P6.27.f^Chapter 27 Ending^882^891^,,^37255^37725%
WHAT WE HAVE LEARNED
EXAM STUDY GUIDE
Page 884
KEY TERMS
permanent magnets, p. 865 north magnetic pole, p. 865 south magnetic pole, p. 865 magnetic field, p. 866 magnetic field lines, 866 Van Allen radiation belts, p. 866 aurora borealis, p. 867 aurora australis, p. 867 magnetic declination, p. 867 tesla, p. 869 cyclotron frequency, p. 874 mass spectrometer, p. 875 magnetic levitation, p. 875 commutator, p. 880 magnetic dipole moment, p. 881 Hall effect, 882 Hall potential difference, 882
NEW SYMBOLS AND EQUATIONS

, magnetic field

, magnetic force on a charged particle

, magnetic force on a current-carrying wire

, magnetic dipole moment

, Hall potential difference

ANSWERS TO SELF-TEST OPPORTUNITIES
PROBLEM-SOLVING PRACTICE
Problem-Solving Guidelines
  1. FIGURE 27.32
    (a) A rectangular loop carrying a current in a magnetic field. (b) View of the rectangular loop looking down on the xy-plane. The magnetic dipole moment is perpendicular to the plane of the loop, with a direction determined by right-hand rule 2.

    When working with magnetic fields and forces, you need to sketch a clear diagram of the problem situation in three dimensions. Often, a separate sketch of the velocity and magnetic field vectors (or the length and field vectors) is useful to visualize the plane in which they lie, since the magnetic force will be perpendicular to that plane.

  2. Remember that the right-hand rules apply for positive charges and currents. If a charge or a current is negative, you can use the right-hand rule but the force will then be in the opposite direction.

  3. A particle in both an electric and a magnetic field experiences an electric force, , and a magnetic force, . Be sure you take the vector sum of the individual forces.

SOLVED PROBLEM27.3 Torque on a Rectangular Current-Carrying Loop

A rectangular loop with height h = 6.50 cm and width w = 4.50 cm is in a uniform magnetic field of magnitude B = 0.250 T, which points in the negative y-direction (Figure 27.32a). The loop makes an angle of θ = 33.0° with the y-axis, as shown in the figure. The loop carries a current of magnitude i = 9.00 A in the direction indicated by the arrows.

PROBLEM

What is the magnitude of the torque on the loop around the z-axis?

SOLUTION
THINK

The torque on the loop is equal to the vector cross product of the magnetic dipole moment and the magnetic field. The magnetic dipole moment is perpendicular to the plane of the loop, with the direction given by right-hand rule 2.

SKETCH

Figure 27.32b is a view of the loop looking down on the xy-plane.

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RESEARCH

The magnitude of the magnetic dipole moment of the loop is

(i)

The magnitude of the torque on the loop is

(ii)

where θμB is the angle between the magnetic dipole moment and the magnetic field. From Figure 27.32b we can see that

(iii)
SIMPLIFY

We can combine equations (i), (ii), and (iii) to obtain

CALCULATE

Putting in the numerical values, we get

ROUND

We report our result to three significant figures:

DOUBLE-CHECK

The magnitude of the force on each of the vertical segments of the loop is

The magnitude of the torque is then the magnitude of the force on the vertical segment that is not along the z-axis times the moment arm (which is w) times the sine of the angle between the force and the moment arm:

This is the same as the result calculated above.

MULTIPLE-CHOICE QUESTIONS
PROBLEMS

A blue problem number indicates a worked-out solution is available in the Student Solutions Manual. One and two •• indicate increasing level of problem difficulty.

Section 27.2
  • 27.20 A proton moving with a speed of 4.0 · 105 m/s in the positive y-direction enters a uniform magnetic field of 0.40 T pointing in the positive x-direction. Calculate the magnitude of the force on the proton.

  • 27.21 The magnitude of the magnetic force on a particle with charge −2e moving with speed v = 1.0 · 105 m/s is 3.0 · 10−18 N. What is the magnitude of the magnetic field component perpendicular to the direction of motion of the particle?

    Answer

  • 27.22 A particle with a charge of +10.0 μC is moving at 300. m/s in the positive z-direction.

    1. Find the minimum magnetic field required to keep it moving in a straight line at constant speed if there is a uniform electric field of magnitude 100. V/m pointing in the positive y-direction.

    2. Find the minimum magnetic field required to keep the particle moving in a straight line at constant speed if there is a uniform electric field of magnitude 100. V/m pointing in the positive z-direction.

  • 27.23 A particle with a charge of 20.0 μC moves along the x-axis with a speed of 50.0 m/s. It enters a magnetic field given by , in teslas. Determine the magnitude and the direction of the magnetic force on the particle.

    Answer

  • ••27.24 The magnetic field in a region in space (where x > 0 and y > 0) is given by , where a and b are positive constants. An electron moving with a constant velocity, , enters this region. What are the coordinates of the points at which the net force acting on the electron is zero?

Section 27.3
  • 27.25 A proton is accelerated from rest by a potential difference of 400. V. The proton enters a uniform magnetic field and follows a circular path of radius 20.0 cm. Determine the magnitude of the magnetic field.

    Answer

  • 27.26 An electron with a speed of 4.0 · 105 m/s enters a uniform magnetic field of magnitude 0.040 T at an angle of 35° to the magnetic field lines. The electron will follow a helical path.

    1. Determine the radius of the helical path.

    2. How far forward will the electron have moved after completing one circle?

  • 27.27 A particle with mass m and charge q is moving within both an electric field and a magnetic field, and . The particle has velocity , momentum , and kinetic energy, K. Find general expressions for and dK/dt, in terms of these seven quantities.

    Answer

  • 27.28 The Earth is showered with particles from space known as muons. They have a charge identical to that of an electron but are many times heavier (m = 1.88 · 10−28 kg). Suppose a strong magnetic field is established in a lab (B = 0.50 T) and a muon enters this field with a velocity of 3.0 · 106 m/s at a right angle to the field. What will be the radius of the resulting orbit of the muon?

  • 27.29 An electron in a magnetic field moves counterclockwise on a circle in the xy-plane, with a cyclotron frequency of ω = 1.2 · 1012 Hz. What is the magnetic field, ?

    Answer

  • 27.30 An electron with energy equal to 4.00 · 102 eV and an electron with energy equal to 2.00 · 102 eV are trapped in a uniform magnetic field and move in circular paths in a plane perpendicular to the magnetic field. What is the ratio of the radii of their orbits?

  • 27.31 A proton with an initial velocity given by (1.0 + 2.0 + 3.0)(105 m/s) enters a magnetic field given by (0.50 T). Describe the motion of the proton.

    Answer

  • 27.32 Initially at rest, a small copper sphere with a mass of 3.00 · 10−6 kg and a charge of 5.00 · 10−4 C is accelerated through a 7000.-V potential difference before entering a magnetic field of magnitude 4.00 T, directed perpendicular to its velocity. What is the radius of curvature of the sphere's motion in the magnetic field?

  • •27.33 Two particles with masses m1 and m2 and charges q and 2q travel with the same velocity, v, and enter a magnetic field of strength B at the same point, as shown in the figure. In the magnetic field, they move in semicircles with radii R and 2R. What is the ratio of their masses? Is it possible to apply an electric field that would cause the particles to move in a straight line in the magnetic field? If yes, what would be the magnitude and direction of the field?

    Answer

  • 27.34 The figure shows a schematic diagram of a simple mass spectrometer, consisting of a velocity selector and a particle detector and being used to separate singly ionized atoms (q = +e = 1.60 · 10−19 C) of gold (Au) and molybdenum (Mo). The electric field inside the velocity selector has magnitude E = 1.789 · 104 V/m and points toward the top of the page, and the magnetic field has magnitude B1 = 1.00 T and points out of the page.

     
    Page 888
    1. Draw the electric force vector, , and the magnetic force vector, , acting on the ions inside the velocity selector.

    2. Calculate the velocity, v0, of the ions that make it through the velocity selector (those that travel in a straight line). Does v0 depend on the type of ion (gold versus molybdenum), or is it the same for both types of ions?

    3. Write the equation for the radius of the semicircular path of an ion in the particle detector: R = R(m, v0, q, B2).

    4. The gold ions (represented by the black circles) exit the particle detector at a distance d2 = 40.00 cm from the entrance slit, while the molybdenum ions (represented by the gray circles) exit the particle detector at a distance d1 = 19.81 cm from the entrance slit. The mass of a gold ion is mgold = 3.27 · 10−25 kg. Calculate the mass of a molybdenum ion.

  • ••27.35 A small particle accelerator for accelerating 3He+ ions is shown in the figure. The 3He+ ions exit the ion source with a kinetic energy of 4.00 keV. Regions 1 and 2 contain magnetic fields directed into the page, and region 3 contains an electric field directed from left to right. The 3He+ ion beam exits the accelerator from a hole on the right that is 7.00 cm below the ion source, as shown in the figure.

    1. If B1 = 1.00 T and region 3 is 50.0 cm long with E = 60.0 kV/m, what value should B2 have to cause the ions to move straight through the exit hole after being accelerated twice in region 3?

    2. What minimum width X should region 1 have?

    3. What is the velocity of the ions when they leave the accelerator?

    Answer

Section 27.4
  • 27.36 A straight wire of length 2.00 m carries a current of 24.0 A. It is placed on a horizontal tabletop in a uniform horizontal magnetic field. The wire makes an angle of 30.0° with the magnetic field lines. If the magnitude of the force on the wire is 0.500 N, what is the magnitude of the magnetic field?

  • 27.37 As shown in the figure, a straight conductor parallel to the x-axis can slide without friction on top of two horizontal conducting rails that are parallel to the y-axis and a distance of L = 0.200 m apart, in a vertical magnetic field of 1.00 T. A 20.0-A current is maintained through the conductor. If a string is connected exactly at the center of the conductor and passes over a frictionless pulley, what mass m suspended from the string allows the conductor to be at rest?

    Answer

  • 27.38 A copper wire of radius 0.500 mm is carrying a current at the Earth's Equator. Assuming that the magnetic field of the Earth has magnitude 0.500 G at the Equator and is parallel to the surface of the Earth and that the current in the wire flows toward the east, what current is required to allow the wire to levitate?

  • 27.39 A copper sheet with length 1.0 m, width 0.50 m, and thickness 1.0 mm is oriented so that its largest surface area is perpendicular to a magnetic field of strength 5.0 T. The sheet carries a current of 3.0 A across its length. What is the magnitude of the force on this sheet? How does this magnitude compare to that of the force on a thin copper wire carrying the same current and oriented perpendicularly to the same magnetic field?

    Answer

  • 27.40 A conducting rod of length L slides freely down an inclined plane, as shown in the figure. The plane is inclined at an angle θ from the horizontal. A uniform magnetic field of strength B acts in the positive y-direction. Determine the magnitude and the direction of the current that would have to be passed through the rod to hold it in position on the inclined plane.

  • •27.41 A square loop of wire, with side length d = 8.0 cm, carries a current of magnitude i = 0.15 A and is free to rotate. It is placed between the poles of an electromagnet that produce a uniform magnetic field of 1.0 T. The loop is initially placed so that its normal vector, , is at a 35.0° angle relative to the direction of the magnetic field vector, with the angle θ defined as shown in the figure. The wire is copper (with a density of ρ = 8960 kg/m3), and its diameter is 0.50 mm. What is the magnitude of the initial angular acceleration of the loop when it is released?

    Answer

  • Page 889
  • 27.42 A rail gun accelerates a projectile from rest by using the magnetic force on a current-carrying wire. The wire has radius r = 5.1 · 10−4 m and is made of copper having a density of ρ = 8960 kg/m3 The gun consists of rails of length L = 1.0 m in a constant magnetic field of magnitude B = 2.0 T, oriented perpendicular to the plane defined by the rails. The wire forms an electrical connection across the rails at one end of the rails. When triggered, a current of 1.00 · 104 A flows through the wire, which accelerates the wire along the rails. Calculate the final speed of the wire as it leaves the rails. (Neglect friction.)

Sections 27.5 and 27.6
  • 27.43 A square loop of wire of side length lies in the xy-plane, with its center at the origin and its sides parallel to the x- and y-axes. It carries a current, i, in the counterclockwise direction, as viewed looking down the z-axis from the positive direction. The loop is in a magnetic field given by , where B0 is a constant field strength, a is a constant with the dimension of length, and and are unit vectors in the positive x-direction and positive z-direction. Calculate the net force on the loop.

    Answer

  • 27.44 A rectangular coil with 20 windings carries a current of 2.00 mA flowing in the counterclockwise direction. It has two sides that are parallel to the y-axis and have length 8.00 cm and two sides that are parallel to the x-axis and have length 6.00 cm. A uniform magnetic field of 50.0 μT acts in the positive x-direction. What torque must be applied to the loop to hold it steady?

  • 27.45 A coil consists of 120 circular loops of wire of radius 4.8 cm. A current of 0.49 A runs through the coil, which is oriented vertically and is free to rotate about a vertical axis (parallel to the z-axis). It experiences a uniform horizontal magnetic field in the positive x-direction. When the coil is oriented parallel to the x-axis, a force of 1.2 N applied to the edge of the coil in the positive y-direction can keep it from rotating. Calculate the strength of the magnetic field.

    Answer

  • 27.46 Twenty loops of wire are tightly wound around a round pencil that has a diameter of 6.00 mm. The pencil is then placed in a uniform 5.00-T magnetic field, as shown in the figure. If a 3.00-A current is present in the coil of wire, what is the magnitude of the torque on the pencil?

  • 27.47 A copper wire with density ρ = 8960 kg/m3 is formed into a circular loop of radius 50.0 cm. The cross-sectional area of the wire is 1.00 · 10−5 m2, and a potential difference of 0.012 V is applied to the wire. What is the maximum angular acceleration of the loop when it is placed in a magnetic field of magnitude 0.25 T? The loop rotates about an axis through a diameter.

    Answer

  • 27.48 A simple galvanometer is made from a coil that consists of N loops of wire of area A. The coil is attached to a mass, M, by a light rigid rod of length L. With no current in the coil, the mass hangs straight down, and the coil lies in a horizontal plane. The coil is in a uniform magnetic field of magnitude B that is oriented horizontally. Calculate the angle from the vertical of the rigid rod as a function of the current, i, in the coil.

  • 27.49 Show that the magnetic dipole moment of an electron orbiting in a hydrogen atom is proportional to its angular momentum, L: μ = –eL/2m, where –e is the charge of the electron and m is its mass.

  • 27.50 The figure shows a top view of a current-carrying ring of wire having a diameter d = 8.00 cm, which is suspended from the ceiling via a thin string. A 1.00-A current flows in the ring in the direction indicated in the figure. The ring is connected to one end of a spring with a spring constant of 100. N/m. When the ring is in the position shown in the figure, the spring is at its equilibrium length, . Determine the extension of the spring when a magnetic field of magnitude B = 2.00 T is applied parallel to the plane of the ring as shown in the figure.

  • 27.51 A coil of wire consisting of 40 rectangular loops, with width 16.0 cm and height 30.0 cm, is placed in a constant magnetic field given by . The coil is hinged to a fixed thin rod along the y-axis (along segment da in the figure) and is originally located in the xy-plane. A current of 0.200 A runs through the wire.

    Page 890
    1. What are the magnitude and the direction of the force, , that exerts on segment ab of the coil?

    2. What are the magnitude and the direction of force, , that exerts on segment bc of the coil?

    3. What is the magnitude of the net force, Fnet, that exerts on the coil?

    4. What are the magnitude and the direction of the torque, , that exerts on the coil?

    5. In what direction, if any, will the coil rotate about the y-axis (viewed from above and looking down that axis)?

    Answer

Section 27.7
  • 27.52 A high electron mobility transistor (HEMT) controls large currents by applying a small voltage to a thin sheet of electrons. The density and mobility of the electrons in the sheet are critical for the operation of the HEMT. HEMTs consisting of AlGaN/GaN/Si are being studied because they promise better performance at higher powers, temperatures, and frequencies than conventional silicon HEMTs can achieve. In one study, the Hall effect was used to measure the density of electrons in one of these new HEMTs. When a current of 10.0 μ A flows through the length of the electron sheet, which is 1.00 mm long, 0.300 mm wide, and 10.0 nm thick, a magnetic field of 1.00 T perpendicular to the sheet produces a voltage of 0.680 mV across the width of the sheet. What is the density of electrons in the sheet?

  • •27.53 The figure shows schematically a setup for a Hall effect measurement using a thin film of zinc oxide of thickness 1.50 μm. The current, i, across the thin film is 12.3 mA and the Hall potential, VH, is −20.1 mV when the magnetic field of magnitude B = 0.90 T is applied perpendicular to the current flow.

    1. What are the charge carriers in the thin film? [Hint: They can be either electrons with charge –e or electron holes (missing electrons) with charge +e.]

    2. Calculate the density of charge carriers in the thin film.

    Answer

Additional Problems
  • 27.54 A cyclotron in a magnetic field of 9 T is used to accelerate protons to 50% of the speed of light. What is the cyclotron frequency of these protons? What is the radius of their trajectory in the cyclotron? What is the cyclotron frequency and trajectory radius of the same protons in the Earth's magnetic field? Assume that the Earth's magnetic field is about 0.5 G.

  • 27.55 A straight wire carrying a current of 3.41 A is placed at an angle of 10.0° to the horizontal between the pole tips of a magnet producing a field of 0.220 T upward. The poles' tips each have a 10.0 cm diameter. The magnetic force causes the wire to move out of the space between the poles. What is the magnitude of that force?

    Answer

  • 27.56 An electron is moving at v = 6.00 · 107m/s perpendicular to the Earth's magnetic field. If the field strength is 0.500 · 10−4 T, what is the radius of the electron's circular path?

  • 27.57 A straight wire with a constant current running through it is in Earth's magnetic field, at a location where the magnitude is 0.43 G. What is the minimum current that must flow through the wire for a 10.0-cm length of it to experience a force of 1.0 N?

    Answer

  • 27.58 A small aluminum ball with a mass of 5.00 g and a charge of 15.0 C is moving northward at 3000. m/s. You want the ball to travel in a horizontal circle with a radius of 2.00 m, in a clockwise sense when viewed from above. Ignoring gravity, what is the magnitude and the direction of the magnetic field that must be applied to the aluminum ball to cause it to have this motion?

  • 27.59 The velocity selector described in Solved Problem 27.2 is used in a variety of devices to produce a beam of charged particles of uniform velocity. Suppose the fields in such a selector are given by and . Find the velocity in the z-direction with which a charged particle can travel through the selector without being deflected.

    Answer

  • 27.60 A circular coil with a radius of 10.0 cm has 100. turns of wire and carries a current, i = 100. mA. It is free to rotate in a region with a constant horizontal magnetic field given by . If the unit normal vector to the plane of the coil makes an angle of 30.0° with the horizontal, what is the magnitude of the net magnetic torque acting on the coil?

  • 27.61 At t = 0 an electron crosses the positive y-axis (so x = 0) at 60.0 cm from the origin with velocity 2.00 · 105 m/s in the positive x-direction. It is in a uniform magnetic field.

    1. Find the magnitude and the direction of the magnetic field that will cause the electron to cross the x-axis at x = 60.0 cm.

    2. What work is done on the electron during this motion?

    3. How long will the trip take from y-axis to x-axis take?

    Answer

  • 27.62 A 12.0-V battery is connected to a 3.00-Ω resistor in a rigid rectangular loop of wire measuring 3.00 m by 1.00 m. As shown in the figure, a length = 1.00 m of wire at the end of the loop extends into a 2.00 m by 2.00 m region with a magnetic field of magnitude 5.00 T, directed into the page. What is the net force on the loop?

  • 27.63 An alpha particle (m = 6.6 · 10−27kg, q = +2e) is accelerated by a potential difference of 2700 V and moves in a plane perpendicular to a constant magnetic field of magnitude 0.340 T, which curves the trajectory of the alpha particle. Determine the radius of curvature and the period of revolution.

    Answer

  • 27.64 In a certain area, the electric field near the surface of the Earth is given by , and the Earth's magnetic field is given by where is a unit vector pointing vertically upward and N is a horizontal unit vector pointing due north. What velocity, , will allow an electron in this region to move in a straight line at constant speed?

  • •27.65 A helium leak detector uses a mass spectrometer to detect tiny leaks in a vacuum chamber. The chamber is evacuated with a vacuum pump and then sprayed with helium gas on the outside. If there is any leak, the helium molecules pass through the leak and into the chamber, whose volume is sampled by the leak detector. In the spectrometer, helium ions are accelerated and released into a tube, where their motion is perpendicular to an applied magnetic field, and they follow a circular orbit of radius r and then hit a detector. Estimate the velocity required if the orbital radius of the ions is to be no more than 5 cm, the magnetic field is 0.15 T, and the mass of a helium-4 atom is about 6.6 · 10−27kg. Assume that each ion is singly ionized (has one electron less than the neutral atom). By what factor does the required velocity change if helium-3 atoms, which have about as much mass as helium-4 atoms, are used?

    Answer

  • Page 891
  • 27.66 In your laboratory, you set up an experiment with an electron gun that emits electrons with energy of 7.50 keV toward an atomic target. What deflection (magnitude and direction) would Earth's magnetic field (0.300 G) produce in the beam of electrons if the beam is initially directed due east and covers a distance of 1.00 m from the gun to the target? (Hint: First calculate the radius of curvature, and then determine how far away from a straight line the electron beam has deviated after 1.00 m.)

  • 27.67 A proton enters the region between the two plates shown in the figure moving in the x-direction with a speed v =1.35 · 106 m/s. The potential of the top plate is 200. V, and the potential of the bottom plate is 0 V. What is the direction and the magnitude of the magnetic field, , that is required between the plates for the proton to continue traveling in a straight line along the x-direction?

    Answer

  • 27.68 An electron moving at a constant velocity, enters a region in space where a magnetic field is present. The magnetic field, is constant and points in the z-direction. What is the magnitude and direction of the magnetic force acting on the electron? If the width of the region where the magnetic field is present is d, what is the minimum velocity the electron must have in order to escape this region?

  • •27.69 A 30-turn square coil with a mass of 0.250 kg and a side length of 0.200 m is hinged along a horizontal side and carries a 5.00-A current. It is placed in a magnetic field pointing vertically downward and having a magnitude of 0.00500 T. Determine the angle that the plane of the coil makes with the vertical when the coil is in equilibrium. Use g = 9.81 m/s2.

    Answer

  • 27.70 A semicircular loop of wire of radius R is in the xy-plane, centered about the origin. The wire carries a current, i, counterclockwise around the semicircle, from x = –R to x = +R on the x-axis. A magnetic field, , is pointing out of the plane, in the positive z-direction. Calculate the net force on the semicircular loop.

  • 27.71 A proton moving at speed v = 1.00 · 10 6 m/s enters a region in space where a magnetic field given by = (−0.500 T) exists. The velocity vector of the proton is at an angle θ = 60.0° with respect to the positive z-axis.

    1. Analyze the motion of the proton and describe its trajectory (in qualitative terms only).

    2. Calculate the radius, r, of the trajectory projected onto a plane perpendicular to the magnetic field (in the xy-plane).

    3. Calculate the period, T, and frequency, f, of the motion in that plane.

    4. Calculate the pitch of the motion (the distance traveled by the proton in the direction of the magnetic field in 1 period).

    Answer