27.65}\)). (a) Find the magnitude and direction of the net force on the conducting bar. Ignore friction, air resistance, and electrical resistance. (b) If the bar has mass \(m\), find the distance \(d\) that the bar must move along the rails from rest to attain speed \(v\). (c) It has been suggested that rail guns based on this principle could accelerate payloads into earth orbit or beyond. Find the distance the bar must travel along the rails if it is to reach the escape speed for the earth (11.2 km/s). Let \(B =\) 0.80 T, \(I =\) 2.0 \(\times\) 10\(^3\) A, \(m =\) 25 kg, and \(L =\) 50 cm. For simplicity assume the net force on the object is equal to the magnetic force, as in parts (a) and (b), even though gravity plays an important role in an actual launch in space.

A wire 25.0 cm long lies along the \(z\)-axis and carries a current of 7.40 A in the \(+z\)-direction. The magnetic field is uniform and has components \(B_x =\) -0.242 T, \(B_y =\) -0.985 T, and \(B_z\) = -0.336 T. (a) Find the components of the magnetic force on the wire. (b) What is the magnitude of the net magnetic force on the wire?

A circular loop of wire with area \(A\) lies in the \(xy\)-plane. As viewed along the \(z\)-axis looking in the -\(z\)-direction toward the origin, a current \(I\) is circulating clockwise around the loop. The torque produced by an external magnetic field \(\overrightarrow{B}\) is given by \(\vec{\tau}\) = D(4\(\hat{\imath}\) - 3\(\hat{\jmath}\)), where \(D\) is a positive constant, and for this orientation of the loop the magnetic potential energy \(U = -\vec{\mu}\) \(\cdot\) \(\overrightarrow{B}\) is negative. The magnitude of the magnetic field is \(B_0 = 13D/IA\). (a) Determine the vector magnetic moment of the current loop. (b) Determine the components \(B_x\), \(B_y\), and \(B_z\) of \(\overrightarrow{B}\).

If a proton is exposed to an external magnetic field of 2 T that has a direction perpendicular to the axis of the proton's spin, what will be the torque on the proton? (a) 0; (b) 1.4 \(\times\) 10\(^{-26}\) N \(\cdot\) m; (c) 2.8 \(\times\) 10\(^{-26}\) N \(\cdot\) m; (d) 0.7 \(\times\) 10\(^{-26}\) N \(\cdot\) m.