Fundamentals of Physics

David Halliday, Robert Resnick, Jearl Walker

Chapter 32

Maxwell's Equations; Magnetism of Matter - all with Video Answers

Educators

Chapter Questions

Problem 1

The magnetic flux through each of five faces of a die (singular of "dice") is given by $\Phi_{B}=\pm N \mathrm{~Wb}$, where $N(=1$ to 5$)$ is the number of spots on the face. The flux is positive (outward) for $N$ even and negative (inward) for $N$ odd. What is the flux through the sixth face of the die?

Prabhu Ramji

Prabhu Ramji

Numerade Educator

Problem 2

Figure $32-27$ shows a closed surface. Along the flat top face, which has a radius of $2.0 \mathrm{~cm}, \mathrm{a}$ perpendicular magnetic field $\vec{B}$ of magnitude $0.30 \mathrm{~T}$ is directed outward. Along the flat bottom face, a magnetic flux of $0.70 \mathrm{mWb}$ is directed outward. What are the (a) magnitude and (b) direction (inward or outward) of the magnetic flux through the curved part of the surface?

Prabhu Ramji

Numerade Educator

Problem 3

A Gaussian surface in the shape of a right circular cylinder with end caps has a radius of $12.0 \mathrm{~cm}$ and a length of $80.0 \mathrm{~cm}$. Through one end there is an inward magnetic flux of $25.0 \mu \mathrm{Wb}$. At the other end there is a uniform magnetic field of $1.60 \mathrm{mT}$, normal to the surface and directed outward. What are the (a) magnitude and (b) direction (inward or outward) of the net magnetic flux through the curved surface?

Prabhu Ramji

Numerade Educator

Problem 4

Two wires, parallel to a $z$ axis and a distance $4 r$ apart, carry equal currents $i$ in opposite directions, as shown in Fig. 32-28. A circular cylinder of radius $r$ and length $L$ has its axis on the $z$ axis, midway between the wires. Use Gauss' law for magnetism to derive an expression for the net outward magnetic flux through the half of the cylindrical surface above the $x$ axis. (Hint: Find the flux through the portion of the $x z$ plane that lies within the cylinder.)

Prabhu Ramji

Numerade Educator

Problem 5

The induced magnetic field at radial distance $6.0 \mathrm{~mm}$ from the central axis of a circular parallel-plate capacitor is $2.0 \times$ $10^{-7} \mathrm{~T}$. The plates have radius $3.0 \mathrm{~mm}$. At what rate $d \vec{E} / d t$ is the electric field between the plates changing?

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator

Problem 6

A capacitor with square plates of edge length $L$ is being discharged by a current of $0.75$ A. Figure $32-29$ is a head-on view of one of the plates from inside the capacitor. A dashed rectangular path is shown. If $L=12 \mathrm{~cm}, W=4.0 \mathrm{~cm}$, and $H=2.0 \mathrm{~cm}$
what is the value of $\oint \vec{B} \cdot d \vec{s}$ around the dashed path?

Sai Chaitanya Tadepalli

Sai Chaitanya Tadepalli

Numerade Educator

Problem 7

Figure $32-30$ shows a circular region of radius $R=3.00 \mathrm{~cm}$ in which a uniform electric flux is directed out of the plane of the page. The total electric flux through the region is given by $\Phi_{E}=(3.00 \mathrm{mV} \cdot \mathrm{m} / \mathrm{s}) t$, where $t$ is in seconds.
What is the magnitude of the magnetic field that is induced at radial distances (a) $2.00 \mathrm{~cm}$ and (b) $5.00 \mathrm{~cm}$ ?

Prabhu Ramji

Numerade Educator

Problem 8

Figure $32-30$ shows a circular region of radius $R=3.00 \mathrm{~cm}$ in which an electric flux is directed out of the plane of the page. The flux encircled by a concentric circle of radius $r$ is given by $\Phi_{E, \text { enc }}=$ $(0.600 \mathrm{~V} \cdot \mathrm{m} / \mathrm{s})(r / R) t$, where $r \leq R$ and $t$ is in seconds. What is the magnitude of the induced magnetic field at radial distances
(a) $2.00 \mathrm{~cm}$ and
(b) $5.00 \mathrm{~cm}$ ?

Prabhu Ramji

Numerade Educator

Problem 9

In Fig. 32-30, a uniform electric field is directed out of the page within a circular region of radius $R=3.00$ $\mathrm{cm} .$ The field magnitude is given by $E=\left(4.50 \times 10^{-3} \mathrm{~V} / \mathrm{m} \cdot \mathrm{s}\right) t$, where $t$ is in seconds. What is the magnitude of the induced magnetic field at radial distances (a) $2.00 \mathrm{~cm}$ and (b) $5.00 \mathrm{~cm}$ ?

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator

Problem 10

In Fig. $32-30$, an electric field is directed out of the page within a circular region of radius $R=3.00$ $\mathrm{cm}$. The field magnitude is $E=(0.500 \mathrm{~V} / \mathrm{m} \cdot \mathrm{s})(1-r / R) t$, where $t$ is in seconds and $r$ is the radial distance $(r \leq R)$. What is the magnitude of the induced magnetic field at radial distances (a) $2.00 \mathrm{~cm}$ and (b) $5.00 \mathrm{~cm}$ ?

Prabhu Ramji

Numerade Educator

Problem 11

Suppose that a parallel-plate capacitor has circular plates with radius $R=30 \mathrm{~mm}$ and a plate separation of $5.0 \mathrm{~mm}$. Suppose also that a sinusoidal potential difference with a maximum value of $150 \mathrm{~V}$ and a frequency of $60 \mathrm{~Hz}$ is applied across the plates; that is,
$$
V=(150 \mathrm{~V}) \sin [2 \pi(60 \mathrm{~Hz}) t]
$$
(a) Find $B_{\max }(R)$, the maximum value of the induced magnetic field that occurs at $r=R$. (b) Plot $B_{\max }(r)$ for $0<r<10 \mathrm{~cm}$.

Keshav Singh

Numerade Educator

Problem 12

A parallel-plate capacitor with circular plates of radius $40 \mathrm{~mm}$ is being discharged by a current of $6.0 \mathrm{~A}$. At what radius (a) inside and (b) outside the capacitor gap is the magnitude of the induced magnetic field equal to $75 \%$ of its maximum value? (c) What is that maximum value?

Prabhu Ramji

Numerade Educator

Problem 13

At what rate must the potential difference between the plates of a parallel-plate capacitor with a $2.0 \mu \mathrm{F}$ capacitance be changed to produce a displacement current of $1.5 \mathrm{~A}$ ?-13 At what rate must the potential difference between the plates of a parallel-plate capacitor with a $2.0 \mu \mathrm{F}$ capacitance be changed to produce a displacement current of $1.5 \mathrm{~A}$ ?

Prabhu Ramji

Numerade Educator

Problem 14

A parallel-plate capacitor with circular plates of radius $R$ is being charged. Show that the magnitude of the current density of the displacement current is $J_{d}=\varepsilon_{0}(d E / d t)$ for $r \leq R$

Sai Chaitanya Tadepalli

Sai Chaitanya Tadepalli

Numerade Educator

Problem 15

Prove that the displacement current in a parallel-plate capacitor of capacitance $C$ can be written as $i_{d}=C(d V / d t)$, where $V$ is the potential difference between the plates.

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator

Problem 16

A parallel-plate capacitor with circular plates of radius $0.10 \mathrm{~m}$ is being discharged. A circular loop of radius $0.20 \mathrm{~m}$ is concentric with the capacitor and halfway between the plates. The displacement current through the loop is $2.0 \mathrm{~A}$. At what rate is the electric field between the plates changing?

Prabhu Ramji

Numerade Educator

Problem 17

A silver wire has resistivity $\rho=1.62 \times 10^{-8} \Omega \cdot \mathrm{m}$ and a cross-sectional area of $5.00 \mathrm{~mm}^{2}$. The current in the wire is uniform and changing at the rate of $2000 \mathrm{~A} / \mathrm{s}$ when the current is 100 A. (a) What is the magnitude of the (uniform) electric field in the wire when the current in the wire is $100 \mathrm{~A} ?$ (b) What is the displacement current in the wire at that time? (c) What is the ratio of the magnitude of the magnetic field due to the displacement current to that due to the current at a distance $r$ from the wire?

Prabhu Ramji

Numerade Educator

Problem 18

The circuit in Fig. $32-31$ consists of switch $\mathrm{S}$, a $12.0 \mathrm{~V}$ ideal battery, a $20.0 \mathrm{M} \Omega$ resistor, and an air-filled capacitor. The capacitor has parallel circular plates of radius $5.00 \mathrm{~cm}$, separated by $3.00 \mathrm{~mm} .$ At time $t=0$, switch $\mathrm{S}$ is closed to begin charging the capacitor. The electric field between the plates is uniform. At $t=250 \mu \mathrm{s}$, what is the magnitude of the magnetic field within the capacitor, at radial distance $3.00 \mathrm{~cm} ?$

Prabhu Ramji

Numerade Educator

Problem 19

Figure $32-30$ shows a circular region of radius $R=3.00 \mathrm{~cm}$ in which a displacement current is directed out of the page. The displacement current has a uniform density of magnitude $J_{d}=6.00 \mathrm{~A} / \mathrm{m}^{2}$. What is the magnitude of the magnetic field due to the displacement current at radial distances (a) $2.00 \mathrm{~cm}$ and $(\mathrm{b}) 5.00 \mathrm{~cm} ?$

Prabhu Ramji

Numerade Educator

Problem 20

Figure 32-30 shows a circular region of radius $R=3.00 \mathrm{~cm}$ in which a uniform displacement current $i_{d}=0.500 \mathrm{~A}$ is out of the page. What is the magnitude of the magnetic field due to the displacement current at radial distances
(a) $2.00 \mathrm{~cm}$ and
(b) $5.00 \mathrm{~cm} ?$

Prabhu Ramji

Numerade Educator

Problem 21

Figure $32-30$ shows a circular region of radius $R=3.00 \mathrm{~cm}$ in which a displacement current is directed out of the page. The magnitude of the density of this displacement current is $J_{d}=\left(4.00 \mathrm{~A} / \mathrm{m}^{2}\right)(1-r / R)$, where $r$ is the radial distance $(r \leq R) .$ What is the magnitude of the magnetic field due to the displacement current at (a) $r=2.00 \mathrm{~cm}$ and (b) $r=5.00 \mathrm{~cm}$ ?

Prabhu Ramji

Numerade Educator

Problem 22

Figure 32-30 shows a circular region of radius $R=3.00 \mathrm{~cm}$ in which a displacement current $i_{d}$ is directed out of the figure. The magnitude of the displacement current is $i_{d}=(3.00 \mathrm{~A})(r / R)$, where $r$ is the radial distance $(r \leq R)$ from the center. What is the magnitude of the magnetic field due to $i_{d}$ at radial distances (a) $2.00 \mathrm{~cm}$ and $(\mathrm{b}) 5.00 \mathrm{~cm} ?$

Prabhu Ramji

Numerade Educator

Problem 23

In Fig. 32-32, a parallel-plate capacitor has square plates of edge length $L=1.0 \mathrm{~m} .$ A current of $2.0$ A charges the capacitor, producing a uniform electric field $\vec{E}$ between the plates, with $\vec{E}$ perpendicular to the plates. (a) What is the displacement current $i_{d}$ through the region between the plates? (b) What is $d E / d t$ in this region? (c) What is the displacement current encircled by the square dashed path of edge length $d=0.50 \mathrm{~m} ?$ (d) What is the value of $\oint \vec{B} \cdot d \vec{s}$ around this square dashed path?

Prabhu Ramji

Numerade Educator

Problem 24

The magnitude of the electric field between the two circular parallel plates in Fig. $32-33$ is $E=\left(4.0 \times 10^{5}\right)-\left(6.0 \times 10^{4} t\right)$, with $E$ in volts per meter and $t$ in seconds. At $t=0, \vec{E}$ is upward. The plate area is $4.0 \times$ $10^{-2} \mathrm{~m}^{2}$. For $t \geq 0$, what are the (a) magnitude and (b) direction (up or down) of the displacement current between the plates and (c) is the direction of the induced magnetic field clockwise or counterclockwise in the figure?

Sai Chaitanya Tadepalli

Sai Chaitanya Tadepalli

Numerade Educator

Problem 25

As a parallel-plate capacitor with circular plates $20 \mathrm{~cm}$ in diameter is being charged, the current density of the displacement current in the region between the plates is uniform and has a magnitude of $20 \mathrm{~A} / \mathrm{m}^{2}$. (a) Calculate the magnitude $B$ of the magnetic field at a distance $r=50 \mathrm{~mm}$ from the axis of symmetry of this region. (b) Calculate $d E / d t$ in this region.

Prabhu Ramji

Numerade Educator

Problem 26

A capacitor with parallel circular plates of radius $R=1.20$ $\mathrm{cm}$ is discharging via a current of $12.0 \mathrm{~A}$. Consider a loop of radius $R / 3$ that is centered on the central axis between the plates. (a) How much displacement current is encircled by the loop? The maximum induced magnetic field has a magnitude of $12.0 \mathrm{mT}$. At what radius
(b) inside and (c) outside the capacitor gap is the magnitude of the induced magnetic field $3.00 \mathrm{mT}$ ?

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator

Problem 27

In Fig. $32-34$, a uniform electric field $\vec{E}$ collapses. The vertical axis scale is set by $E_{s}=6.0 \times 10^{5}$ $\mathrm{N} / \mathrm{C}$, and the horizontal axis scale is set by $t_{s}=12.0 \mu \mathrm{s} .$ Calculate the magnitude of the displacement current through a $1.6 \mathrm{~m}^{2}$ area perpendicular to the field during each of the time intervals $a, b$, and $c$ shown on the graph. (Ignore the behavior at the ends of the intervals.)

Prabhu Ramji

Numerade Educator

Problem 28

Figure $32-35 a$ shows the current $i$ that is produced in a wire of resistivity $1.62 \times 10^{-8} \Omega \cdot \mathrm{m}$. The magnitude of the current versus time $t$ is shown in Fig. $32-35 b$. The vertical axis scale is set by $i_{s}=10.0$ A, and the horizontal axis scale is set by $t_{s}=50.0 \mathrm{~ms}$. Point $P$ is at radial distance $9.00 \mathrm{~mm}$ from the wire's center. Determine the magnitude of the magnetic field $\vec{B}_{i}$ at point $P$ due to the actual current $i$ in the wire at (a) $t=20 \mathrm{~ms}$, (b) $t=$ $40 \mathrm{~ms}$, and (c) $t=60 \mathrm{~ms}$. Next, assume that the electric field driving the current is confined to the wire. Then determine the magnitude of the magnetic field $\vec{B}_{i d}$ at point $P$ due to the displacement current $i_{d}$ in the wire at (d) $t=20 \mathrm{~ms}$, (e) $t=$ $40 \mathrm{~ms}$, and (f) $t=60 \mathrm{~ms}$. At point $P$ at $t=20 \mathrm{~s}$, what is the direction (into or out of the page) of (g) $\vec{B}_{i}$ and (h) $\vec{B}_{i d}$ ?

Keshav Singh

Numerade Educator

Problem 29

In Fig. $32-36$, a capacitor with circular plates of radius $R=18.0 \mathrm{~cm}$ is connected to a source of $\operatorname{emf} \mathscr{E}=\mathscr{E}_{m} \sin \omega t$, where $\mathscr{E}_{m}=220 \mathrm{~V}$
and $\omega=130 \mathrm{rad} / \mathrm{s}$. The maximum value of the displacement current is $i_{d}=7.60 \mu \mathrm{A} .$ Neglect fringing of the electric field at the edges of the plates. (a) What is the maximum value of the current $i$ in the circuit? (b) What is the maximum value of $d \Phi_{E} / d t$, where $\Phi_{E}$ is the electric flux through the region between the plates? (c) What is the separation $d$ between the plates? (d) Find the maximum value of the magnitude of $\vec{B}$ between the plates at a distance $r=11.0 \mathrm{~cm}$ from the center

Keshav Singh

Numerade Educator

Problem 30

Assume the average value of the vertical component of Earth's magnetic field is $43 \mu \mathrm{T}$ (downward) for all of Arizona, which has an area of $2.95 \times 10^{5} \mathrm{~km}^{2}$. What then are the (a) magnitude and (b) direction (inward or outward) of the net magnetic flux through the rest of Earth's surface (the entire surface excluding Arizona)?

Paul A.

California State Polytechnic University, Pomona

Problem 31

In New Hampshire the average horizontal component of Earth's magnetic field in 1912 was $16 \mu \mathrm{T}$, and the average inclination or "dip" was $73^{\circ} .$ What was the corresponding magnitude of Earth's magnetic field?

Keshav Singh

Numerade Educator

Problem 32

Figure $32-37 a$ is a one-axis graph along which two of the allowed energy values (levels) of an atom are plotted. When the atom is placed in a magnetic field of $0.500 \mathrm{~T}$, the graph changes to that of Fig. $32-37 b$ because of the energy associated with $\vec{\mu}_{\text {orb }} \cdot \vec{B}$. (We neglect $\vec{\mu}_{s^{*}}$ ) Level $E_{1}$ is unchanged, but level $E_{2}$ splits into a (closely spaced) triplet of levels. What are the allowed values of $m_{\ell}$ associated with (a) energy level $E_{1}$ and (b) energy level $E_{2}$ ? (c) In joules, what amount of energy is represented by the spacing between the triplet levels?

Keshav Singh

Numerade Educator

Problem 33

If an electron in an atom has an orbital angular momentum with $m=0$, what are the components (a) $L_{\mathrm{orb}, z}$ and
(b) $\mu_{\mathrm{orb}, z} ?$ If the atom is in an external magnetic field $\vec{B}$ that has magnitude $35 \mathrm{mT}$ and is directed along the $z$ axis, what are (c) the energy $U_{\text {orb }}$ associated with $\vec{\mu}_{\text {orb }}$ and (d) the energy $U_{\text {spin }}$ associated with $\vec{\mu}_{s} ?$ If, instead, the electron has $m=-3$, what are (e) $L_{\text {orb }, z}$, (f) $\mu_{\text {orb }, z},(g) U_{\text {orb }}$, and (h) $U_{\text {spin }}$ ?

Keshav Singh

Numerade Educator

Problem 34

What is the energy difference between parallel and antiparallel alignment of the $z$ component of an electron's spin magnetic dipole moment with an external magnetic field of magnitude $0.25 \mathrm{~T}$, directed parallel to the $z$ axis?

Keshav Singh

Numerade Educator

Problem 35

What is the measured component of the orbital magnetic dipole moment of an electron with (a) $m_{\ell}=1$ and (b) $m_{\ell}=-2$ ?

Keshav Singh

Numerade Educator

Problem 36

An electron is placed in a magnetic field $\vec{B}$ that is directed along a $z$ axis. The energy difference between parallel and antiparallel alignments of the $z$ component of the electron's spin magnetic moment with $\vec{B}$ is $6.00 \times 10^{-25} \mathrm{~J}$. What is the magnitude of $\vec{B}$ ?

Keshav Singh

Numerade Educator

Problem 37

Figure 32-38 shows a loop model (loop $L$ ) for a diamagnetic material. (a) Sketch the magnetic field lines within and about the material due to the bar magnet. What is the direction of (b) the loop's net magnetic dipole moment $\vec{\mu}$,
(c) the conventional current $i$ in the loop (clockwise or counterclockwise in the figure), and (d) the magnetic force on the loop?

Keshav Singh

Numerade Educator

Problem 38

Assume that an electron of mass $m$ and charge magnitude $e$ moves in a circular orbit of radius $r$ about a nucleus. A uniform magnetic field $\vec{B}$ is then established perpendicular to the plane of the orbit. Assuming also that the radius of the orbit does not change and that the change in the speed of the electron due to field $\vec{B}$ is small, find an expression for the change in the orbital magnetic dipole moment of the electron due to the field.

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator

Problem 39

A sample of the paramagnetic salt to which the magnetization curve of Fig. $32-14$ applies is to be tested to see whether it obeys Curie's law. The sample is placed in a uniform $0.50$ T magnetic field that remains constant throughout the experiment. The magnetization $M$ is then measured at temperatures ranging from 10 to $300 \mathrm{~K}$. Will it be found that Curie's law is valid under these conditions?

Keshav Singh

Numerade Educator

Problem 40

A sample of the paramagnetic salt to which the magnetization curve of Fig. 32-14 applies is held at room temperature $(300 \mathrm{~K})$. At what applied magnetic field will the degree of magnetic saturation of the sample be (a) $50 \%$ and (b) $90 \%$ ? (c) Are these fields attainable in the laboratory?

Keshav Singh

Numerade Educator

Problem 41

A magnet in the form of a cylindrical rod has a length of $5.00 \mathrm{~cm}$ and a diameter of $1.00 \mathrm{~cm}$. It has a uniform magnetization of $5.30 \times 10^{3} \mathrm{~A} / \mathrm{m}$. What is its magnetic dipole moment?

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator

Problem 42

A $0.50 \mathrm{~T}$ magnetic field is applied to a paramagnetic gas whose atoms have an intrinsic magnetic dipole moment of $1.0 \times$ $10^{-23} \mathrm{~J} / \mathrm{T}$. At what temperature will the mean kinetic energy of translation of the atoms equal the energy required to reverse such a dipole end for end in this magnetic field?

Keshav Singh

Numerade Educator

Problem 43

An electron with kinetic energy $K_{e}$ travels in a circular path that is perpendicular to a uniform magnetic field, which is in the positive direction of a $z$ axis. The electron's motion is subject only to the force due to the field. (a) Show that the magnetic dipole moment of the electron due to its orbital motion has magnitude $\mu=$ $K_{e} / B$ and that it is in the direction opposite that of $\vec{B}$. What are the (b) magnitude and (c) direction of the magnetic dipole moment of a positive ion with kinetic energy $K_{i}$ under the same circumstances? (d) An ionized gas consists of $5.3 \times 10^{21}$ electrons $/ \mathrm{m}^{3}$ and the same number density of ions. Take the average electron kinetic energy to be $6.2 \times 10^{-20} \mathrm{~J}$ and the average ion kinetic energy to be $7.6 \times 10^{-21} \mathrm{~J}$. Calculate the magnetization of the gas when it is in a magnetic field of $1.2 \mathrm{~T}$.

Keshav Singh

Numerade Educator

Problem 44

Figure $32-39$ gives the magnetization curve for a paramagnetic material. The vertical axis scale is set by $a=0.15$, and the horizontal axis scale is set by $b=0.2 \mathrm{~T} / \mathrm{K}$. Let $\mu_{\mathrm{sam}}$ be the measured net magnetic moment of a sample of the material and $\mu_{\max }$ be the maximum possible net magnetic moment of that sample. According to Curie's law, what would be the ratio $\mu_{\mathrm{sam}} / \mu_{\max }$ were the sample placed in a uniform magnetic field of magnitude $0.800 \mathrm{~T}$, at a temperature of $2.00 \mathrm{~K}$ ?

Keshav Singh

Numerade Educator

Problem 45

Consider a solid containing $N$ atoms per unit volume, each atom having a magnetic dipole moment $\vec{\mu}$. Suppose the direction of $\vec{\mu}$ can be only parallel or antiparallel to an externally applied magnetic field $\vec{B}$ (this will be the case if $\vec{\mu}$ is due to the spin of a single electron). According to statistical mechanics, the probability of an atom being in a state with energy $U$ is proportional to $e^{-U i k T}$, where $T$ is the temperature and $k$ is Boltzmann's constant. Thus, because energy $U$ is $-\vec{\mu} \cdot \vec{B}$, the fraction of atoms whose dipole moment is parallel to $\vec{B}$ is proportional to $e^{\mu B / k T}$ and the fraction of atoms whose dipole moment is antiparallel to $\vec{B}$ is proportional to $e^{-\mu B / k T}$. (a) Show that the magnitude of the magnetization of this solid is $M=N \mu \tanh (\mu B / k T)$. Here tanh is the hyperbolic tangent function: $\tanh (x)=\left(e^{x}-e^{-x}\right) /\left(e^{x}+e^{-x}\right) .$ (b) Show that the result given in (a) reduces to $M=N \mu^{2} B / k T$ for $\mu B \ll k T$.
(c) Show that the result of (a) reduces to $M=N \mu$ for $\mu B \geqslant k T$. (d) Show that both (b) and (c) agree qualitatively with Fig. $32-14$.

Raj Bala

Numerade Educator

Problem 46

You place a magnetic compass on a horizontal surface, allow the needle to settle, and then give the compass a gentle wiggle to cause the needle to oscillate about its equilibrium position. The oscillation frequency is $0.312 \mathrm{~Hz}$. Earth's magnetic field at the location of the compass has a horizontal component of $18.0 \mu \mathrm{T}$. The needle has a magnetic moment of $0.680 \mathrm{~mJ} / \mathrm{T}$. What is the needle's rotational inertia about its (vertical) axis of rotation?

Keshav Singh

Numerade Educator

Problem 47

The magnitude of the magnetic dipole moment of Earth is $8.0 \times 10^{22} \mathrm{~J} / \mathrm{T}$. (a) If the origin of this magnetism were a magnetized iron sphere at the center of Earth, what would be its radius? (b) What fraction of the volume of Earth would such a sphere occupy? Assume complete alignment of the dipoles. The density of Earth's inner core is $14 \mathrm{~g} / \mathrm{cm}^{3} .$ The magnetic dipole moment of an iron atom is $2.1 \times 10^{-23} \mathrm{~J} / \mathrm{T}$. (Note: Earth's inner core is in fact thought to be in both liquid and solid forms and partly iron, but a permanent magnet as the source of Earth's magnetism has been ruled out by several considerations. For one, the temperature is certainly above the Curie point.)

Keshav Singh

Numerade Educator

Problem 48

The magnitude of the dipole moment associated with an atom of iron in an iron bar is $2.1 \times 10^{-23} \mathrm{~J} / \mathrm{T}$. Assume that all the atoms in the bar, which is $5.0 \mathrm{~cm}$ long and has a cross-sectional area of $1.0 \mathrm{~cm}^{2}$, have their dipole moments aligned. (a) What is the dipole moment of the bar? (b) What torque must be exerted to hold this magnet perpendicular to an external field of magnitude $1.5 \mathrm{~T} ?$ (The density of iron is $7.9 \mathrm{~g} / \mathrm{cm}^{3} .$ )

Keshav Singh

Numerade Educator

Problem 49

The exchange coupling mentioned in Module $32-8$ as being responsible for ferromagnetism is not the mutual magnetic interaction between two elementary magnetic dipoles. To show this, calculate (a) the magnitude of the magnetic field a distance of $10 \mathrm{~nm}$ away, along the dipole axis, from an atom with magnetic dipole moment $1.5 \times 10^{-23} \mathrm{~J} / \mathrm{T}$ (cobalt), and (b) the minimum energy required to turn a second identical dipole end for end in this field. (c) By comparing the latter with the mean translational kinetic energy of $0.040 \mathrm{eV}$, what can you conclude?

Keshav Singh

Numerade Educator

Problem 50

A magnetic rod with length $6.00 \mathrm{~cm}$, radius $3.00 \mathrm{~mm}$, and (uniform) magnetization $2.70 \times 10^{3} \mathrm{~A} / \mathrm{m}$ can turn about its center like a compass needle. It is placed in a uniform magnetic field $\vec{B}$ of magnitude $35.0 \mathrm{mT}$, such that the directions of its dipole moment and $\vec{B}$ make an angle of $68.0^{\circ}$. (a) What is the magnitude of the torque on the rod due to $\vec{B} ?$ (b) What is the change in the orientation energy of the rod if the angle changes to $34.0^{\circ}$ ?

Keshav Singh

Numerade Educator

Problem 51

The saturation magnetization $M_{\max }$ of the ferromagnetic metal nickel is $4.70 \times 10^{5} \mathrm{~A} / \mathrm{m}$. Calculate the magnetic dipole moment of a single nickel atom. (The density of nickel is $8.90 \mathrm{~g} / \mathrm{cm}^{3}$, and its molar mass is $58.71 \mathrm{~g} / \mathrm{mol}$.)

Keshav Singh

Numerade Educator

Problem 52

Measurements in mines and boreholes indicate that Earth's interior temperature increases with depth at the average rate of $30 \mathrm{C}^{\circ} / \mathrm{km}$. Assuming a surface temperature of $10^{\circ} \mathrm{C}$, at what depth does iron cease to be ferromagnetic? (The Curie temperature of iron varies very little with pressure.)

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator

Problem 53

A Rowland ring is formed of ferromagnetic material. It is circular in cross section, with an inner radius of $5.0 \mathrm{~cm}$ and an outer radius of $6.0 \mathrm{~cm}$, and is wound with 400 turns of wire. (a) What current must be set up in the windings to attain a toroidal field of magnitude $B_{0}=0.20 \mathrm{mT} ?$ (b) A secondary coil wound around the toroid has 50 turns and resistance $8.0 \Omega$. If, for this value of $B_{0}$, we have $B_{M}=800 B_{0}$, how much charge moves through the secondary coil when the current in the toroid windings is turned on?

Keshav Singh

Numerade Educator

Problem 54

Using the approximations given in Problem 61 , find (a) the altitude above Earth's surface where the magnitude of its magnetic field is $50.0 \%$ of the surface value at the same latitude; (b) the maximum magnitude of the magnetic field at the core-mantle boundary, $2900 \mathrm{~km}$ below Earth's surface; and the (c) magnitude and (d) inclination of Earth's magnetic field at the north geographic pole. (e) Suggest why the values you calculated for (c) and (d) differ from measured values.

Keshav Singh

Numerade Educator

Problem 55

Earth has a magnetic dipole moment of $8.0 \times 10^{22} \mathrm{~J} / \mathrm{T}$. (a) What current would have to be produced in a single turn of wire extending around Earth at its geomagnetic equator if we wished to set up such a dipole? Could such an arrangement be used to cancel out Earth's magnetism (b) at points in space well above Earth's surface or (c) on Earth's surface?

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator

Problem 56

A charge $q$ is distributed uniformly around a thin ring of radius $r$. The ring is rotating about an axis through its center and perpendicular to its plane, at an angular speed $\omega$. (a) Show that the magnetic moment due to the rotating charge has magnitude $\mu=\frac{1}{2} q \omega r^{2}$. (b) What is the direction of this magnetic moment if the charge is positive?

Keshav Singh

Numerade Educator

Problem 57

A magnetic compass has its needle, of mass $0.050 \mathrm{~kg}$ and length $4.0 \mathrm{~cm}$, aligned with the horizontal component of Earth's magnetic field at a place where that component has the value $B_{h}=$ $16 \mu \mathrm{T}$. After the compass is given a momentary gentle shake, the needle oscillates with angular frequency $\omega=45 \mathrm{rad} / \mathrm{s}$. Assuming that the needle is a uniform thin rod mounted at its center, find the magnitude of its magnetic dipole moment.

Keshav Singh

Numerade Educator

Problem 58

The capacitor in Fig. $32-7$ is being charged with a $2.50$ A current. The wire radius is $1.50 \mathrm{~mm}$, and the plate radius is $2.00 \mathrm{~cm}$. Assume that the current $i$ in the wire and the displacement current $i_{d}$ in the capacitor gap are both uniformly distributed. What is the magnitude of the magnetic field due to $i$ at the following radial distances from the wire's center: (a) $1.00 \mathrm{~mm}$ (inside the wire), (b) $3.00 \mathrm{~mm}$ (outside the wire), and (c) $2.20 \mathrm{~cm}$ (outside the wire)? What is the magnitude of the magnetic field due to $i_{d}$ at the following radial distances from the central axis between the plates: (d) $1.00 \mathrm{~mm}$ (inside the gap), (e) $3.00 \mathrm{~mm}$ (inside the gap), and (f) $2.20 \mathrm{~cm}$ (outside the gap)? (g) Explain why the fields at the two smaller radii are so different for the wire and the gap but the fields at the largest radius are not.

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator

Problem 59

A parallel-plate capacitor with circular plates of radius $R=16 \mathrm{~mm}$ and gap width $d=5.0 \mathrm{~mm}$ has a uniform electric field between the plates. Starting at time $t=0$, the potential difference between the two plates is $V=(100 \mathrm{~V}) e^{-t / \tau}$, where the time constant $\tau=12 \mathrm{~ms}$. At radial distance $r=0.80 R$ from the central axis, what is the magnetic field magnitude (a) as a function of time for $t \geq 0$ and (b) at time $t=3 \tau$ ?

Keshav Singh

Numerade Educator

Problem 60

A magnetic flux of $7.0 \mathrm{mWb}$ is directed outward through the flat bottom face of the closed surface shown in Fig. 32-40. Along the flat top face (which has a radius of $4.2 \mathrm{~cm}$ ) there is a 0.40 T magnetic field $\vec{B}$ directed perpendicular to the face. What are the (a) magnitude and (b) direction (inward or outward) of the magnetic flux through the curved part of the surface?

Keshav Singh

Numerade Educator

Problem 61

The magnetic field of Earth can be approximated as the magnetic field of a dipole. The horizontal and vertical components of this field at any distance $r$ from Earth's center are given by
$$
B_{h}=\frac{\mu_{0} \mu}{4 \pi r^{3}} \cos \lambda_{m}, \quad B_{v}=\frac{\mu_{0} \mu}{2 \pi r^{3}} \sin \lambda_{m},
$$
where $\lambda_{m}$ is the magnetic latitude (this type of latitude is measured from the geomagnetic equator toward the north or south geomagnetic pole). Assume that Earth's magnetic dipole moment has magnitude $\mu=8.00 \times 10^{22} \mathrm{~A} \cdot \mathrm{m}^{2} .$ (a) Show that the magnitude of Earth's field at latitude $\lambda_{m}$ is given by
$$
B=\frac{\mu_{0} \mu}{4 \pi r^{3}} \sqrt{1+3 \sin ^{2} \lambda_{m}}
$$
(b) Show that the inclination $\phi_{i}$ of the magnetic field is related to the magnetic latitude $\lambda_{m}$ by $\tan \phi_{i}=2 \tan \lambda_{m} .$

Keshav Singh

Numerade Educator

Problem 62

Use the results displayed in Problem 61 to predict the (a) magnitude and (b) inclination of Earth's magnetic field at the geomagnetic equator, the (c) magnitude and (d) inclination at geomagnetic latitude $60.0^{\circ}$, and the (e) magnitude and (f) inclination at the north geomagnetic pole.

Keshav Singh

Numerade Educator

Problem 63

A parallel-plate capacitor with circular plates of radius $55.0 \mathrm{~mm}$ is being charged. At what radius (a) inside and (b) outside the capacitor gap is the magnitude of the induced magnetic field equal to $50.0 \%$ of its maximum value?

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator

Problem 64

A sample of the paramagnetic salt to which the magnetization curve of Fig. 32-14 applies is immersed in a uniform magnetic field of $2.0 \mathrm{~T}$. At what temperature will the degree of magnetic saturation of the sample be (a) $50 \%$ and (b) $90 \%$ ?

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator

Problem 65

A parallel-plate capacitor with circular plates of radius $R$ is being discharged. The displacement current through a central circular area, parallel to the plates and with radius $R / 2$, is $2.0 \mathrm{~A}$. What is the discharging current?

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator

Problem 66

Figure $32-41$ gives the variation of an electric field that is perpendicular to a circular area of $2.0 \mathrm{~m}^{2}$. During the time period shown, what is the greatest displacement current through the area?

Keshav Singh

Numerade Educator

Problem 67

In Fig. 32-42, a parallel-plate capacitor is being discharged by a current $i=5.0 \mathrm{~A} .$ The plates are square with edge length $L=8.0 \mathrm{~mm}$. (a) What is the rate at which the electric field between the plates is changing? (b) What is the value of $\oint \vec{B} \cdot d \vec{s}$ around the dashed path, where $H=2.0 \mathrm{~mm}$ and $W=3.0 \mathrm{~mm} ?$

Keshav Singh

Numerade Educator

Problem 68

What is the measured component of the orbital magnetic dipole moment of an electron with the values (a) $m_{\ell}=3$ and (b) $m_{\ell}=-4$ ?

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator

Problem 69

In Fig. 32-43, a bar magnet lies near a paper cylinder. (a) Sketch the magnetic field lines that pass through the surface of the cylinder. (b) What is the sign of $\vec{B} \cdot d \vec{A}$ for every area $d \vec{A}$ on the surface? (c) Does this contradict Gauss' law for magnetism? Explain.

Thomas Lau

Numerade Educator

Problem 70

In the lowest energy state of the hydrogen atom, the most probable distance of the single electron from the central proton (the nucleus) is $5.2 \times 10^{-11} \mathrm{~m}$. (a) Compute the magnitude of the proton's electric field at that distance. The component $\mu_{s, z}$ of the proton's spin magnetic dipole moment measured on a $z$ axis is $1.4 \times 10^{-26} \mathrm{~J} / \mathrm{T}$. (b) Compute the magnitude of the proton's magnetic field at the distance $5.2 \times 10^{-11} \mathrm{~m}$ on the $z$ axis. (Hint: Use Eq. $29-27 .$ ) (c) What is the ratio of the spin magnetic dipole moment of the electron to that of the proton?

Keshav Singh

Numerade Educator

Problem 71

Figure 32.-38 shows a loop model (loop $I$.) for a paramagnetic material. (a) Sketch the field lines through and about the material due to the magnet. What is the direction of (b) the loop's net magnetic dipole moment $\vec{\mu},(\mathrm{c})$ the conventional current $i$ in the loop (clockwise or counterclockwise in the figure), and (d) the magnetic force acting on the loop?

Keshav Singh

Numerade Educator

Problem 72

Two plates (as in Fig. $32-7$ ) are being discharged by a constant current. Each plate has a radius of $4.00 \mathrm{~cm}$. During the discharging, at a point between the plates at radial distance $2.00 \mathrm{~cm}$ from the central axis, the magnetic field has a magnitude of $12.5 \mathrm{nT}$. (a) What is the magnitude of the magnetic field at radial distance $6.00 \mathrm{~cm} ?$ (b) What is the current in the wires attached to the plates?

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator

Problem 73

If an electron in an atom has orbital angular momentum with $m_{\ell}$ values limited by $\pm 3$, how many values of (a) $L_{\text {orb } z}$ and (b) $\mu_{\text {orb } z}$ can the electron have? In terms of $h, m$, and $e$, what is the greatest allowed magnitude for (c) $L_{\text {orb }, z}$ and $($ d $) \mu_{\text {orb }, z} ?$ (e) What is the greatest allowed magnitude for the $z$ component of the electron's $n e t$ angular momentum (orbital plus spin)? (f) How many values (signs included) are allowed for the $z$ component of its net angular momentum?

Keshav Singh

Numerade Educator

Problem 74

A parallel-plate capacitor with circular plates is being charged. Consider a circular loop centered on the central axis and located between the plates. If the loop radius of $3.00 \mathrm{~cm}$ is greater than the plate radius, what is the displacement current between the plates when the magnetic field along the loop has magnitude $2.00 \mu \mathrm{T} ?$

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator

Problem 75

Suppose that $\pm 4$ are the limits to the values of $m_{\ell}$ for an electron in an atom. (a) How many different values of the electron's $\mu_{\text {orb }, z}$ are possible? (b) What is the greatest magnitude of those possible values? Next, if the atom is in a magnetic field of magnitude $0.250 \mathrm{~T}$, in the positive direction of the $z$ axis, what are (c) the maximum energy and (d) the minimum energy associated with those possible values of $\mu_{\text {orb }, z}$ ?

Keshav Singh

Numerade Educator

Problem 76

What are the measured components of the orbital magnetic dipole moment of an electron with (a) $m_{\ell}=3$ and (b) $m_{\ell}=-4$ ?

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator