Understanding Physics

Karen Cummings, Priscilla W. Laws, Edward F. Redish

Chapter 32

Inductors and Magnetic Materials - all with Video Answers

Educators

Chapter Questions

Problem 1

The inductance of a close-packed coil of 400 turns is $8.0 \mathrm{mH}$. Calculate the magnetic flux through the coil when the current is $5.0 \mathrm{~mA}$.

Vishal Gupta

Vishal Gupta

Numerade Educator

Problem 2

A circular coil has a $10.0 \mathrm{~cm}$ radius and consists of $30.0$ closely wound turns of wire. An externally produced magnetic field of magnitude $2.60 \mathrm{mT}$ is perpendicular to the coil.
(a) If no current is in the coil, what is the magnitude of the magnetic flux that links its turns? (b) When the current in the coil is $3.80 \mathrm{~A}$ in a certain direction, the net flux through the coil is found to vanish. What is the inductance of the coil?

Katie Mcalpine

Numerade Educator

Problem 3

Two long parallel wires, both of radius $a$ and whose centers are a distance $d$ apart, carry equal currents in opposite directions. Show that, neglecting the flux within the wires, the inductance of a length $l$ of such a pair of wires is given by
$$
L=\frac{\mu_{0} l}{\pi} \ln \frac{d-a}{a}
$$
(Hint: Calculate the flux through a rectangle of which the wires form two opposite sides.)

Vishal Gupta

Numerade Educator

Problem 4

A wide copper strip of width $W$ is bent to form a tube of radius $R$ with two parallel planar extensions, as shown in Fig. $32-25 .$ There is a current $i$ through the strip, distributed uniformly over its width. In this way a "one-turn solenoid" is formed. (a) Derive an expression for the magnitude of the magnetic field $\vec{B}$ in the FIGURE $32-25=$ Problem 4 .

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 5

A $12 \mathrm{H}$ inductor carries a steady current of $2.0 \mathrm{~A}$. How can a $60 \mathrm{~V}$ self-induced emf be made to appear in the inductor?

Vishal Gupta

Numerade Educator

Problem 6

At a given instant the current and selfinduced emf in an inductor are directed as indicated in Fig. $32-26 .$ (a) Is the current increasing or decreasing? (b) The induced emf is $17 \mathrm{~V}$ and the rate of change of the FIGURE $32-26=$ Problem $6 .$ current is $25 \mathrm{kA} / \mathrm{s} ;$ find the inductance.

Vishal Gupta

Numerade Educator

Problem 7

Two inductors $L_{1}$ and $L_{2}$ are connected in series and are separated by a large distance. (a) Show that the equivalent inductance is given by
$$
L_{\mathrm{eq}}=L_{1}+L_{2}
$$
(Hint: Review the derivations for resistors in series and capacitors in series. Which is similar here?) (b) Why must their separation be large for this relationship to hold? (c) What is the generalization of
(a) for $N$ inductors in series?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 8

The current $i$ through a $4.6 \mathrm{H}$ inductor varies with time $t$ as shown by the graph of Fig. $32-27$. The inductor has a resistance of $12 \Omega$. Find the magnitude of the induced emf $\mathscr{C}$ during the time intervals (a) $t_{1}=0$ to $t_{2}=2 \mathrm{~ms}$, (b) $t_{2}=2 \mathrm{~ms}$ to $t_{3}=5 \mathrm{~ms}$, (c) $t_{3}=5 \mathrm{~ms}$ to $t_{4}=6 \mathrm{~ms}$. (Ignore the behavior at the ends of the intervals.)

Vishal Gupta

Numerade Educator

Problem 9

At time $t=0 \mathrm{~ms}$, a $45 \mathrm{~V}$ potential difference is suddenly applied to the leads of a coil with inductance $L=50 \mathrm{mH}$ and resistance $R=180 \Omega$. At what rate is the current through the coil increasing at $t=1.2 \mathrm{~ms}$ ?

Katie Mcalpine

Numerade Educator

Problem 10

Two inductors $L_{1}$ and $L_{2}$ are connected in parallel and separated by a large distance. (a) Show that the equivalent inductance is given by
$$
\frac{1}{L_{\mathrm{eq}}}=\frac{1}{L_{1}}+\frac{1}{L_{2}}
$$
(Hint: Review the derivations for resistors in parallel and capacitors in parallel. Which is similar here?) (b) Why must their separation be large for this relationship to hold? (c) What is the generalization of
(a) for $N$ inductors in parallel?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 11

The inductance of a closely wound coil is such that an emf of $3.0 \mathrm{mV}$ is induced when the current changes at the rate of $5.0 \mathrm{~A} / \mathrm{s}$. A steady current of $8.0 \mathrm{~A}$ produces a magnetic flux of $40 \mu$ Wb through each turn. (a) Calculate the inductance of the coil.
(b) How many turns does the coil have?

Vishal Gupta

Numerade Educator

Problem 12

Coil 1 in Fig. $32-4$ has $L_{1}=25 \mathrm{mH}$ and $N_{1}=$ 100 turns. Coil 2 has $L_{2}=40 \mathrm{mH}$ and $N_{2}=200$ turns. The coils are rigidly positioned with respect to each other; their mutual inductance $M$ is $3.0 \mathrm{mH}$. A $6.0 \mathrm{~mA}$ current in coil 1 is changing at the rate of $4.0 \mathrm{~A} / \mathrm{s}$. (a) What magnetic flux $\Phi_{1 \rightarrow 2}$ links coil 2 , and what self-induced emf appears there? (b) What magnetic flux $\Phi_{2 \rightarrow 1}$ links coil 1, and what mutually induced emf appears there?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 13

Two coils are at fixed locations. When coil 1 has no current and the current in coil 2 increases at the rate $15.0 \mathrm{~A} / \mathrm{s}$, the emf in coil 1 is $25.0 \mathrm{mV}$. (a) What is their mutual inductance? (b) When coil 2 has no current and coil 1 has a current of $3.60 \mathrm{~A}$, what is the flux linkage in coil $2 ?$

Vishal Gupta

Numerade Educator

Problem 14

are part of the spark coil of an automobile. When the current in one solenoid falls from $6.0$ A to zero in $2.5 \mathrm{~ms}$, an emf of $30 \mathrm{kV}$ is induced in the other solenoid. What is the mutual inductance $M$ of the solenoids?

Vishal Gupta

Numerade Educator

Problem 15

Two coils. connected as shown in Fig. $32-28$, separately have inductances $L_{1}$ and $L_{2}$. Their mutual inductance is $M$.
(a) Show that this combination can be replaced by a single coil of equivalent inductance given by
$$
L_{\mathrm{eq}}=L_{1}+L_{2}+2 M
$$
(b) How could the coils in Fig. $32-28$ be reconnected to yield an equivalent inductance of
$$
L_{\mathrm{eq}}=L_{1}+L_{2}-2 M ?
$$
(This problem is an extension of Problem 7, but the requirement that the coils be far apart has been removed.)

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 16

A coil $\mathrm{C}$ of $N$ turns is placed around a long solenoid $\mathrm{S}$ of radius $R$ and $n$ turns per unit length as in Fig. $32-29 .$ Show that the mutual inductance for the coil-solenoid combination is given by $M=\mu_{0} \pi R^{2} n N .$ Explain why $M$ does not depend on the shape, size, or possible lack of close-packing of the coil.

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 17

Figure $32-30$ shows, in cross section, two coaxial solenoids. Show that the mutual inductance $M$ for a length $l$ of this solenoid-solenoid combination is given by $M=\pi R_{1}^{2} l \mu_{0} n_{1} n_{2}$ in which $n_{1}$ and $n_{2}$ are the respective numbers of turns per unit length and $R_{1}$ is the radius of the inner solenoid. Why does $M$ depend on $R_{1}$ and not $\quad$ FIGURE $32-30=$ Problem 17 . on $R_{2} ?$

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 18

Figure 32-31 shows a coil of $N_{2}$ turns wound as shown around part of a toroid of $N_{1}$ turns. The toroid's inner radius is $a$, its outer radius is $b$, and its height is $h$. Show that the mutual inductance $M$ for the toroid-coil combination is
$$
M=\frac{\mu_{0} N_{1} N_{2} h}{2 \pi} \ln \frac{b}{a}
$$

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 19

A rectangular loop of $N$ close-packed turns is positioned near a long straight wire as shown in Fig. $32-32$.
(a) What is the mutual inductance $\bar{M}$ for the loop-wire combination? (b) Evaluate $M$ for $N=100, a=1.0$ $\mathrm{cm}, b=8.0 \mathrm{~cm}$, and $l=30 \mathrm{~cm} .$

Vishal Gupta

Numerade Educator

Problem 20

The current in an $R L$ circuit builds up to one third of its steady-state value in $5.00 \mathrm{~s}$. Find the inductive time constant.

Averell Hause

Carnegie Mellon University

Problem 21

Consider the $R L$ circuit of Fig. $32-6 .$ In terms of the battery emf $\mathscr{E}$, (a) what is the self-induced emf $\Delta V_{2}$ when the switch has just been closed on $a$, and (b) what is $\Delta V_{2}$ when $t=2.0 \tau_{L} ?$ (c) In terms of $\tau_{L}$, when will $\Delta V_{2}$ be just one-half the battery emf $\mathscr{E}$ ?

Vishal Gupta

Numerade Educator

Problem 22

Consider the $R L$ circuit of Fig. 32-6. In terms of the battery emf $\mathscr{E}$, (a) what is the self-induced emf $\Delta V_{2}$ when the switch has just been closed on $a$, and (b) what is $\Delta V_{2}$ when $t=2.0 \tau_{L} ?$ (c) In terms of $\tau_{L}$, when will $\Delta V_{2}$ be just one-half the battery emf $\mathscr{E}$ ?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 23

The current in an $R L$ circuit drops from $1.0 \mathrm{~A}$ to $10 \mathrm{~mA}$ in the first second following removal of the battery from the circuit. If $L$ is $10 \mathrm{H}$, find the resistance $R$ in the circuit.

Vishal Gupta

Numerade Educator

Problem 24

Time Suppose the emf of the battery in the circuit of Fig. $32-7$ varies with time $t$ so that the current is given by $i(t)=3.0 \mathrm{~A}+(5.0 \mathrm{~A} / \mathrm{s}) t$, where $i$ is in amperes and $t$ is in seconds. Take $R=4.0 \Omega$ and $L=6.0 \mathrm{H}$, and find an expression for the battery emf as function of time. (Hint: Apply the loop rule.)

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 25

A solenoid having an inductance of $6.30 \mu \mathrm{H}$ is connected in series with a $1.20 \mathrm{k} \Omega$ resistor. (a) If a $14.0 \mathrm{~V}$ battery is inserted into the circuit, how long will it take for the current through the resistor to reach $80.0 \%$ of its final value? (b) What is the current through the resistor at time $t=1.0 \tau_{l} ?$

Vishal Gupta

Numerade Educator

Problem 26

A wooden torodial core with a square cross section has an inner radius of $10 \mathrm{~cm}$ and an outer radius of $12 \mathrm{~cm}$. It is wound with one layer of wire (of diameter $1.0 \mathrm{~mm}$ and resistance per meter $0.020 \Omega / \mathrm{m}$ ). What are (a) the inductance and
(b) the inductive time constant of the resulting toroid? Ignore the thickness of the insulation on the wire.

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 27

At time $t=0 \mathrm{~ms}$, a $45.0 \mathrm{~V}$ potential difference is suddenly applied to a coil with $L=50.0 \mathrm{mH}$ and $R=$ $180 \Omega$. At what rate is the current increasing at $t=1.20 \mathrm{~ms}$ ?

Vishal Gupta

Numerade Educator

Problem 28

In the Circuit In the circuit of Fig. $32-33, \mathscr{E}=10 \mathrm{~V}, R_{1}=5.0 \Omega, R_{2}$
$=10 \Omega$, and $L=5.0 \mathrm{H}$. For the two separate conditions (I) switch S just closed and (II) switch S closed for a long time, calculate (a) the current $i_{1}$ through $R_{1}$, (b) the current $i_{2}$ through $R_{2},(\mathrm{c})$ the current $i$ through the switch, (d) the potential difference across $R_{2}$, (e) the potential difference across $L$, and $(\mathrm{f})$ the rate of change $d i_{2} / d t$.

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 29

. In the Figure In Fig. $32-34, \mathscr{E}=$ $100 \mathrm{~V}, R_{1}=10.0 \Omega, R_{2}=20.0 \Omega, R_{3}$
$=30.0 \Omega$, and $L=2.00 \mathrm{H}$. Find the values of $i_{1}$ and $i_{2}$ (a) immediately after closing of switch $\mathrm{S},(\mathrm{b})$ a long time later, (c) immediately after the reopening of switch $\mathrm{S}$, and (d) a long time after the reopening.

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 29

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

Keshav Singh

Numerade Educator

Problem 30

Figure $32-35 a$ shows a circuit consisting of an ideal battery with emf $\mathscr{E}=6.00 \mu \mathrm{V}$, a resistance $R$, and a small wire loop of area $5.0 \mathrm{~cm}^{2}$. For the time interval $t_{1}=10$ to $t_{2}=$ $20 \mathrm{~s}$, an external magnetic field is set up throughout the loop. The field is uniform, its direction is into the page in Fig. $32-35 a$, and the field magnitude is given by $B=a t$, where $B$ is in teslas, $a$ is a constant with units of teslas per second, and $t$ is in seconds. Figure $32-35 b$ gives the current $i$ in the circuit before, during, and after the external field is set up. Find $a$.

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 31

Once the switch $\mathrm{S}$ is closed in Fig. $32-36$ the time required for the current to reach any obtainable value depends, in part, on the value of resistance $R$. Suppose the emf $\mathscr{E}$ of the ideal battery is $12 \mathrm{~V}$ and the inductance of the ideal (resistanceless) inductor is $18 \mathrm{mH}$. FIGURE $32-36=$ How much time is needed for the current Problems 31,34 , to reach $2.00 \mathrm{~A}$ if $R$ is (a) $1.00 \Omega$, (b) $5.00$ and 61 . $\Omega$, and $(\mathrm{c}) 6.00 \Omega ?(\mathrm{~d}) \mathrm{Why}$ is there a huge
jump between the answers to (b) and (c)? (e) For what value of $R$ is the time required for the current to reach $2.00 \mathrm{~A}$ least? (f) What is that least time? (Hint: Rethink Eq. 32-21.)

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 32

In the circuit shown in Fig. $32-37$, switch $S$ is closed at time $t=0$. Thereafter, the constant current source, by varying its emf, maintains a constant current $i$ out of its upper terminal. (a) Derive an expression for the current through the inductor as a function of time. (b) Show that the current through the resistor equals the current through the inductor at time $t=(L / R) \ln 2 .$

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 33

In Fig. $32-38 a$, switch $\mathrm{S}$ has been closed on $A$ long enough to establish a steady current in the inductor of inductance $L_{1}=5.00 \mathrm{mH}$ and the resistor of resistance $R_{1}=$ $25 \Omega$. Similarly, in Fig. $32-38 b$, switch $\mathrm{S}$ has been closed on $A$ long enough to establish a steady current in the inductor of inductance $L_{2}=3.00 \mathrm{mH}$ and the resistor of resistance $R_{2}=30 \Omega$. The ratio $\Phi_{02} / \Phi_{01}$ of the magnetic flux through a turn in inductor 2 to that in inductor 1 is $1.5$. At time $t=0$, the two switches are closed on $B$. At what time $t$ is the flux through a turn in the two inductors equal?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 34

In Fig. $32-38 a$, switch $\mathrm{S}$ has been closed on $A$ long enough to establish a steady current in the inductor of inductance $L_{1}=5.00 \mathrm{mH}$ and the resistor of resistance $R_{1}=$ $25 \Omega$. Similarly, in Fig. $32-38 b$, switch $\mathrm{S}$ has been closed on $A$ long enough to establish a steady current in the inductor of inductance $L_{2}=3.00 \mathrm{mH}$ and the resistor of resistance $R_{2}=30 \Omega$. The ratio $\Phi_{02} / \Phi_{01}$ of the magnetic flux through a turn in inductor 2 to that in inductor 1 is $1.5$. At time $t=0$, the two switches are closed on $B$. At what time $t$ is the flux through a turn in the two inductors equal?

Vishal Gupta

Numerade Educator

Problem 35

A transformer has 500 primary turns and 10 secondary turns. (a) If $\Delta V_{p}$ is $120 \mathrm{~V}$ (rms), what is $\Delta V_{s}$ with an open circuit? (b) If the secondary now has a resistive load of $15 \Omega$, what are the currents in the primary and secondary?

Vishal Gupta

Numerade Educator

Problem 36

A generator supplies $100 \mathrm{~V}$ to the primary coil of a transformer of 50 turns. If the secondary coil has 500 turns, what is the secondary voltage?

Vishal Gupta

Numerade Educator

Problem 37

In Fig. 32-39 let the rectangular box on the left represent the (high-impedance) output of an audio amplifier, with $r=$ $1000 \Omega$. Let $R=10 \Omega$ represent the (low-impedance) coil of a loudspeaker. For maximum transfer of energy to the load $R$ we must have $R=r$, and that is not true in this FIGURE $32-39=$ case. However, a transformer can be $\quad$ Problem 37 . used to "transform" resistances, making them behave electrically as if they were larger or smaller than they actually are. Sketch the primary and secondary coils of a transformer that can be introduced between the amplifier and the speaker in Fig. $32-39$ to match the impedances. What must he the turns ratio?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 38

Figure $32-40$ shows an "autotransformer." It consists of a single coil (with an iron core). Three taps $T_{N}$ are provided. Between taps $T_{1}$ and $T_{2}$ there are 200 turns, and between taps $T_{2}$ and $T_{3}$ there are 800 turns. Any two taps can be considered the "primary terminals" and any two taps can be considered the "secondary terminals" List all the ratios by which the primary voltage may be changed to a secondary voltage.

Vishal Gupta

Numerade Educator

Problem 40

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?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 41

If an electron in an atom has an orbital angular momentum with $m_{l}=0$, (a) what is the component $\mu_{z}^{\text {orb }}$ ? If the atom is in an external magnetic field $\vec{B}$ of magnitude $35 \mathrm{mT}$ and directed along $z$ axis, what are the potential energies associated with the orientations of (b) the electron's orbital magnetic dipole moment and (c) the electron's spin magnetic dipole moment?
(d) Repeat (a) through (c) for $m_{l}=-3$.

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 42

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 $\vec{B}$ ?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 43

Suppose that $\pm 4$ are the limits to the values of $m$ for an electron in an atom. (a) How many different values of the z-component $\mu_{z}^{\text {orb }}$ of the electron's orbital magnetic dipole moment are possible? (b) What is the greatest magnitude of those possible values? Next, suppose that 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 potential energy and (d) the minimum potential energy associated with those possible values of $\mu_{z}^{\text {orb }}$ ?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 44

Nuclear Magnetic Resonance (NMR) and Magnetic Resonance Imaging (MRI) exploit the interactions between charged particles and very strong magnetic fields in order to produce images (including images of soft tissue). The magnetic field in a certain MRI machine is $0.5$ Tesla. What is the maximum difference in energy that one might measure for a single electron placed in this field?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 45

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

Vishal Gupta

Numerade Educator

Problem 46

The dipole moment associated with an atom of iron in an iron bar has magnitude $2.1 \times 10^{-23} \mathrm{~J} / \mathrm{T}$. 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 magnitude of the dipole moment of the bar? (b) What is the magnitude of the torque that must be exerted to hold this magnet perpendicular to an external field of $1.5 \mathrm{~T}$ ? (The density of iron is $7.9 \mathrm{~g} / \mathrm{cm}^{3} .$ )

Vishal Gupta

Numerade Educator

Problem 47

The magnetic dipole moment of Earth has magnitude $8.0 \times 10^{22} \mathrm{~J} / \mathrm{T}$. (a) If the origin of this magnetism were a magnetized iron sphere at the center of the Earth, what would be its radius? (b) What fraction of the volume of the Earth would such a sphere occupy? Assume complete alignment of the dipoles. The density of the 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: The 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 the Earth's magnetism has been ruled out by several considerations. For one, the temperature is certainly above the Curie point.)

Vishal Gupta

Numerade Educator

Problem 48

Measurements in mines and boreholes indicate that the 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 49

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 50

l Figure $32-41$ shows a loop model (loop $L$ ) for a diamagnetic material. (a) Sketch the magnetic field lines through and about the material due to the bar magnet. (b) What are the directions of the loop's net magnetic dipole moment $\vec{\mu}$ and the conventional current $i$ in the loop? (c) What is the direction of the magnetic force on the loop?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 51

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 the magnitude of its magnetic dipole moment?

Vishal Gupta

Numerade Educator

Problem 52

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

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 53

A sample of the paramagnetic salt to which the magnetization curve of Fig. $32-19$ 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 Curie's law be valid under these conditions?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 54

Repeat Problem 50 for the case in which loop $L$ is the model for a paramagnetic material.

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 55

An electron with kinetic energy $K$ travels in a circular path that is perpendicular to a uniform magnetic field, 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 $\mu=K /|\vec{B}|$ and that it is in the direction opposite that of $\vec{B}$. (b) What are the magnitude and direction of the magnetic dipole moment of a positive ion with kinetic energy $K_{\text {ion }}$ under the same circumstances? (c) 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 $1.2 \mathrm{~T}$.

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 56

A sample of the paramagnetic salt to which the magnetization curve of Fig. $32-17$ 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?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 57

In New Hampshire the average horizontal component of the 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 the Earth's magnetic field?

Vishal Gupta

Numerade Educator

Problem 58

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

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 59

Use the results of Problem 60 to predict the Earth's magnetic field (both magnitude and inclination) at (a) the geomagnetic equator, (b) a point at geomagnetic latitude $60^{\circ}$, and
(c) the north geomagnetic pole.

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 60

The magnetic field of the Earth can be approximated as the magnetic field of a dipole, with horizontal and vertical components, at a point a distance $r$ from the Earth's center, 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 the Earth's magnetic dipole moment is $\mu=8.00 \times 10^{22} \mathrm{~A} \cdot \mathrm{m}^{2} .$ (a) Show that the magnitude of the 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
$$
\tan \phi_{i}=2 \tan \lambda_{m}
$$

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 61

In Fig. 32-36, a $12.0 \mathrm{~V}$ ideal battery, a $20 \Omega$ resistor, and an ideal inductor are connected by a switch at time $t=0 \mathrm{~s}$. At what rate is the battery transferring energy to the inductor's field at $t=1.61 \tau_{L}$ ?

Katie Mcalpine

Numerade Educator

Problem 62

You place a magnetic compass on a horizontal surface, allow the needle to settle into equilibrium position, and then give the compass a gentle wiggle to cause the needle to oscillate about that equilibrium position. The frequency of oscillation is $0.312 \mathrm{~Hz}$. The 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?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 63

A long narrow coil is surrounded by a short wide coil as shown in Fig. $32-42$. Both coils have negligible resistance. The short wide coil has a diameter $d_{S}, n_{S}$ turns per unit length, and a length $S .$ Its ends are connected through a resistor of resistance $R$. The long narrow inner coil has a diameter $d_{L}, n_{L}$ turns per unit length, and a length $L$. Its ends are connected across a variable power source.

For each of the partial sentences below, indicate whether they are correctly completed by the phrase greater than $(>)$, less than $(<)$, or the same as $(=)$. If you cannot determine which is the case from the information given, indicate not sufficient information (NSI).

Vishal Gupta

Numerade Educator

Problem 64

Figure $32-43$ shows a solenoid and two hoops. When the switch is closed, the solenoid carries a current in the direction indicated. The planes of the small loops are parallel to the planes of the hoops of the solenoid.

Vishal Gupta

Numerade Educator