|| A long wire carrying a 5.0 A current perpendicular to the xy-plane intersects the x-axis at $x=-2.0\text{cm}$. A second, parallel wire carrying a 3.0 A current intersects the x-axis at $x=+2.0\text{ cm}$. At what point or points on the x-axis is the magnetic field zero if (a) the two currents are in the same direction and (b) the two currents are in opposite directions?

||| The two insulated wires in FIGURE P29.42 cross at a $30\text{°}$ angle but do not make electrical contact. Each wire carries a 5.0 A current. Points 1 and 2 are each 4.0 cm from the intersection and equally distant from both wires. What are the magnitude and direction of the magnetic fields at points 1 and 2?

FIGURE P29.42

|| What are the strength and direction of the magnetic field at the center of the loop in FIGURE P29.43?

FIGURE P29.43

|| When seen from the end, three long, parallel wires form an equilateral triangle 6.0 cm on a side. The wires each carry a 5.0 A current, with one current direction opposite the other two. What is the magnetic field strength at the center of the triangle?

| Find an expression for the magnetic field strength at the center (point P) of the circular arc in FIGURE P29.45.

FIGURE P29.45

|| What are the strength and direction of the magnetic field at point P in FIGURE P29.46?

FIGURE P29.46

|| Each turn of a solenoid is a current loop with a magnetic dipole moment. Consider a 200-turn cylindrical solenoid that has an interior volume of $40{\text{cm}}^3$ and for which each turn is a magnetic dipole moment with magnitude $8.0\times {10}^{-4}\text{A}{\text{m}}^2$. What is the magnetic field strength inside the solenoid?

|| The magnetic field strength at the north pole of a 2.0-cm-diameter, 8-cm-long Alnico magnet is 0.10 T. To produce the same field with a solenoid of the same size, carrying a current of 2.0 A, how many turns of wire would you need?

|| Your employer asks you to build a 20-cm-long solenoid with an interior field of 5.0 mT. The specifications call for a single layer of wire, wound with the coils as close together as possible. You have two spools of wire available. Wire with a #18 gauge has a diameter of 1.02 mm and has a maximum current rating of 6 A. Wire with a #26 gauge is 0.41 mm in diameter and can carry up to 1 A. Which wire should you use, and what current will you need?

|| The earth’s magnetic field, with a magnetic dipole moment of $8.0\times {10}^{22}\text{A}{\text{m}}^2$, is generated by currents within the molten iron of the earth’s outer core. Suppose we model the core current as a 3000-km-diameter current loop made from a 1000-km-diameter “wire.” The loop diameter is measured from the centers of this very fat wire.

1. What is the current in the current loop?

2. What is the current density J in the current loop?

3. To decide whether this is a large or a small current density, compare it to the current density of a 1.0 A current in a 1.0-mm-diameter wire.

|| BIO Weak magnetic fields can be measured at the surface of the brain. Although the currents causing these fields are quite complicated, we can estimate their size by modeling them as a current loop around the equator of a 16-cm-diameter (the width of a typical head) sphere. What current is needed to produce a 3.0 pT field—the strength measured for one subject—at the pole of this sphere?

| BIO The heart produces a weak magnetic field that can be used to diagnose certain heart problems. It is a dipole field produced by a current loop in the outer layers of the heart.

1. It is estimated that the field at the center of the heart is 90 pT. What current must circulate around an 8.0-cm-diameter loop, about the size of a human heart, to produce this field?

2. What is the magnitude of the heart’s magnetic dipole moment?

| What is the magnetic field strength at the center of the semicircle in FIGURE P29.53?

FIGURE P29.53

|| The toroid of FIGURE P29.54 is a coil of wire wrapped around a doughnut-shaped ring (a torus). Toroidal magnetic fields are used to confine fusion plasmas.

1. From symmetry, what must be the shape of the magnetic field in this toroid? Explain.

2. Consider a toroid with N closely spaced turns carrying current I. Use Ampère’s law to find an expression for the magnetic field strength at a point inside the torus at distance r from the axis.

3. Is a toroidal magnetic field a uniform field? Explain.

FIGURE P29.54

|| The coaxial cable shown in FIGURE P29.55 consists of a solid inner conductor of radius $R_1$ surrounded by a hollow, very thin outer conductor of radius $R_2$. The two carry equal currents I, but in opposite directions. The current density is uniformly distributed over each conductor.

1. Find expressions for three magnetic fields: within the inner conductor, in the space between the conductors, and outside the outer conductor.

2. Draw a graph of B versus r from $r=0$ to $r=2R_2$ if $R_1=\frac{1}{3}{R}_2$.

FIGURE P29.55

|| CALC A flat, circular disk of radius R is uniformly charged with total charge Q. The disk spins at angular velocity $\omega$ about an axis through its center. What is the magnetic field strength at the center of the disk?

|| A long, hollow wire has inner radius $R_1$ and outer radius $R_2$. The wire carries current I uniformly distributed across the area of the wire. Use Ampère’s law to find an expression for the magnetic field strength in the three regions $0<r<R_1$, $R_1<r<R_2$, and $R_2<r$.

|| A proton moving in a uniform magnetic field with $\vec{v}{}_1=1.00\times {10}^6î\text{m}\text{/}\text{s}$ experiences force $\vec{F}{}_1=1.20\times {10}^{-16}\widehat{k}\text{N}$. A second proton with $\vec{v}{}_2=2.00\times {10}^6ĵ\text{m}\text{/}\text{s}$ experiences $\vec{F}{}_2=\text{−}4.16\times {10}^{-16}\widehat{k}\text{N}$ in the same field. What is $\vec{B}?$ Give your answer as a magnitude and an angle measured ccw from the $+x\text{-}\text{axis}$.

|| INT An electron travels with speed $1.0\times {10}^7\text{m}\text{/}\text{s}$ between the two parallel charged plates shown in FIGURE P29.59. The plates are separated by 1.0 cm and are charged by a 200 V battery. What magnetic field strength and direction will allow the electron to pass between the plates without being deflected?

FIGURE P29.59

|| INT An electron in a cathode-ray tube is accelerated through a potential difference of 10 kV, then passes through the 2.0-cm-wide region of uniform magnetic field in FIGURE P29.60. What field strength will deflect the electron by $10\text{°}?$

FIGURE P29.60

|| An antiproton (same properties as a proton except that $q=-e)$ is moving in the combined electric and magnetic fields of FIGURE P29.61. What are the magnitude and direction of the antiproton’s acceleration at this instant?

FIGURE P29.61

|| An antiproton is identical to a proton except it has the opposite charge, $-e$. To study antiprotons, they must be confined in an ultrahigh vacuum because they will annihilate—producing gamma rays—if they come into contact with the protons of ordinary matter. One way of confining antiprotons is to keep them in a magnetic field. Suppose that antiprotons are created with a speed of $1\text{.}5\times {10}^4\text{m}\text{/}\text{s}$ and then trapped in a 2.0 mT magnetic field. What minimum diameter must the vacuum chamber have to allow these antiprotons to circulate without touching the walls?

||

1. A 65-cm-diameter cyclotron uses a 500 V oscillating potential difference between the dees. What is the maximum kinetic energy of a proton if the magnetic field strength is 0.75 T?

2. How many revolutions does the proton make before leaving the cyclotron?

|| INT FIGURE P29.64 shows a mass spectrometer, an analytical instrument used to identify the various molecules in a sample by measuring their charge-to-mass ratio $q\text{/}m$. The sample is ionized, the positive ions are accelerated (starting from rest) through a potential difference $\text{Δ}V$, and they then enter a region of uniform magnetic field. The field bends the ions into circular trajectories, but after just half a circle they either strike the wall or pass through a small opening to a detector. As the accelerating voltage is slowly increased, different ions reach the detector and are measured. Consider a mass spectrometer with a 200.00 mT magnetic field and an 8.0000 cm spacing between the entrance and exit holes. To five significant figures, what accelerating potential differences $\text{Δ}V$ are required to detect the ions (a) ${\text{O}}_2{}^{+}$ (b) ${\text{N}}_2{}^{+}$ and (c) ${\text{CO}}^{+}?$ See Exercise 29 for atomic masses; the mass of the missing electron is less than 0.001 u and is not relevant at this level of precision. Although ${\text{N}}_2{}^{+}$ and ${\text{CO}}^{+}$ both have a nominal molecular mass of 28, they are easily distinguished by virtue of their slightly different accelerating voltages. Use the following constants: $1\text{u}=1.6605\times {10}^{-27}\text{kg}\text{,}e=1.6022\times {10}^{-19}\text{C}$.

FIGURE P29.64

|| The uniform 30 mT magnetic field in FIGURE P29.65 points in the positive z-direction. An electron enters the region of magnetic field with a speed of $5.0\times {10}^6\text{m}\text{/}\text{s}$ and at an angle of $30\text{°}$ above the xy-plane. Find the radius r and the pitch p of the electron’s spiral trajectory.

FIGURE P29.65

||| Particle accelerators, such as the Large Hadron Collider, use magnetic fields to steer charged particles around a ring. Consider a proton ring with 36 identical bending magnets connected by straight segments. The protons move along a 1.0-m-long circular arc as they pass through each magnet. What magnetic field strength is needed in each magnet to steer protons around the ring with a speed of $2\text{.}5\times {10}^7\text{m}\text{/}\text{s}?$ Assume that the field is uniform inside the magnet, zero outside.

|| INT Controlled fusion is a possible future energy source that would harness the same nuclear fusion reactions that power the sun. The simplest fusion reaction is ${}^2{\text{H}}^{+}+{}^2\text{H}{}^{+}\to {}^3\text{H}{\text{e}}^{++}+\text{n}+\text{energy}$, in which nuclei of two deuterium atoms fuse into a ${}^3\text{He}$ nucleus while ejecting a neutron and releasing a substantial amount of energy. Deuterium is not an element but is the name given to “heavy hydrogen,” in which the nucleus is not simply a proton but a proton and a neutron, with atomic mass 2 u. Two positive deuterium nuclei, which repel each other, can get close enough to fuse only if they have very high speeds. This can be achieved by creating a plasma of ionized deuterium gas at a temperature of $1.0\times {10}^8\text{K}$. No material substance can contain a plasma at this temperature, so the idea is to contain the plasma with magnetic fields. Consider the simplest model of using a solenoid to confine the ions to cyclotron motion around the field lines. The plasma ions have a range of speeds, and it’s necessary to contain all the ions with speeds up to three times the rms speed at the plasma temperature. What magnetic field strength is needed to keep the fastest ions in 20-cm-diameter cyclotron orbits? The actual magnetic fields are considerably more complex, but your answer is a reasonable estimate of the required field strengths.

|| A proton moves in the uniform fields $\vec{E}=2500\widehat{k}\text{V}\text{/}\text{m}$ and $\vec{B}=0.50\widehat{k}\text{T}$. At $t=0\text{s}$ the proton is moving in a 1.0-cm-diameter circle in the xy-plane.

1. At what time does the proton have a speed of $5.0\times {10}^5\text{m}\text{/}\text{s}?$

2. How many revolutions will the proton have made during this time interval?

|| A Hall-effect probe to measure magnetic field strengths needs to be calibrated in a known magnetic field. Although it is not easy to do, magnetic fields can be precisely measured by measuring the cyclotron frequency of protons. A testing laboratory adjusts a magnetic field until the proton’s cyclotron frequency is 10.0 MHz. At this field strength, the Hall voltage on the probe is 0.543 mV when the current through the probe is 0.150 mA. Later, when an unknown magnetic field is measured, the Hall voltage at the same current is 1.735 mV. What is the strength of this magnetic field?

||| CALC It is shown in more advanced courses that charged particles in circular orbits radiate electromagnetic waves, called cyclotron radiation. As a result, a particle undergoing cyclotron motion with speed v is actually losing kinetic energy at the rate

How long does it take (a) an electron and (b) a proton to radiate away half its energy while spiraling in a 2.0 T magnetic field?

|| INT The two springs in FIGURE P29.71 each have a spring constant of $10\text{N}\text{/}\text{m}$. They are compressed by 1.0 cm when a current passes through the wire. How big is the current?

FIGURE P29.71

|| INT The 10-turn loop of wire shown in FIGURE P29.72 lies in a horizontal plane, parallel to a uniform horizontal magnetic field, and carries a 2.0 A current. The loop is free to rotate about a nonmagnetic axle through the center. A 50 g mass hangs from one edge of the loop. What magnetic field strength will prevent the loop from rotating about the axle?

FIGURE P29.72

|| INT Magnetic fields are sometimes measured by balancing magnetic forces against known mechanical forces. Your task is to measure the strength of a horizontal magnetic field using a 12-cm-long rigid metal rod that hangs from two nonmagnetic springs, one at each end, with spring constants 1.3 N/m. You first position the rod to be level and perpendicular to the field, whose direction you determined with a compass. You then connect the ends of the rod to wires that run parallel to the field and thus experience no forces. Finally, you measure the downward deflection of the rod, stretching the springs, as you pass current through it. Your data are as follows:

Use an appropriate graph of the data to determine the magnetic field strength.

|| INT A conducting bar of length l and mass m rests at the left end of the two frictionless rails of length d in FIGURE P29.74. A uniform magnetic field of strength B points upward.

1. In which direction, into or out of the figure, will a current I through the conducting bar cause the bar to experience a force to the right?

2. Find an expression for the bar’s speed as it leaves the rails at the right end.

FIGURE P29.74

||

1. INT In FIGURE P29.75, a long, straight, current-carrying wire of linear mass density $\mu$ is suspended by threads. A magnetic field perpendicular to the wire exerts a horizontal force that deflects the wire to an equilibrium angle $\theta$. Find an expression for the strength and direction of the magnetic field $\vec{B}$.

2. What $\vec{B}$ deflects a 55 g/m wire to a $12\text{°}$ angle when the current is 10 A?

   FIGURE P29.75

   

   FIGURE P29.75 Full Alternative Text

|| INT CALC An electromagnetic rail gun uses magnetic forces to launch projectiles. FIGURE P29.76 shows a 10-cm-long, 10 g metal wire that can slide without friction along 1.0-m-long horizontal rails. The rails are connected to a 300 V source, and a 0.10 T magnetic field fills the space between the rails. Each rail has linear resistivity $\text{λ}=0.10\text{Ω}\text{/}\text{m}$, which means that the resistance is $\text{λ}$ multiplied by the length of rail through which current flows. Assume that the sliding wire and the left end, where the voltage source is, have zero resistance. The wire is initially placed at $x_0=5.0\text{cm,}$ then the switch is closed. What is the wire’s speed as it leaves the rails?

FIGURE P29.76

||| INT CALC A wire along the x-axis carries current I in the negative x-direction through the magnetic field

1. Draw a graph of B versus x over the interval $-\frac{3}{2}l<x<\frac{3}{2}l$.

2. Find an expression for the net force $\vec{F}{}_{\text{net}}$ on the wire.

3. Find an expression for the net torque on the wire about the point $x=0$.

||| A scientist measuring the resistivity of a new metal alloy left her ammeter in another lab, but she does have a magnetic field probe. So she creates a 6.5-m-long, 2.0-mm-diameter wire of the material, connects it to a 1.5 V battery, and measures a 3.0 mT magnetic field 1.0 mm from the surface of the wire. What is the material’s resistivity?

||| FIGURE CP29.79 is an edge view of a 2.0 kg square loop, 2.5 m on each side, with its lower edge resting on a frictionless, horizontal surface. A 25 A current is flowing around the loop in the direction shown. What is the strength of a uniform, horizontal magnetic field for which the loop is in static equilibrium at the angle shown?

FIGURE CP29.79

||| You have a 1.0-m-long copper wire. You want to make an N-turn current loop that generates a 1.0 mT magnetic field at the center when the current is 1.0 A. You must use the entire wire. What will be the diameter of your coil?

||| CALC

1. Derive an expression for the magnetic field strength at distance d from the center of a straight wire of finite length l that carries current I.

2. Determine the field strength at the center of a current-carrying square loop having sides of length 2R.

3. Compare your answer to part b to the field at the center of a circular loop of diameter 2R. Do so by computing the ratio $B_{\text{square}}\text{/}{B}_{\text{circle}}$.

||| CALC A long, straight conducting wire of radius R has a nonuniform current density $J=J_{0}r\text{/}R$, where $J_0$ is a constant. The wire carries total current I.

1. Find an expression for $J_0$ in terms of I and R.

2. Find an expression for the magnetic field strength inside the wire at radius r.

3. At the boundary, $r=R$, does your solution match the known field outside a long, straight current-carrying wire?

||| An infinitely wide flat sheet of charge flows out of the figure in FIGURE CP29.83. The current per unit width along the sheet (amps per meter) is given by the linear current density $J_s$.

1. What is the shape of the magnetic field? To answer this question, you may find it helpful to approximate the current sheet as many parallel, closely spaced current-carrying wires. Give your answer as a picture showing magnetic field vectors.

2. Find the magnetic field strength at distance d above or below the current sheet.

FIGURE CP29.83