Fundamentals of Physics

David Halliday, Robert Resnick , Jearl Walker

Chapter 30

Induction and Inductance - all with Video Answers

Educators

Chapter Questions

Problem 1

In Fig. $30-33,$ a circular loop of wire 10 $\mathrm{cm}$
in diameter (seen edge-on) is placed with its
normal $\vec{N}$ at an angle $\theta=30^{\circ}$ with the direction
of a uniform magnetic field $\vec{B}$ of magnitude
0.50 T. The loop is then rotated such that $\vec{N}$ rotates in a cone about the field direction at the
rate 100 rev/min; angle $\theta$ remains unchanged
during the process. What is the emf induced in
the loop?

Katie Mcalpine

Katie Mcalpine

Numerade Educator

Problem 2

A certain clastic conducting material is stretched into a circular loop of 12.0 $\mathrm{cm}$ radius. It is placed with its plane perpendicular to a uniform 0.800 $\mathrm{T}$ magnetic field. When released, the radius of the loop starts to shrink at an instantaneous rate of
75.0 $\mathrm{cm} / \mathrm{s} .$ What emf is induced in the loop at that instant?

Averell Hause

Carnegie Mellon University

Problem 3

SSM WWW In Fig. $30-34,$ a 120 -
turn coil of radius 1.8 $\mathrm{cm}$ and resist-
ance 5.3$\Omega$ is coaxial with a solenoid
of 220 turns/cm and diameter 3.2 $\mathrm{cm}$ .
The solenoid current drops from.
1.5 $\mathrm{A}$ to zero in time interval $\Delta t=$
25 $\mathrm{ms}$ . What current is induced in the
coil during $\Delta t ?$

Prabhu Ramji

Numerade Educator

Problem 4

A wire loop of radius 12 $\mathrm{cm}$ and resistance 8.5$\Omega$ is located in a uniform
magnetic field $\vec{B}$ that changes in magnitude as given in Fig. $30-35 .$ The vertical
axis scale is set by $B_{s}=0.50 \mathrm{T}$ , and the
horizontal axis scale is set by $t_{s}=6.00 \mathrm{s}$
The loop's plane is perpendicular to $\vec{B}$ .
What emf is induced in the loop during time intervals (a) 0 to 2.0 $\mathrm{s}$
(b) 2.0 s to 4.0 s, and (c) 4.0 s to 6.0 s?

Averell Hause

Carnegie Mellon University

Problem 5

In Fig. $30-36,$ a wire forms a closed circular loop, of radius
$R=2.0 \mathrm{m}$ and resistance 4.0$\Omega .$ The circle is centered on a long
straight wire; at time $t=0,$ the current in the long straight wire
is 5.0 $\mathrm{A}$ rightward. Thereafter, the current changes according to
$i=5.0 \mathrm{A}-\left(2.0 \mathrm{A} / \mathrm{s}^{2}\right) t^{2} .$ (The straight wire is insulated; so there
is no electrical contact between it and the wire of the loop.)
What is the magnitude of the current induced in the loop at
times $t>0 ?$

Katie Mcalpine

Numerade Educator

Problem 6

Figure $30-37 a$ shows a circuit consisting of an ideal battery
with emf $\mathscr{G}=6.00 \mu \mathrm{V}$ , a resistance $R,$ and a small wire loop of area
5.0 $\mathrm{cm}^{2} .$ For the time interval $t=10 \mathrm{s}$ to $t=20 \mathrm{s},$ an external magnetic field is set up throughout the loop. The field is uniform, its
direction is into the page in Fig. $30-37 a$ , and the field magnitude is
given by $B=a t,$ where $B$ is in teslas, $a$ is a constant, and $t$ is in
seconds. Figure $30-37 b$ gives the current $i$ in the circuit before, during, and after the external field is set up. The vertical axis scale is
set by $i_{s}=2.0 \mathrm{m}$ . Find the constant $a$ in the equation for the field
magnitude.

Averell Hause

Carnegie Mellon University

Problem 7

In Fig. $30-38,$ the magnetic flux
through the loop increases according to
the relation $\Phi_{B}=6.0 t^{2}+7.0 t,$ where $\Phi_{B}$ is
in milliwebers and $t$ is in seconds. (a) What
is the magnitude of the emf induced in the
loop when $t=2.0 \mathrm{s} ?$ (b) Is the direction of
the current through $R$ to the right or left?

Katie Mcalpine

Numerade Educator

Problem 8

A uniform magnetic field $\vec{B}$ is perpendicular to the plane of a circular loop
of diameter 10 $\mathrm{cm}$ formed from wire of
diameter 2.5 $\mathrm{mm}$ and resistivity 1.69 $\mathrm{x}$
$10^{-8} \Omega \cdot \mathrm{m} .$ At what rate must the magnitude of $B$ change to induce a 10 A current in the loop?

Averell Hause

Carnegie Mellon University

Problem 9

A small loop of area 6.8 $\mathrm{mm}^{2}$ is placed inside a long solenoid
that has 854 turns/cm and carries a sinusoidally varying current $i$ of
amplitude 1.28 $\mathrm{A}$ and angular frequency 212 rad/s. The central axes
of the loop and solenoid coincide. What is the amplitude of the emf
induced in the loop?

Katie Mcalpine

Numerade Educator

Problem 10

Figure $30-39$ shows a closed
loop of wire that consists of a pair of
equal semicircles, of radius $3.7 \mathrm{cm},$
lying in mutually perpendicular
planes. The loop was formed by folding a flat circular loop along a diameter until the two halves became
perpendicular to each other. A uniform magnetic field $\vec{B}$ of magnitude
76 $\mathrm{mT}$ is directed perpendicular to
the fold diameter and makes equal
angles (of $45^{\circ}$ ) with the planes of the
semicircles. The magnetic field is reduced to zero at a uniform rate
during a time interval of 4.5 $\mathrm{ms}$ . During this interval, what are
the (a) magnitude and (b) direction (clockwise or counterclockwise when viewed along the direction of $\vec{B}$ ) of the emf induced in
the loop?

Averell Hause

Carnegie Mellon University

Problem 11

A rectangular coil of $N$ turns and of length $a$ and width $b$ is
rotated at frequency $f$ in a uniform magnetic field $B,$ as indicated in
Fig. $30-40 .$ The coil is connected to co-rotating cylinders, against
which metal brushes slide to make contact. (a) Show that the emf
induced in the coil is given (as a function of time $t )$ by
$$\mathscr{E}_{0}=2 \pi f N a b B \sin (2 \pi f t)=\mathscr{E}_{0} \sin (2 \pi f t)$$
This is the principle of the commercial alternating-current generator. (b) What value of $N a b$ gives an emf with $\mathscr{E}_{0}=150 \mathrm{V}$
when the loop is rotated at 60.0 rev/s in a uniform magnetic field
of 0.500 $\mathrm{T} ?$

Katie Mcalpine

Numerade Educator

Problem 12

In Fig. $30-41,$ a wire loop of lengths $L=40.0 \mathrm{cm}$ and $W=$
25.0 $\mathrm{cm}$ lies in a magnetic field $\vec{B}$ . What are the (a) magnitude $\mathscr{E}$ and
(b) direction (clockwise or counterclockwise $-$ or "none" if $\mathscr{E}$=0 )
of the emf induced in the loop if $\vec{B}=(4.00 \times$
$10^{-2} \mathrm{T} / \mathrm{m} ) y \hat{\mathrm{k}} ?$ What are
(c) $\mathscr{E}$ and $(\mathrm{d})$ the
direction if $\vec{B}=\left(6.00 \times 10^{-2} \mathrm{T} / \mathrm{s}\right) t \hat{\mathrm{k}} ?$ What are $(\mathrm{e}) \mathscr{E}$ and $(\mathrm{f})$ the direction if $\vec{B}=(8.00 \times$ $10^{-2} \mathrm{T} / \mathrm{m} \cdot \mathrm{s} ) y t \hat{\mathrm{k}} ?$ What are $(g)\mathscr{E}$ and $(h)$ the direction if $\vec{B}=\left(3.00 \times 10^{-2} \mathrm{T} / \mathrm{m} \cdot \mathrm{s}\right) x t \hat{\mathrm{j}}$ ? What
are (i) $\mathscr{E}$ and $(\mathrm{j})$ the direction if $\vec{B}=(5.00 \times$
$10^{-2} \mathrm{T} / \mathrm{m} \cdot \mathrm{s} )$ yti?

Averell Hause

Carnegie Mellon University

Problem 13

ILW One hundred turns of (insulated) copper wire are
wrapped around a wooden cylindrical core of cross-sectional area
$1.20 \times 10^{-3} \mathrm{m}^{2} .$ The two ends of the wire are connected to a resistor. The total resistance in the circuit is 13.0$\Omega .$ If an externally applied uniform longitudinal magnetic field in the core changes from
1.60 $\mathrm{T}$ in one direction to 1.60 $\mathrm{T}$ in the opposite direction, how
much charge flows through a point in the circuit during the change?

Katie Mcalpine

Numerade Educator

Problem 14

In Fig. $30-42 a,$ a uniform magnetic field $\vec{B}$ increases in
magnitude with time $t$ as given by Fig. $30-42 b,$ where the vertical
axis scale is set by $B_{s}=9.0 \mathrm{mT}$ and the horizontal scale is set by
$t_{s}=3.0 \mathrm{s}$ s. A circular conducting loop of area $8.0 \times 10^{-4} \mathrm{m}^{2}$ lies in the field, in the plane of the page. The amount of charge $q$ passing
point $A$ on the loop is given in Fig. $30-42 c$ as a function of $t$ , with
the vertical axis scale set by $q_{s}=6.0 \mathrm{mC}$ and the horizontal axis
scale again set by $t_{s}=3.0 \mathrm{s}$ . What is the loop's resistance?

Averell Hause

Carnegie Mellon University

Problem 15

A square wire loop with
2.00 $\mathrm{m}$ sides is perpendicular to a
uniform magnetic field, with half the
area of the loop in the field as
shown in Fig. $30-43 .$ The loop contains an ideal battery with emf $8=$
20.0 V. If the magnitude of the field
varies with time according to $B=$
$0.0420-0.870 t,$ with $B$ in teslas and
$t$ in seconds, what are (a) the net emf
in the circuit and (b) the direction of
the (net) current around the loop?

Katie Mcalpine

Numerade Educator

Problem 16

Figure 30.44$a$ shows a wire that forms a rectangle
$(W=20 \mathrm{cm}, H=30 \mathrm{cm})$ and has a resistance of 5.0 $\mathrm{m} \Omega$ . Its interior is split into three equal areas, with magnetic fields $\vec{B}_{1}, \vec{B}_{2}$
and $\vec{B}_{3}$ . The fields are uniform within each region and directly out
of or into the page as indicated. Figure $30-44 b$ gives the change in the $z$ components $B_{z}$ of the three fields with time vertical axis
scale is set by $B_{s}=4.0 \mu \mathrm{T}$ and $B_{b}=-2.5 B_{s}$ , and the horizontal axis
scale is set by $t_{s}=2.0 \mathrm{s}$ . What are the (a) magnitude and (b) direc-
tion of the current induced in the wire?

Averell Hause

Carnegie Mellon University

Problem 17

A small circular loop of area 2.00 $\mathrm{cm}^{2}$ is placed in the plane
of, and concentric with, a large circular loop of radius 1.00 $\mathrm{m}$ . The
current in the large loop is changed at a constant rate from 200 $\mathrm{A}$
to $-200 \mathrm{A}$ (a change in direction) in a time of 1.00 $\mathrm{s}$ , starting at
$t=0 .$ What is the magnitude of the magnetic field $\vec{B}$ at the center
of the small loop due to the current in the large loop at $(a) t=0,$
(b) $t=0.500$ s, and $(c) t=1.00$ s? (d) From $t=0$ to $t=1.00$ s, is $B$
reversed? Because the inner loop is small, assume $\vec{B}$ is uniform over
its area, (e) What emf is induced in the small loop at $t=0.500$ s?

Katie Mcalpine

Numerade Educator

Problem 18

In Fig. $30-45,$ two straight conducting rails form a right angle. A
conducting bar in contact with the
rails starts at the vertex at time $t=0$
and moves with a constant velocity
of 5.20 $\mathrm{m} / \mathrm{s}$ along them. A magnetic
field with $B=0.350 \mathrm{T}$ is directed
out of the page. Calculate (a) the
flux through the triangle formed by the rails and bar at $t=3.00 \mathrm{s}$
and (b) the emf around the triangle at that time. (c) If the emf is
$\mathscr{E}=a t^{n},$ where $a$ and $n$ are constants, what is the value of $n ?$

Averell Hause

Carnegie Mellon University

Problem 19

ILW An electric generator contains a coil of 100 turns of wire,
each forming a rectangular loop 50.0 $\mathrm{cm}$ by 30.0 $\mathrm{cm}$ . The coil is
placed entirely in a uniform magnetic field with magnitude $B=$
3.50 $\mathrm{T}$ and with $\vec{B}$ initially perpendicular to the coil's plane. What
is the maximum value of the emf produced when the coil is spun at
1000 rev/min about an axis perpendicular to $\vec{B} ?$

Katie Mcalpine

Numerade Educator

Problem 20

At a certain place, Earth's magnetic field has magnitude
$B=0.590$ gauss and is inclined downward at an angle of $70.0^{\circ}$ to
the horizontal. A flat horizontal circular coil of with a radius
of 10.0 $\mathrm{cm}$ has 1000 turns and a total resistance of 85.0$\Omega .$ It is connected in series to a meter with 140$\Omega$ resistance. The coil is flipped
through a half-revolution about a diameter, so that it is again horizontal. How much charge flows
through the meter during the flip?

Averell Hause

Carnegie Mellon University

Problem 21

In Fig. $30-46,$ a stiff wire bent
into a semicircle of radius $a=2.0$
$\mathrm{cm}$ is rotated at constant angular
speed 40 $\mathrm{rev} / \mathrm{s}$ in a uniform 20 $\mathrm{mT}$
magnetic field. What are the (a) frequency and (b) amplitude of the emf
induced in the loop?

Katie Mcalpine

Numerade Educator

Problem 22

A rectangular loop (area $=$
0.15 $\mathrm{m}^{2}$ ) turns in a uniform magnetic
field, $B=0.20 \mathrm{T}$ . When the angle between the field and the normal to
the plane of the loop is $\pi / 2$ rad and
increasing at $0.60 \mathrm{rad} / \mathrm{s},$ what emf is
induced in the loop?

Averell Hause

Carnegie Mellon University

Problem 23

SSM Figure $30-47$ shows two
parallel loops of wire having a common axis. The smaller loop (radius $r )$
is above the larger loop (radius $R )$
by a distance $x \gg R .$ Consequently, the magnetic field due to the
counterclockwise current $i$ in the larger loop is nearly uniform
throughout the smaller loop. Suppose that $x$ is increasing at the
constant rate $d x / d t=v .$ (a) Find an expression for the magnetic
flux through the area of the smaller loop as a function of $x .$ Hint:
See Eq. $29-27 .$ In the smaller loop, find (b) an expression for the
induced emf and (c) the direction of the induced current.

Katie Mcalpine

Numerade Educator

Problem 24

A wire is bent into three
circular segments, each of radius $r=$
$10 \mathrm{cm},$ as shown in Fig. $30-48 .$ Each
segment is a quadrant of a circle,
$a b$ lying in the $x y$ plane, $b c$ lying in
the $y z$ plane, and ca lying in the zx
plane. (a) If a uniform magnetic
plane. (a) If a uniform magnetic
field $\vec{B}$ points in the positive $x$ direction, what is the magnitude of the
emf developed in the wire when $B$
increases at the rate of 3.0 $\mathrm{mT} / \mathrm{s} ?$
(b) What is the direction of the
current in segment $b c ?$

Averell Hause

Carnegie Mellon University

Problem 25

Two long, parallel copper wires of diameter 2.5 $\mathrm{mm}$ carry
currents of 10 $\mathrm{A}$ in opposite directions. (a) Assuming that their
central axes are 20 $\mathrm{mm}$ apart, calculate the magnetic flux per meter
of wire that exists in the space between those axes. (b) What percentage of this flux lies inside the wires? (c) Repeat part (a) for
parallel currents.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 26

For the wire arrangement In Fig. $30-49, a=12.0 \mathrm{cm}$ and $b=$
16.0 $\mathrm{cm} .$ The current in the long
straight wire is $i=4.50 t^{2}-10.0 t$ straight wire is $i=4.50 t^{2}-10.0 t$
where $i$ is in amperes and $t$ is in seconds. (a) Find the emf in the square
loop at $t=3.00 \mathrm{s}$ . (b) What is the
direction of the induced current in
the loop?

Averell Hause

Carnegie Mellon University

Problem 27

ILW As seen in Fig. $30-50,$ a
square loop of wire has sides of
length 2.0 $\mathrm{cm}$ . A magnetic field is directed out of the page; its magnitude
is given by $B=4.0 t^{2} y,$ where $B$ is in
teslas, $t$ is in seconds, and $y$ is in meters. At $t=2.5 \mathrm{s},$ what are the
(a) magnitude and (b) direction of
the emf induced in the loop?

Katie Mcalpine

Numerade Educator

Problem 28

In Fig. $30-51$ , a rectangular
loop of with length $a=2.2 \mathrm{cm},$ width $b=0.80 \mathrm{cm},$ and resistance $R=0.40 \mathrm{m} \Omega$ is placed near an infinitely long wire carrying
current $i=4.7 \mathrm{A}$ . The loop is then moved away from the wire at
constant speed $v=3.2 \mathrm{mm} / \mathrm{s}$ . When the center of the loop is at
distance $r=1.5 b,$ what are (a) the magnitude of the magnetic flux
through the loop and (b) the current induced in the loop?

Averell Hause

Carnegie Mellon University

Problem 29

In Fig. $30-52,$ a metal rod is
forced to move with constant velocity $\vec{v}$ along two parallel metal rails,
connected with a strip of metal at
one end. A magnetic field of magnitude $B=0.350 \mathrm{T}$ points out of the
page. (a) If the rails are separated
by $L=25.0 \mathrm{cm}$ and the speed of the
rod is $55.0 \mathrm{cm} / \mathrm{s},$ what emf is generated? (b) If the rod has a resistance of 18.0$\Omega$ and the rails and connector have negligible resistance, what is the current in the rod?
(c) At what rate is energy being transferred to thermal energy?

Katie Mcalpine

Numerade Educator

Problem 30

In Fig. $30-53 a$ a circular loop of wire is concentric with a solenoid and lies in a plane perpendicular to the solenoid's central axis.
The loop has radius 6.00 $\mathrm{cm} .$ The solenoid has radius $2.00 \mathrm{cm},$ consists of 8000 turns/m, and has a current $i_{\text { sol }}$ varying with time $t$ as
given in Fig. $30-53 b,$ where the vertical axis scale is set by $i_{s}=1.00$
A and the horizontal axis scale is set by $t_{y}=2.0 \mathrm{s}$ . Figure $30-53 c$
shows, as a function of time, the energy $E_{\text { th }}$ that is transferred
to thermal energy of the loop; the vertical axis scale is set by $E_{s}=$
100.0 $\mathrm{nJ} .$ What is the loop's resistance?

Averell Hause

Carnegie Mellon University

Problem 31

SSM ILW If 50.0 $\mathrm{cm}$ of copper wire (diameter $=1.00 \mathrm{mm} )$ is
formed into a circular loop and placed perpendicular to a uniform
magnetic field that is increasing at the constant rate of 10.0 $\mathrm{mT} / \mathrm{s}$ , at
what rate is thermal energy generated in the loop?

Katie Mcalpine

Numerade Educator

Problem 32

A loop antenna of area 2.00 $\mathrm{cm}^{2}$ and resistance 5.21$\mu \Omega$ is
perpendicular to a uniform magnetic field of magnitude 17.0$\mu \mathrm{T}$ .
The field magnitude drops to zero in 2.96 $\mathrm{ms}$ . How much thermal
energy is produced in the loop by the change in field?

Averell Hause

Carnegie Mellon University

Problem 33

Figure $30-54$ shows a rod of
length $L=10.0 \mathrm{cm}$ that is forced to
move at constant speed $v=5.00 \mathrm{m} / \mathrm{s}$
along horizontal rails. The rod, rails,
and connecting strip at the right
form a conducting loop. The rod has
resistance $0.400 \Omega ;$ the rest of the
loop has negligible resistance. A current $i=100 \mathrm{A}$ through the long
straight wire at distance $a=10.0 \mathrm{mm}$
from the loop sets up a (nonuniform)
magnetic field through the loop. Find
the (a) emf and (b) current induced
in the loop. (c) At what rate is thermal energy generated in the
rod? (d) What is the magnitude of the force that must be applied to
the rod to make it move at constant speed? (e) At what rate does
this force do work on the rod?

Katie Mcalpine

Numerade Educator

Problem 34

In Fig. $30-55,$ a long rectangular conducting loop, of width $L$
resistance $R,$ and mass $m,$ is hung in a horizontal, uniform magnetic
field $\vec{B}$ that is directed into the page
and that exists only above line aa.
The loop is then dropped; during its
fall, it accelerates until it reaches a
certain terminal speed $v_{r}$ . Ignoring
air drag, find an expression for $v_{t}$

Averell Hause

Carnegie Mellon University

Problem 35

The conducting rod shown in
Fig. $30-52$ has length $L$ and is being
pulled along horizontal, frictionless
conducting rails at a constant velocity $\vec{v} .$ The rails are connected at one
end with a metal strip. A uniform
magnetic field $\vec{B},$ directed out of
the page, fills the region in which the rod moves. Assume that $L=$
$10 \mathrm{cm}, v=5.0 \mathrm{m} / \mathrm{s}$ and $B=1.2 \mathrm{T}$ . What are the (a) magnitude and
(b) direction (up or dow the page) of the emf induced in the rod?
What are the (c) size and (d) direction of the current in the conducting loop? Assume that the resistance of the rod is 0.40$\Omega$ and
that the resistance of the rails and metal strip is negligibly small.
(e) At what rate is thermal energy being generated in the rod?
(f) What external force on the rod is needed to maintain $\vec{v} ?(\mathrm{g})$ At
what rate does this force do work on the rod?

Katie Mcalpine

Numerade Educator

Problem 36

Figure $30-56$ shows two circular regions $R_{1}$ and $R_{2}$ with radii $r_{1}=$
20.0 $\mathrm{cm}$ and $r_{2}=30.0 \mathrm{cm} .$ In $R_{1}$
there is a uniform magnetic field of
magnitude $B_{1}=50.0 \mathrm{mT}$ directed
into the page, and in $R_{2}$ there is a a
uniform magnetic field of magnitude $B_{2}=75.0 \mathrm{mT}$ directed out of
the page (ignore fringing). Both
fields are decreasing at the rate of
8.50 $\mathrm{mT} / \mathrm{s}$ . Calculate $\ \vec{E} \cdot d \vec{s}$ for
(a) path $1,$ (b) path $2,$ and (c) path 3 .

Averell Hause

Carnegie Mellon University

Problem 37

SSM ILW A long solenoid has a diameter of 12.0 $\mathrm{cm} .$ When a
current $i$ exists in its windings, a uniform magnetic field of magnitude $B=30.0 \mathrm{mT}$ is produced in its interior. By decreasing $i,$ the
field is caused to decrease at the rate of 6.50 $\mathrm{mT} / \mathrm{s}$ . Calculate the
magnitude of the induced electric ficld (a) 2.20 $\mathrm{cm}$ and (b) 8.20 $\mathrm{cm}$
from the axis of the solenoid.

Katie Mcalpine

Numerade Educator

Problem 38

A circular region in an $x y$ plane is penetrated by a uniform
magnetic field in the positive direction of the $z$ axis. The field's magnitude $B$ (in teslas) increases with time $t($ in seconds ) according to $B=$
at, where $a$ is a constant. The magnitude $E$ of the electric field set up by
that increase in the magnetic field is
given by Fig. $30-57$ versus radial distance $r ;$ the vertical axis scale is set by $E_{s}=300 \mu \mathrm{N} / \mathrm{C},$ and the
horizontal axis scale is set by $r_{j}=4.00 \mathrm{cm} .$ Find $a .$

Averell Hause

Carnegie Mellon University

Problem 39

The magnetic field of a cylindrical magnet that has a
pole-face diameter of 3.3 $\mathrm{cm}$ can be varied sinusoidally between
29.6 $\mathrm{T}$ and 30.0 $\mathrm{T}$ at a frequency of 15 $\mathrm{Hz}$ . (The current in a wire
wrapped around a permanent magnet is varied to give this variation in the net field.) At a radial distance of $1.6 \mathrm{cm},$ what is the
amplitude of the electric field induced by the variation?

Katie Mcalpine

Numerade Educator

Problem 40

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

Averell Hause

Carnegie Mellon University

Problem 41

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 magnetic flux 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 42

Figure $30-58$ shows a copper strip of
width $W=16.0 \mathrm{cm}$ that has been bent to form
a shape that consists of a tube of radius.
$R=1.8 \mathrm{cm}$ plus two parallel flat extensions.
Current $i=35 \mathrm{mA}$ is distributed uniformly across the width so that the tube is effectively
a one-turn solenoid. Assume that the magnetic
field outside the tube is negligible and the
field inside the tube is uniform. What are (a)
the magnetic field magnitude inside the tube
and (b) the inductance of the tube (excluding
the flat extensions)?

Averell Hause

Carnegie Mellon University

Problem 43

Two identical long wires of radius
$a=1.53 \mathrm{mm}$ are parallel and carry identical
currents in opposite directions. Their center-to-center separation is
$d=14.2 \mathrm{cm} .$ Neglect the flux within the wires but consider the flux
in the region between the wires. What is the inductance per unit
length of the wires?

Katie Mcalpine

Numerade Educator

Problem 44

A 12 $\mathrm{H}$ inductor carries a current of 2.0 $\mathrm{A}$ At what rate must
the current be changed to produce a 60 $\mathrm{V}$ emf in the inductor?

Averell Hause

Carnegie Mellon University

Problem 45

At a given instant the current
and self-induced emf in an inductor
are directed as indicated in Fig. $30-59$ .
(a) Is the current increasing or decreasing? (b) The induced emf is
17 V, and the rate of change of the current is $25 \mathrm{kA} / \mathrm{s} ;$ find the
inductance.

Katie Mcalpine

Numerade Educator

Problem 46

The current $i$ through a 4.6 $\mathrm{H}$ inductor varies with time $t$ as shown
by the graph of Fig. $30-60$ , where the
vertical axis scale is set by $i_{s}=8.0 \mathrm{A}$
and the horizontal axis scale is set by
$t_{s}=6.0 \mathrm{ms}$ . The inductor has a resistance of 12$\Omega .$ Find the magnitude of
the induced emf $\mathscr{E}$ during time intervals $(a) 0$ to $2 \mathrm{ms},(\mathrm{b}) 2 \mathrm{ms}$ to $5 \mathrm{ms},$ and
(c) 5 $\mathrm{ms}$ to 6 $\mathrm{ms}$ . (Ignore the behavior
at the ends of the intervals.)

Averell Hause

Carnegie Mellon University

Problem 47

Inductors in series. Two inductors $L_{1}$ and $L_{2}$ are connected in
series and are separated by a large distance so that the magnetic
field of one cannot affect the other. (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) What is the generalization of
(a) for $N$ inductors in series?

Katie Mcalpine

Numerade Educator

Problem 48

Inductors in parallel. Two inductors $L_{1}$ and $L_{2}$ are connected
in parallel and separated by a large distance so that the magnetic
field of one cannot affect the other. (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) What is the generalization of (a) for $N$ inductors in parallel?

Averell Hause

Carnegie Mellon University

Problem 49

The inductor arrangement of Fig. $30-61,$ with $L_{1}=30.0 \mathrm{mH}, L_{2}=$
$50.0 \mathrm{mH}, L_{3}=20.0 \mathrm{mH},$ and $L_{4}=$
$15.0 \mathrm{mH}, \quad \mathrm{is}$ to be connected to a a
varying current source. What is the
equivalent inductance of the
arrangement? (First see Problems
47 and $48 . )$

Katie Mcalpine

Numerade Educator

Problem 50

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

Averell Hause

Carnegie Mellon University

Problem 51

ILW 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.

Katie Mcalpine

Numerade Educator

Problem 52

The switch in Fig. $30-15$ is closed on $a$ at time $t=0 .$ What is
the ratio $\mathscr{E}_{L} /\mathscr{E}$ of the inductor's self-induced emf to the battery's
emf (a) just after $t=0$ and $(b)$ at $t=2.00 \tau_{L} ?(\mathrm{c})$ At what multiple
of $\tau_{L}$ will $\mathscr{E}_{L} /\mathscr{E}=0.500 ?$

Averell Hause

Carnegie Mellon University

Problem 53

SSM 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
connected across the pair, 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 $t=1.0 \tau_{L} ?$

Katie Mcalpine

Numerade Educator

Problem 54

In Fig. $30-62,\mathscr{E}$=100 \mathrm{V}, $R_{1}=$ $10.0 \Omega, R_{2}=20.0 \Omega, R_{3}=30.0 \Omega,$ and
$L=2.00 \mathrm{H} .$ Immediately after switch
S is closed, what are (a) $i_{1}$ and (b) $i_{2} ?$
(Let currents in the indicated
directions have positive values and
currents in the opposite directions
have negative values.) A long time
later, what are $(\mathrm{c}) i_{1}$ and $(\mathrm{d}) \dot{i}_{2}$ ? The
switch is then reopened. Just then, what are ( e ) $i_{1}$ and $(\mathrm{f}) i_{2} ?$ A long time later, what are $(\mathrm{g}) i_{1}$ and $(\mathrm{h}) i_{2} ?$

Averell Hause

Carnegie Mellon University

Problem 55

SSM A battery is connected to a series $R L$ circuit at time
$t=0 .$ At what multiple of $\tau_{L}$ will the current be 0.100$\%$ less than
its equilibrium value?

Katie Mcalpine

Numerade Educator

Problem 56

In Fig. $30-63$ , the inductor has 25 turns and the ideal battery
has an emf of 16 $\mathrm{V}$ . Figure $30-64$ gives the magnetic flux $\Phi$ through
each turn versus the current $i$ through the inductor. The vertical axis scale is set by $\Phi_{s}=4.0 \times 10^{-4} \mathrm{T} \cdot \mathrm{m}^{2},$ and the horizontal axis
scale is set by $i_{s}=2.00 \mathrm{A}$ . If switch $\mathrm{S}$ is closed at time $t=0,$ at what rate $di/dt$ will the current be changing at $t=1.5 \tau_{L} ?$

Averell Hause

Carnegie Mellon University

Problem 57

In Fig. $30-65, R=15 \Omega$
$L=5.0 \mathrm{H},$ the ideal battery has
$\mathscr{E}$=10 \mathrm${V},$ and the fuse in the upper
branch is an ideal 3.0 A fuse. It has
zero resistance as long as the current through it remains less than
3.0 A. If the current reaches 3.0 $\mathrm{A}$
the fuse "blows" and thereafter has
infinite resistance. Switch S is closed
at time $t=0 .$ (a) When does the fuse blow? (Hint: Equation $30-41$ .
does not apply. Rethink Eq. $30-39 .$ ) (b) Sketch a graph of the current i through the inductor as a function of time. Mark the time at
which the fuse blows.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 58

Suppose the emf of the battery in the circuit shown in
Fig. $30-16$ varies with time $t$ so that the current is given by $i(t)=$
$3.0+5.0 t$ , 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 a
function of $t .($Hint$:$ Apply the loop rule.)

Averell Hause

Carnegie Mellon University

Problem 59

SSM WWW In Fig. $30-66$
after switch $\mathrm{S}$ is closed at time $t=0$
the emf of the source is automatically adjusted to maintain a constant
current $i$ through $\mathrm{S}$ . (a) Find the current through the inductor as a function of time. (b) At what time is the
current through the resistor equal to
the current through the inductor?

Katie Mcalpine

Numerade Educator

Problem 60

A wooden toroidal 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.

Averell Hause

Carnegie Mellon University

Problem 61

SSM A coil is connected in series with a 10.0 $\mathrm{k} \Omega$ resistor. An
ideal 50.0 $\mathrm{V}$ battery is applied across the two devices, and the current reaches a value of 2.00 $\mathrm{m} \mathrm{A}$ after 5.00 $\mathrm{ms}$ (a) Find the inductance of the coil. (b) How much energy is stored in the coil at this
same moment?

Katie Mcalpine

Numerade Educator

Problem 62

A coil with an inductance of 2.0 $\mathrm{H}$ and a resistance of 10$\Omega$ is
suddenly connected to an ideal battery with $\mathscr{E}=100 \mathrm{V}$ . At 0.10 $\mathrm{s}$
after the connection is made, what is the rate at which (a) energy is
being stored in the magnetic field, (b) thermal energy is appearing
in the resistance, and (c) energy is being delivered by the battery?

Averell Hause

Carnegie Mellon University

Problem 63

ILW At $t=0,$ a battery is connected to a series arrangement
of a resistor and an inductor. If the inductive time constant is 37.0 .
ms, at what time is the rate at which energy is dissipated in the resistor equal to the rate at which energy is stored in the inductor's magnetic field?

Katie Mcalpine

Numerade Educator

Problem 64

At $t=0,$ a battery is connected to a series arrangement of a
resistor and an inductor. At what multiple of the inductive time
constant will the energy stored in the inductor's magnetic field be
0.500 its steady-state value?

Averell Hause

Carnegie Mellon University

Problem 65

For the circuit of Fig. $30-16,$ assume that $\mathscr{E}=10.0 \mathrm{V}, R=$
$6.70 \Omega,$ and $L=5.50 \mathrm{H}$ . The ideal battery is connected at time $t=0$
(a) How much energy is delivered by the battery during the first
2.00 s? (b) How much of this energy is stored in the magnetic field
of the inductor? (c) How much of this energy is dissipated in the
resistor?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 66

A circular loop of wire 50 $\mathrm{mm}$ in radius carries a current of
100 A. Find the (a) magnetic field strength and (b) energy density
at the center of the loop.

Averell Hause

Carnegie Mellon University

Problem 67

SSM A solenoid that is 85.0 $\mathrm{cm}$ long has a cross-sectional
area of 17.0 $\mathrm{cm}^{2} .$ There are 950 turns of wire carrying a current of
6.60 A. (a) Calculate the energy density of the magnetic field inside the solenoid. (b) Find the total energy stored in the magnetic
field there (neglect end effects).

Katie Mcalpine

Numerade Educator

Problem 68

A toroidal inductor with an inductance of 90.0 $\mathrm{mH}$ encloses
a volume of 0.020 $\mathrm{m}^{3} .$ If the average energy density in the toroid is
$70.0 \mathrm{J} / \mathrm{m}^{3},$ what is the current through the inductor?

Averell Hause

Carnegie Mellon University

Problem 69

ILW What must be the magnitude of a uniform electric field if
it is to have the same energy density as that possessed by a 0.50 $\mathrm{T}$
magnetic field?

Katie Mcalpine

Numerade Educator

Problem 70

Figure $30-67 a$ shows, in
cross section, two wires that are
straight, parallel, and very long.
The ratio $i_{1} / i_{2}$ of the current carried by wire 1 to that carried by
wire 2 is 1$/ 3 .$ Wire 1 is fixed in
place. Wire 2 can be moved along
the positive side of the $x$ axis so as
to change the magnetic energy
density $u_{B}$ set up by the two currents at the origin. Figure $30-67 b$
gives $u_{B}$ as a function of the position $x$ of wire $2 .$ The curve has an
asymptote of $u_{B}=1.96 \mathrm{nJ} / \mathrm{m}^{3}$ as
$x \rightarrow \infty,$ and the horizontal axis
scale is set by $x_{s}=60.0 \mathrm{cm} .$ What is
the value of $(\mathrm{a}) i_{1}$ and $(\mathrm{b}) i_{2} ?$

Averell Hause

Carnegie Mellon University

Problem 71

A length of copper wire carries a current of 10 A uniformly
distributed through its cross section. Calculate the energy density
of (a) the magnetic field and (b) the electric field at the surface of
the wire. The wire diameter is $2.5 \mathrm{mm},$ and its resistance per unit
length is 3.3$\Omega / \mathrm{km}$ .

Katie Mcalpine

Numerade Educator

Problem 72

Coil 1 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 fixed in place; their mutual inductance $M$ is 3.0 $\mathrm{mH}$ . $\mathrm{A} 6.0 \mathrm{mA}$ current in coil 1 is changing
at the rate of 4.0 $\mathrm{A} / \mathrm{s}$ (a) What magnetic flux $\Phi_{12}$ links coil $1,$ and
(b) what self-induced emf appears in that coil? (c) What magnetic
flux $\Phi_{21}$ links coil $2,$ and $(\mathrm{d})$ what mutually induced emf appears in
that coil?

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 73

SSM Two coils are at fixed locations. When coil I 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?

Katie Mcalpine

Numerade Educator

Problem 74

Two solenoids 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?

Averell Hause

Carnegie Mellon University

Problem 75

ILW A rectangular loop of $N$
closely packed turns is positioned
near a long straight wire as shown in
Fig. $30-68$ . What is the mutual inductance $M$ for the loop-wire combination if $N=100, a=1.0 \mathrm{cm}, b=$
$8.0 \mathrm{cm},$ and $l=30 \mathrm{cm} ?$

Katie Mcalpine

Numerade Educator

Problem 76

A coil $C$ of $N$ turns is placed
around a long solenoid S of radius
$R$ and $n$ turns per unit length, as in
Fig. $30-69$ . (a) Show that the mutual
inductance for the coil-solenoid
combination is given by $M=$
$\mu_{0} \pi R^{2} n N .$ (b) Explain why $M$ does
not depend on the shape, size, or possible lack of close packing of the coil.

Averell Hause

Carnegie Mellon University

Problem 77

SSM Two coils connected as shown in Fig. $30-70$ 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. $30-70$ be reconnected to yield an
equivalent inductance of
$$L_{\mathrm{cq}}=L_{1}+L_{2}-2 M ?$$
(This problem is an extension of Problem $47,$ but the requirement
that the coils be far apart has been removed.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 78

At time $t=0,$ a a 12.0 $\mathrm{V}$ potential difference is suddenly
applied to the leads of a coil of inductance 23.0 $\mathrm{mH}$ and a certain
resistance $R .$ At time $t=0.150 \mathrm{ms}$ , the current through the inductor is changing at the rate of 280 $\mathrm{A} / \mathrm{s}$ . Evaluate $R .$

Averell Hause

Carnegie Mellon University

Problem 79

SSM In Fig. $30-71,$ the battery is ideal and $\mathscr{E}=10 \mathrm{V}, \quad R_{1}=5.0 \quad \Omega$
$R_{2}=10 \Omega,$ and $L=5.0 \mathrm{H} .$ Switch $\mathrm{S}$ is closed at time $t=0 .$ Just
afterwards, what are (a) $i_{1},$ (b) $i_{2}$
(c) the current $i$ s through the switch,
(d) the potential difference $V_{2}$
across resistor $2,$ (e) the potential
difference $V_{L}$ across the inductor,
and $(\mathrm{f})$ the rate of change $d i_{2} / d t ? \mathrm{A}$
long time later, what are $(\mathrm{g}) i_{1},(\mathrm{h}) i_{2}$
$(\mathrm{i}) i_{\mathrm{S}},(\mathrm{j}) V_{2,}(\mathrm{k}) V_{L},$ and $(1) d i_{2} / d t ?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 80

In Fig. $30-63, R=4.0 \mathrm{k} \Omega, L=8.0 \mu \mathrm{H},$ and the ideal battery
has $\mathscr{g}=20 \mathrm{V} .$ How long after switch $\mathrm{S}$ is closed is the current
2.0 $\mathrm{mA}$ ?

Averell Hause

Carnegie Mellon University

Problem 81

SSM Figure $30-72 a$ shows a
rectangular conducting loop of
resistance $R=0.020 \quad \Omega, \quad$ height
$H=1.5 \mathrm{cm},$ and length $D=2.5$
$\mathrm{cm}$ being pulled at constant speed
$v=40 \mathrm{cm} / \mathrm{s}$ through two regions
of uniform magnetic field. Figure
$30-72 b$ gives the current $i$ induced
in the loop as a function of the
position $x$ of the right side of the
loop. The vertical axis scale is set
by $i_{s}=3.0 \mu$ A. For example, a current equal to $i_{s}$ is induced clockwise as the loop enters region 1 .
What are the (a) magnitude and (b) direction (into or out of the page) of the magnetic field in
region 1$?$ What are the $(\mathrm{c})$ magnitude and (d) direction of the magnetic field in region 2$?$

Katie Mcalpine

Numerade Educator

Problem 82

A uniform magnetic field $\vec{B}$ is perpendicular to the plane of a
circular wire loop of radius $r$ . The magnitude of the field varies
with time according to $B=B_{0} e^{-l / \tau},$ where $B_{0}$ and $\tau$ are constants.
Find an expression for the emf in the loop as a function of time.

Averell Hause

Carnegie Mellon University

Problem 83

Switch $S$ in Fig. $30-63$ is closed at time $t=0,$ initiating the
buildup of current in the 15.0 $\mathrm{mH}$ inductor and the 20.0$\Omega$ resistor.
At what time is the emf across the inductor equal to the potential
difference across the resistor?

Katie Mcalpine

Numerade Educator

Problem 84

Figure $30-73 a$ shows two concentric circular regions in
which uniform magnetic fields can
change. Region 1, with radius $r_{1}=$
$1.0 \mathrm{cm},$ has an outward magnetic
field $\vec{B}_{1}$ that is increasing in magnitude. Region $2,$ with radius $r_{2}=$
$2.0 \mathrm{cm},$ has an outward magnetic
field $\vec{B}_{2}$ that may also be changing.
Imagine that a conducting ring of
radius $R$ is centered on the two regions and then the emf $\mathscr{E}$ around
the ring is determined. Figure
$30-73 b$ gives emf $\mathscr{E}$ as a function of the square $R^{2}$ of the ring's radius,
to the outer edge of region $2 .$ The
vertical axis scale is set by$\mathscr{E}_{s}=$
20.0 nV. What are the rates
(a) $d B_{1} / d t$ and $(b) d B_{2} / d t ?$ (c) Is
the magnitude of $\vec{B}_{2}$ increasing,
decreasing, or remaining constant?

Averell Hause

Carnegie Mellon University

Problem 85

SSM Figure $30-74$ shows a uniform magnetic field $\vec{B}$ confined to
a cylindrical volume of radius $R$ .
The magnitude of $\vec{B}$ is decreasing at a constant rate of 10 $\mathrm{mT} / \mathrm{s}$ . In
unit-vector notation, what is the
initial acceleration of an electron
released at (a) point $a$ (radial distance $r=5.0 \mathrm{cm},(b)$ point $b(r=$
$0 ),$ and $(c)$ point $c(r=5.0 \mathrm{cm}) ?$

Katie Mcalpine

Numerade Educator

Problem 86

In Fig. $30-75 a$ , 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 $R_{1}=25.0$ $\Omega$
Similarly, in Fig. $30-75 b$ , switch $S$ 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 $R_{2}=30.0 \Omega$ . The ratio $\Phi_{\mathrm{oz}} / \Phi_{\mathrm{ol}}$ of the magnetic flux through a turn in inductor 2 to
that in inductor 1 is $1.50 .$ 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?

Averell Hause

Carnegie Mellon University

Problem 87

SSM A square wire loop 20 $\mathrm{cm}$ on a side, with resistance
$20 \mathrm{m} \Omega,$ has its plane normal to a uniform magnetic field of magnitude $B=2.0 \mathrm{T}$ . If you pull two opposite sides of the loop away
from each other, the other two sides automatically draw toward
each other, reducing the area enclosed by the loop. If the area is reduced to zero in time $\Delta t=0.20$ s, what are (a) the average emf and
(b) the average current induced in the loop during $\Delta t$ ?

Katie Mcalpine

Numerade Educator

Problem 88

A coil with 150 turns has a magnetic flux of 50.0 $\mathrm{nT} \cdot \mathrm{m}^{2}$
through each turn when the current is 2.00 $\mathrm{mA}$ . (a) What is the inductance of the coil? What are the (b) inductance and (c) flux
through each turn when the current is increased to 4.00 $\mathrm{mA}$ ? (d)
What is the maximum emf 8 across the coil when the current
through it is given by $i=(3.00 \mathrm{mA}) \cos (377 t),$ with $t$ in seconds?

Averell Hause

Carnegie Mellon University

Problem 89

A coil with an inductance of 2.0 $\mathrm{H}$ and a resistance of 10$\Omega$ is
suddenly connected to an ideal battery with $\mathscr{E}=100 \mathrm{V}$ (a) What is
the equilibrium current? (b) How much energy is stored in the
magnetic field when this current exists in the coil?

Katie Mcalpine

Numerade Educator

Problem 90

How long would it take, following the removal of the battery,
for the potential difference across the resistor in an $R L$ circuit
(with $L=2.00 \mathrm{H}, R=3.00 \Omega )$ to decay to 10.0$\%$ of its initial
value?

Averell Hause

Carnegie Mellon University

Problem 91

SSM In the circuit of Fig. $30-76$
$R_{1}=20 \mathrm{k} \Omega, R_{2}=20 \Omega, L=50 \mathrm{mH}$
and the ideal battery has $8=40 \mathrm{V}$
Switch $\mathrm{S}$ has been open for a long
time when it is closed at time $t=0$ .
Just after the switch is closed, what
are (a) the current $i_{\text { bat }}$ through the
battery and (b) the rate $d i_{\text { bat }} / d t ?$
At $t=3.0 \mu \mathrm{s},$ what are $(\mathrm{c}) i_{\mathrm{bat}}$ and
(d) $d i_{\text { bat }} / d t ?$ A long time later, what are $(\mathrm{e})$ $i_{\text { bat }}$ and $(\mathrm{f}) d i_{\mathrm{bat}} / d t ?$

Katie Mcalpine

Numerade Educator

Problem 92

The flux linkage through a certain coil of 0.75$\Omega$ resistance
would be 26 $\mathrm{mWb}$ if there were a current of 5.5 $\mathrm{A}$ in it. (a)
Calculate the inductance of the coil. (b) If a 6.0 $\mathrm{V}$ ideal battery
were suddenly connected across the coil, how long would it take
for the current to rise from 0 to 2.5 $\mathrm{A}$ ?

Averell Hause

Carnegie Mellon University

Problem 93

In Fig. $30-63,$ a 12.0 $\mathrm{V}$ ideal battery, a 20.0$\Omega$ resistor, and an
inductor are connected by a switch at time $t=0 .$ At what rate is the
battery transferring energy to the inductor's field at $t=1.61 \tau_{L} ?$

Katie Mcalpine

Numerade Educator

Problem 94

A long cylindrical solenoid with 100 turns/cm has a radius of
1.6 $\mathrm{cm} .$ Assume that the magnetic field it produces is parallel to its
axis and is uniform in its interior. (a) What is its inductance per
meter of length? (b) If the current changes at the rate of $13 \mathrm{A} / \mathrm{s},$
what emf is induced per meter?

Averell Hause

Carnegie Mellon University

Problem 95

In Fig. $30-77, R_{1}=8.0 \Omega, R_{2}=10 \Omega, L_{1}=0.30 \mathrm{H}, L_{2}=0.20 \mathrm{H}$
and the ideal battery has $\mathscr{E}=6.0 \mathrm{V}$ . (a) Just after switch $\mathrm{S}$ is closed, at what rate is the current in inductor 1 changing? (b) When the
circuit is in the steady state, what is the current in inductor 1$?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 96

A square loop of wire is held in a uniform 0.24 T magnetic
field directed perpendicular to the plane of the loop. The length of
each side of the square is decreasing at a constant rate of 5.0 $\mathrm{cm} / \mathrm{s}$ .
What emf is induced in the loop when the length is 12 $\mathrm{cm} ?$

Averell Hause

Carnegie Mellon University

Problem 97

At time $t=0,$ 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 98

The inductance of a closely wound coil is such that an emf of
3.00 $\mathrm{mV}$ is induced when the current changes at the rate of 5.00
A/s. A steady current of 8.00 A produces a magnetic flux of 40.0
$\mu \mathrm{Wb}$ through each turn. (a) Calculate the inductance of the coil.
(b) How many turns does the coil have?

Averell Hause

Carnegie Mellon University

Problem 99

The magnetic field in the interstellar space of our galaxy has a
magnitude of about $10^{-10} \mathrm{T}$ . How much energy is stored in this
field in a cube 10 light-years on edge? (For scale, note that the
nearest star is 4.3 light-years distant and the radius of the galaxy is
about $8 \times 10^{4}$ light-years.)

Katie Mcalpine

Numerade Educator

Problem 100

Figure $30-78$ shows a wire that has been bent into a circular
arc of radius $r=24.0 \mathrm{cm},$ centered at $O .$ A straight wire $O P$ can be
rotated about $O$ and makes sliding contact with the arc at $P$ .
Another straight wire $O Q$ completes the conducting loop. The
three wires have cross-sectional area 1.20 $\mathrm{mm}^{2}$ and resistivity
$1.70 \times 10^{-8} \Omega \cdot \mathrm{m},$ and the apparatus lies in a uniform magnetic
field of magnitude $B=0.150 \mathrm{T}$ directed out of the figure. Wire $O P$
begins from rest at angle $\theta=0$ and has constant angular acceleration of 12 rad/s? As functions of $\theta$ (in rad), find (a) the loop's
resistance and (b) the magnetic flux through the loop.(c) For what
$\theta$ is the induced current maximum and (d) what is that maximum?

Averell Hause

Carnegie Mellon University

Problem 101

A toroid has a 5.00 $\mathrm{cm}$ square cross section, an inside radius
of $15.0 \mathrm{cm}, 500$ turns of wire, and a current of 0.800 A. What is the
magnetic flux through the cross section?

Katie Mcalpine

Numerade Educator