Rz
10.0 ,f), and the ideal batFlc. 27-4q Pro
4.0 V. What are the (a) size and
computer?
tive? ssM
reaches
# the safe level of 100 V?
of the current through
resistance
o.44
tery
has emf,'8 : 720V.What is
In Fig. 27-54a,
If you
wrist
is connected
to of 78
wear a conducting(b)
strap that (up
T
direction
or down)
the Figure
27-70
shows
(c)
1, the
stzea porand (d) direction
72 An automobile gasoline
youin v l across
increase
as much
does
battery t has emf ground, your potential
the
point a if we close
switch
-47b isin
current
atwhich
S innot
Fig. 27
closed,
what is when
the1,
change
ideal
battery
tion
of
a
circuit
through
(.)
current
battery
the
size
$-I
: stand
(left
or
of the currentthrough the has emf 8as
nal resistance
resistor 1,in
or
does Vl
remain because
the same? the resisgaugerr is
more
rapidly
shown
schematically
also discharge
Zrl!/ up; you
there
is a(u)
current
I : switch
6.00 A. S,, (b) only
onlyright)
FlG. 27-21
battery
has emf Question
8z: I2.06.tance
(d) direction
and
the
of
current
in
grounding
is
much
less
through
connection
than
the
throughareresistance
(on the
Fig.
27-69.
The indicator
The resistances
R r : Rz : 2, and the (e)
and internal
resistance
12:Rl0.012
f). Zrl
:20
7V Initially,
resistor
is wired
to- o a battery.
Then
(c)
a single
Tank
switches
Sr
and
?nd
all
52,
is
set
by
R,.,
O.Th
(c)
your
you
when
through
and
shoes.
Suppose
that
body
(e)
(f)
2, and the
srze and
: 4.00
The batteries are connected in series with
2.00R1 :size
2.00R
What
(f) g1.
Ff
27-48
direction
of the cur+
G.and
resistor
R2 is added in parallel.
Are (a) the potential
difference
Problem3T.
has
of
dashboard)
a resistance battery
an external resistance R. (u) What R stand up, your potential is 1400 V and the chair-to-you capaci- unrt is
rc.What are (a) re
three
switches?
R-r
across R1 and (b) the current i1 through Rr now more than,
the
current
direction
of
the
it
through
current
in
battery
FtG.
27-37
value makes the terminal-to-terminal
rent through resistance
po- tance
3? (g)
C). The
a float
con- in that wrist-strap grounding
unit
pF, is
is equivalent
10
resistance
What
(.)tank
previously?
Is the
less than, or the same as 10
FlG. 27-70
Problem 78. RrR2ry
resistor 1?
Problem 21.
tential difference of one of the batteries
(g) What
3?discharge
is the
difFig.
80 isInthe
27 -72.,
the idealat
more than, less
than, orvariable
resistance Rp of R1 and R2nected
equal
V inpotential
you to
to 100
0.30 s,
connection
willtoallowresistor
potential
What
electric
to
a
zero? (b) Which battery is that? ssM
(")needV,,
R2 together more
79 In Fig.
Rr? (d) Is the total current through R1 and
27-77, Rt: 20.0 O,
ILW (b)
for, say,your
ference
is less time than you would
to reach
which
batteries
have emfs Zt:20.0
- Vt?
cooll
FlG. 27-71 Probl
4o,,r.Rz : 10.0
generates
point,4?
solar
a potential
difference
of 0.10
V
resistance
R1 previously?
whose
varies
linearly
less A
or cell
to the
current
through
than,
than,
equal
,f), and the S
ideal batFlG. 27-41 Problem 30.
#
when a 500 ,f) resistor is connected acrosscomputer?
it, and a potential
dif8z:
10.0 V'
8t:5.00
:
:
..38
V
-22,
V
27
all
the
resistors
have
a
8ference
Res-monster maze. In Fig.
tery
has
emf,'8
720V.What
is
-49, Z6|is io:1.40
Incurrent
Fig. in27resistance
6.00A V o.39
gasoline.
theis volume
with
of are
.o31The
of 0.15 V when a 1000
,f) resistor
:
substituted. What
In
Fig. 27-42, the
Ina Fig.
27-50,8
12.0Y,
gasoline
72 An
have anautomobile
resistance of 4.0 f) and all the (ideal) batteries
emf of
: Rz:
the: current
if 5.00
at point
we close
: 16.0
and
Et:
the resisthe (a) internal resistance and (b) emf of the solar cell? (c) The
and thethe
resistances
: are: RrV,
f,), R+
V and
RrRt:: 2.00
100
Ez
O,
RzC),
is
140
f) when
you can
R?is(If
4.0 V. What is the currentresistance
through resistor
: 2000
gauge
in and I2.0
shown
Rrtances
f),
Rz:
3000 O' and
area of the cell is 5.0 cm2, and the rate per
unit area
Rs : 8.00 C),
at
which itschematically
Ro
4.00 O. What is the emf of the ideal
(u)
(b)
only
switch
only
S,,
FlG" 27-5A Probl
proper loop through this maze, you can answer the
find
the
are
2.00
O.
each
What
receives energy from light istank
(on 200
2.0 mWcm2.
The indicator
What27-69.
is the efficiency
the
R: :ProblemT2.
300
is Fig.
empty
and
20 Obattery?
when
e f),
FlG.and
27-69
(c) all are the potenTank O. One
switchesR:
Sr :and
52, ?nd
question with a few seconds of mental calculation.)
4000
0. What
of the cell for converting light energy to dashboard)
thermal energy in
the a resistance of
(b)
(u)
are
the
size
and
direchas
unrtgrounded
point of the circuit is
three switches?
the tank is full. Find the current
1000 C) external resistor?
(a) V,q
tial
differences
V a, (b)
-RrR2ry
nRl 10 C). The tank unit is a float con(left or right)
tion
of current
i
(u)(c)
(a) -empty,
0). What
are theand
size
(b) half-full,
in the circuit
is(V
the
tankresistor
Fig.
80 In Vr-Vc,
27 -72., the ideal
sec. 27-7 Multiloop Circuits
nectedwhen
to
variable
a
(.)
(d)
Vc-Vr,
and
.23 In Fig. 27-38, Rr : Rz:
(.)
(d)
and
the
size
and
direcbatteries
have
Zt:20.0
emfs
(b)
FlG.
(up
4o,,r.
27-71
Problem
79.
and
direction
or
down)
full.Tieatwhose
ideal.
the resistance
battery as
varies
linearly
(u)
4.00 f) and Rr : 2.50 0. Find the
- V'Vc?
Flc.
27-4q
10.0
8t:5.00
Problem
38. can be
Vi2? (This
V 8z:VA
tion
of
curr ent
the volume of gasoline. The
equivalent resistance between Flc.with
of
the
current
through
resistance
27-38 ProblemZ3.
FlG.
Problem
27-42
31. consists of
game
and
Et:
5.00
and
the
resisA
arcade
73
controller
on
an
electronic
V
points D and E. (Hint: Imagine
: 100calresistance is 140 f) when the
In Fig.
FlG"
27-51,,
answered
with
mental
O'
only Rr
(c)
1,Fig.the
stze and (d) direction
'.40
that a battery is connected across those points.)
each 2.00 O. What
..32
Inacross
27-43,
the
ideal
is emptyconnected
tankresistor
and 20
O when
plates
the
of a 0.220 p.Ftances are
a variable
FlG.
27-69
ProblemT2.
:
:
(")
Rz:
R:
50.0
f,),
R+
culation.)
At
what
rate
is
75.0
O'
:
(b)
.24 When resistors 1 and 2 are connected
(u)
batteries
have
are
the
emfs
size
and
direcZt
10.0
(left to
right) of the current
in series,
or
the tank
is full.theFind the
current
(8 ".45 You are given
capacitor.
is
charged
5.00
capacitor
V and
%z:0.500%1,
&nd
the re-Y then dischargedtion (left
equivalent resistance is 16.0
0. When theyThe
are connected
in
or right)
ofbeing
current
i transferred
energy
and
the ideal
has emf in
battery
(c) the (e)
(a) empty,
(b)
inO.the
andand
circuit
when
thesistances
tank is
resistance
through
- of dissipating only
are each
4.00 0.
Whathalf-full, 2,
parallel, the equivalent resistance
(a)
is
3.0
What
are
the
for
potential
the
resistor.
The
time
the
difference
through
(.)
ble
(d)
and the6.00
size
and
direcfull.Tieat
as
ideal.
the
battery
is
the
current
(a)
in
resistan
(f)
2
ce
(a)
4'
V.
battery
and is the
is the
What
energy
equivalent
smaller resistance and (b) the larger resistance of these two
(f) direction of the cursize and
tion of curr ent i2? (This can be
and (b)to
resistance
3? V is measured by a clock
resistors?
plates
across the
to
decrease
0.800
is
numb
FrG. 27-51
arcade game consists of
73 A controller onoo33
an electronic
being
supplied
What
is iabsorbed
by the minimum
in (b) resisanswered
with only
mental
cal-or
(g) resistance?
through
3?
.25 Four 18.0 C) resistorsinside
Inrent
Fig. 27-44.,
the ideal resistance
are connected
in parallel
across
a the range
Problems
FlG.
27-43
32.
game.
can
If
of
discharge
times
that
the
combine
in
series
p.F
plates
0.220
resistor
the
of
a
a
variable
connected
across
FlG. 27-22 Question 8.
Problems 40or
an
batteries have emfs Et :5.0 V
(") battery?
culation.)
At
whatresistance
rate is
the
1, (c)
2, (d) resis25.0 V ideal battery. What is the
current through the battery?
43,potential
and 98.
the
electric
at tance
: What
capacitor.
The capacitor
isEzcharged
tois
5.00
then
discharged
Y
and
12 V, the
resistances
ps
ms,
should
handled
is
from
10.0
to
6.00
what
be
effectively
that
is
capable
of dissip
energy
being
transferred
in
.26
Figure
-39
27
shows
(e)
9 A resistor R1 is wired
to a five
battery, then resistor R2 is added
tance
3.,
and
resistance
4?
are
eachz.O
C),
potential
and
the
:
for the potential
through the(b)resistor. The time
Inis the
Fig.energy
27-73,, R
10 f).
C) resistors.
Findpotential
the equivaand (f)
Sthe of difference
R1 and value
the
in5.00
Are (a) the
across
(b)point,4?
series.
higher
value
the
resistancebattery 4'81
bedifference
the (a)
lower
and
is defined
be zero
plates
V is at
measured
by a clock
the
decrease
to to
0.800
In Fig.
lent resistance
points
between
difference
Thus,than,
the
ammeter.
the ratio
of theor
two
lessacross
themeter
..41
Rr
now more
than,
same asto ssM
current
i1 across
through
being supplied
absorbed
orInis
by
".46
What
the
resisgroundedo.39
equivalent
point of the
Fig.
27-52,
two
circuit.
batteries
of emf
V - 27-55,
I2.0 Va
:
:
:
In
Fig.
range
resistor?
27-50,8
12.0Y,
of
the
(u)
readings
(b)
is
not
R
F
but
only
resistance
and
H
anapparent
R'
If
R
and
V'li.
F
and G.
inside the game. If the
of discharge
previously? (c) Is the equivalent
Whatrange
are potentials
(a) V1 and times that can
parallel
resistors is c
the battery?
,f), what
(a) the
85.0
reading, (b) the voltmeter readare
ammeter
(Hint:
For
pair
each
points,
of
points
tance
A
between
and
B?
resistance Rp of R1 and R2 more
:the
ps
is V2
from
10.0
topoints?
6.00 ms,Rz:
what should O' and
be handled effectively(b)
atRr
the
indicated
2000
ing,
and (c)that
R'? (d)
If 28
Ra is is
decreased,
does the difference beimagine
a equal
battery
27-5A
con- (a)
Problem
39.
ratef),
energy is
74
that
at which3000
to a resistor a
Chapter
Fields In Fig. 27-4a, show
less than,
Rt?
than,
or
to
I Magnetic
81 In FlG"
Fig.
27-73,,
R : circuit
10 f). section
(Hint:
value of the resistance
besame?
the (a) lower valuessM
and (b) higher
This
mightseriesFlG.
tween
R' across
and R increase,
decrease. or remain the
Problems
:
nected
the pair.)
27-72
4000
Probl
R:
0.
What
are theRpotenWhat is the equivalent resis10
After
Fig.
27 the electron
switch: rngoes
oo34
ssM
r.
Figure
in
is
a
maximum
when
27-45
dissipated
R
thermal
energy
shows
as
a
battery.
All
the resisto
oo54
range
resistor?
of
the
(Fig.
T.
The
through
a
0.010
0.440
T
28-44).
What
are
In
Fig.
27-60.
Rr
2.00R.
.27
:
-andtance
In Fig.on27-40,
Rra,, there
100 C),
look battery
simpler
if you
first assume
point
15 is
closed
is
: I.40
.60
section
oftial
circuit.
(a)
The
has
(b)
emf VB?
RV,q
MC),
and
the idealpoints
Z - I2.0
A
anaRC
series
circuit,
differences
between
and
V a,
(b)resisV
half-circle
region
and
then
the (a)Inmagnitude
and
directhe
ammeter
resistance
is exits
zero,
Rz:
50 C), and
the
ideal
batteries
(b)R.Show
power
P
maximum
is
that
this
2.0R
'8214r.
same resistance. If
an
:(u)the
(a)
: at
is Find the(Hint: 20.0 V.that
current
resistance
rate
which
energy(b)
i through
In Fig.
27-4a,tances
show
arethat
FlG. 74
27-39
Problem26.
R1right)
2.0
C),
Rzthe points
switch is closed
4.0
p.F.
the
time constant.
Calculate
C
gap
(left
headed
toward
2 across
tion
region
or
of
the
current
and
the
battery
is ideal.
What
Aa and
B are
- 1.80
:6.0
This First,
circuit
section
might
have
emfs
8tgives
%z:5.0
V current
Va
(.)
FlG. 27-72
Problem 80.
Vr-Vc,
Vc-Vr,
FlG.
Problem
27-44
33. (d) long time so that the steady state is
:remove
Figure
27-23
that
f),
andenergy
R:
6.0
C),
will
appear
on the
capacitor
during
and
the
maximum
of ffiz:4.0
25.0
There
isthe
required
multiple
gives
an electric
tocharge
ofcm.
tension
8lR
cur- forpoten- dissipated in R as thermal
R
r.and
is that
a the
maximum
when
pa
sistor
were
added
in
and
(u)
V.
Find
the
cur- Wires A and B, having
look
you
simpler
if
first
assume
m
lengths
of
40.0
and
75
equal
:
connected
to
a
battery.)
four
values
R
:
sets
of
of
and
capaciis
(c)
Then
the
switch
opened
indicated
reached.
long
current
isPi does
How leads?
charging.
6.0 itA.take for the charge to build up
difference
in the supporting
renttial
ammeter?A,V 2000 V across (b)
VA
power
isVc?
Show that this maximum
2.0R
'8214r.
rentininthe
resistor 1, (b) the current in
- differ-connected
that atpoints
B are
(1) with
tance
Rs and
and
Co, (2) ZRo
in
Whatand
is the current
pC?
time / : A
The
potential
electric
16.0
to.40
parallel array, the curr
theC:gap,
a polarity
that the
.o55
Fig. 27-61,
is to such
be
Theare
bent wire
shown in Fig. in series. A po2.60
equal
diameters
of
ffiffi,
(c) theR,potential
resistorIn2,and
dif:
4.00Fig.
ms? €*
Rs and speed
2Co,()2Roand
resistorto
2 ata tbattery.)
Co, (3)
ZCo.
FlG. 28^44
ence
between
Problem
A
andisBclosed
electron's
increases
uniformly 75 Wires A and B,
m and
lengths
of
40.0
having
.61
In
27-74, the ideal
-63
FrG. 27-66bat27magnetic
in Fig.In
Switch
in value
moving
adjusted
by
28-45
lies inequal
: 10039.ofconnected
aSpoints
uniform
ference between
points
a and b.
Fig.
27-51,,the
Rr ends
O' r rS$82
the 4.0R
battery would
goes
Which
which
gap.itRegion
that
with
curve?
connect
FlG.
27-23
10. 60.0
as
itset
traverses
the
2
conthebegin
section
to
Question
:'.40
difference
Vffiffi,
applied
between
tential
/is
0'
charging
an
Problems 67 and 97
at
time
to
Figure -67 displays
the
sliding
contact
across
unFlG.
Problem
54. of
ininifield.
series. A poEach
straight
section
isthe
2.0
equal
diameters
of 2.60
are
connected
a 27-68
:30.0
.28
tery
has
In
Fig.
emf.'8
and the
27 -g,what
:
is the po:5
82
In
Fig.
ideal
27-74,
the
batfield
tains
a
uniform
magnetic
direst
:
of
the
circuit
is
Va
:
Va
Y
4.0R
-between
tially
uncharged
capacitor
of
til points 0 and b are brought
two circuits with a charged
Rz:
R:
50.0
f,),
R+
m long
and
makes
an angle
of
75.0
O'
Icapaci7.25%.
What is the valu
FrG.difference
28-43
ends
of
60.0
V
is
applied
the
tential
of
are
RA
0.I27
O
and
composite
wire.
The
resistances
the
tential
difference
Vawith
rected
out
the page,(One
of
:30.0
(u)
78
Is
the
device
repre- V,magniemf.'8
and
to
potential.
the same
15.0
resistor through
awire
tance
C - the
capacitor
that
is to beYdischarged
60'V.with
x pF
axis,
and the
(8tery has
resistances
Problem 32.
aretheRr : Rz : 14 f,),
points
between
andby
celectron
if. V, - goes the composite wire. The
are
RA
0.I27 O
andemf
resistances
tude
0.020
T.d The
the
has
"Box"
battery
sented
by
absorbing
oris
--+
tests
for this
condition
moR -are
20.0
At what
time
of
resistance
when
switch
:
:
throughare
a resistor
a 14
carries
a and
current
of
2.0 ideal
A.What
FlG./27-63
(a)
too{J
magnitude
of
the
Rn:0.729
f).
For
wire.4,
what
resistances
Rr
Rz
f,),
A copper wir
4.0through
1.0 V,a sensitive
Rr
Rz:leaves region
2. At whatf).
time
I
:In R+
: :6.0cr C), Re c2: 2.0
a half-circle
and: then
V %z:
providing
energyare
to the
circuit,
mentarily
connecting
(a)
Rs Rr
magnitude
/ of equivalent
the
Rn:0.729
For
wire.4,
what
is
across
capacitor
Fig. :
27-67a,
thethe
netpotential
magnetic
is closed.
force
onthe
the
wire
61 and76.
: R3
(a)
V.
6.00
What
isProblems
the
Rs : 6.0 C), Re : 2.0
Problem34.
10does
f), and
27-45
R: : €$
leave?
5.0
O, and
it between
(b)to atthat
and
what
rate?
(b) (b)
(c)what
ammeter
ifthe
0 and
b:current
potential
V?
Of whatR3 : R+
difference
density
and and
the resistor?
equal
across
p,F. In Fig.
in unit-vector
notation
if the mag-V? FlG.
20.0 'C) and Cr :5.00 :
minum jacket of outer
(c) Of
potential
difference
density
FlG.current
Problem2T.
27-4A
:
f),
Rt
and
1.5
O.
What
are
battery
is
ideal?
curf),
Rt
and
1.5
O.
What
are
points
pocurthese
at the same
FrG.
27-51
(a): 4.0k
netic field
is given
T
particular
bywith
O and
Aare
type of fundamental particle decays by
27-67b, Rz: 10.0
Cz:
(b) resisWhat
is wire
i indifference
..62
In
Fig.
27-46,
Rr
2.00
initial
potential
an
of
".33
Aresistance?
capacitor
(see
B.,
26-7)?
For
material
A
made
Table
(see
B.,
type
is wire
26-7)?
For
wire
A f,),..35
made
Table
type
is*. wire
contact
- material
tential.
the ammeter
dei : 2.00 A in the comp
.29
positron
aand
and
T?
intowill
annot
electron
and
transforming
pF.
The
the
8.00 ir,
:is 4.0i
Nine
copper
wires
of length
(b)
/ ;Slicling
(c) i of
diameter
d are
rents (a)
i3,initial
io, ratio
and
R,V(b)
5.00
C), and through
the battery
(a)
(b)
a resistor when
a switch between
100
discharged
rents
ir,(d)
i (d) i3, and
io, (c)
Problems
Problems
40
and
FlG. 27-67 Problem 68.
flect.)
(c)
whento
this
(d)
(f)
tance
1,
resistance
(d)
(e)
B
2,
material
is
resiswhat
type
what
are
/
and
V?
Of
irA.
particle
B
isform
in
magnetic
field
of
decaying
atadrest
a uniform
theShow
is 48.
the
two
capacitors
charges
on
connected
',
inthat
parallel
a
single
composite
conductor
:
:
of
.41
is
ideal.
What
(f)
value
of
R3
(d)
(e)
B
maxipotential
material
is
type
0. At t line
10.0 s, the
difference(e) r'?
power
isAclosed
at / what
horizontal
what
are
/
and
V?
Of
them
the current
',
in (a) the
justment
FlG.
27-73
Problem
81.
following
is
made,,
the
:
ssM
away from
the ..53
magnitude
3.53 mT
and
e- and e*Dmove
r'?At tim e t - 0, both
resistance
R.
be the
the
of a single
copper
eozleu(e)1.50.
mizes
Figure
28-50ofshows
the
dissipation
rate
in
(a) What
is and
1.00
is the time constant
carries
atance
current
capacitor
5000
Aa V.from
across
the
30oWhat
to themust
of diamet ermade?
angle g :holds:
FlG. 27-73 Prob
(e)
3.,
resistance
4? of
:direction
ssM
relation
point /inif R'.
plane perpendicular
paths
to B. How wood
lying
inthe
made?
have
switches are closed. At what time r do the two capacitorspotential
wire
of length
it is toR,R,/Rr.
have
FIG.difference
28-45 Problem
of Earth's
mass
cylinder
resistan
cenorth.
3? (b)
difference
V
south
to
magnetic
is
across 40.
What m
the potential
the
the
circuit?
magnetic
field
ofa same resistance? ssM
uniform
adecay
t4
(R,)
An
resistance
unknown
ideal
of
emf Z - lzY,
76 Figure 27-63 shows
long after the decay do the e- and e+ collide?
the same charge?
Lbattery
and
length
kg an
0.250
(60.0
-directed
field
pT)
is
tot :batteries
77.0
s?0.100
T. In unit-vector
at
magnitude
capacitor
oo30
..36
The 0.50
resistances
in of
Figs.27-4Ia
and b are all 6.0 C), and
..41
Both
inuncharged
Fig. 27-52, two batteries of emf V
k1
can
be measured
in terms
a
t4
Fig.
I2.0 and
:Nthe
V
and
inthe
current,
what
is the
..34
wire
ofIn
4.0
O,turns
an
capacitor
lzY,
10.0
a resistor
resistance
R
and
m, with
ideal
of
emf
Z
76
Figure
27-63
shows
an
battery
ooo$Q
x
ward
is the
torque
actA
source
injects
speed
v
L 5of
107
m/s
north
and
inclined
notation,
what
an
electron
of
downward
at
to
the
horizon70.0"
p,F
1.00
capacitor
conresistor
a
are
A
3.00
MO
..63
the
batteries
are
ideal
IzV
(a)
batteries.
When
switch
in
S
the plates of a leaky
difference
standard (R,) using this device,
27-47a The
are potential
ideal.
Emf
Zt of between
it magnitude
longitudiaround
coil
about
the hinge
ing
on-4Ia
(b) direction
into
field of
magnitude
Belectric
10-3CT. wrapped
tal.
Find
athe
uniform
magnetic
theother)
1.00
ideal battery of emf 8 - 4.00
) R.,
nected in series with an
what
isof
of
capacitance
4.0
switch
S is and
closed,
the
- 1.0 xpotenFig.
27 is
is
:pcF.After
closed,
what
is
change
in the
plate
(meaning
leaks
from
to
the
which
called
Wheatstone
that
amagnetic
battery
t the
has
acharge
fixed
Problems2.0
R7 V. At
value,
27-46
but oneFlG.
-R
resistor
O,m(a)
an
uncharged
capacitor
athe
of
resistance
and
plane
the
of
thedue
so4.0
that
ssM
ssM
line?
The across
force
velocity
an angle
the nally,
on 100
line
field.
of theL,electron
0 : 10o
of the
to Earth's
s after the connection is made, what is the rate at which (a) the
tralVl
resistor
or does makes
(b)with
Vlremain
bridge.
the 27-&1
same?
When
pF
to
one-fourth
initial
value in2.0
What
capacitor
drops
emf
charge
capacitor
%2
of.when
battery
35,99,106,
current
through
thewire
resistor
the
onitsthe
2
can
and I07.
be varied
F$G.
the
long
cencontains
coil
Find the distance C
d from 4.0
the
direction
of the magnetic
field.
+
(b) energy
o48
is being stored
charge of the capacitor is increasing,
.42
current
)
A
single-turn
pcF.After
is
what
is
of
capacitance
switch
S
closed,
the
A
wire
1.80
m
long
A
carries
a
current
of
13.0
and
plates?
resistance
is
between
the
capacitor
the
equivalent
o.56
Problems
55 and
R.,
R7
In of
Fig.injection
27-62, a voltmeter
- field tral.o64
axis of the cylinder. The
pC? the
8.0
point
crosses
at which
the electronisnext
in the capacitor, (c) thermal energy is appearing in the resistor,
of 4.00
loop,
carrying
a current
makes
with awith
an
of
35.0"
uniform
magnetic
field energy
of magC) and an
of resistance
Rv : 300
iswhen
released
oncapacitor
a plane
cylinder
A angle
1.0 pF
an initial
stored
charge
theforce
capacitor
injection point.
current
through
the resistor
the
on
through
the
a right trianand (d) energy is being delivered by the battery?
is inthat
the passes
shape of
A,line
nitude
Ban
1.50 T.
magnetic
the
on the wire.
The starting
77
motor
a-iscar
is0Calculate
turning
too
slowly,
and
of
the through
ammeter of resistance Ra : 3.00
to
inclined
at
angle
(u)
resistor.
a
1.0 MO
0.50
J
discharged
of
+
and
sides 50.0, 120,and
with
R,r'
sec.
28-7
..43 isAwith
Synchrotrons
Cyclotrons
plane
ofreplace
the
f)gle
horizontal,
are
being
used
to measureisa 8.0
Problem 47. has to
pC?
FlG.
28-47
wire
cm
carries
athe
A(b)
50.0
0.500
current
the
is the
Additional ProblemslRr
the
initial
charge
on
the 28-50
capacitor?
What
motor,
What
the mechanic
whether
tolong
decide
Problem
53. in
FlG.
l-30 cm. The loop is in a uniform
Chapter 27
1.
bl 'lal'l
27-19 Question2.
electric field in them,
Rz and then again
ction of the electric
)
I Circuits
I
I
potential difference
al to that across Rr?
more than, less than,
I
1
I
1
e,
nd 4.
R3 in series? (b) Are
12.0 O, R.,:4.00 O,
the
V.
Forequivalent
what valueresisof Ra
:
7-z\,greatest
first.
ansfers energy
to the
possible
mum
) R2,P,,,u*,
Rr, with
Rr rate
to
(d)
What
n,in?
are
P',n*
and then in parallel.
he amount of current
7.,
oltmeter
r: 100 0' have
Rr -3.0
theVcapacitors
meter
resistance
Rvofis
eries have
an emf
(If youinto
?troduce
can the
findmeathe
R'?
Ignore
across
the
answer the question
G.
rLw
0.
o
l-Qtt
\y\h
{
X
R
A-- /
.ffi)
27'56 Problem
50.
.
C).
?,
e
e
e of 1.00 mA. What exts in a deflection of (a)
ll scale? (d) If the ame internal resistance of
27
Supp___ose
k1
s.
l^,1
l^,1 l^,1
f^,1
lr-,
11 1.
I
L
FtG. 27-57
Problem 51.
.o35 REstimate
to the
coil parallel
thethat
total
resistance
in a circuit
alsopath length traveled by a deuteron the current
positive
direction
an x axis
through
magnetic
field Find
Eofincline
a
resistor
through
the
when
the discharge
starts?
12V
is parallel
manual
says
that
the
direction
cable,
car's
magnetic fleld of magnitude
75.0 mT whosethe
or the battery.The
field r,of(c)
magnimagnetic
vertical
uniform
If there
contains
resistance R6
100 O Problem 28-5 during the (entire) plane.
in thea cyclotron
of :Sample
(3.00
(10.0
mT)i is+athat
mT)t.
notation,
is the
unit-vector
what
gives,
asIn
of time
the potena function
magni- have an expression
What is theshould
to the current in the 130 cm side of the loop.battery
internal
resis0.020
more
than
that
i
through
the
coil
is
least
current
T,
what
the
tude
0.500
potential
andacceleration
an ideal battery
of emf
process.
V
Assume
that
the
accelerating
magnetic
forceVon
the wire?
the capacitor, (d) the potential differtial
difference
- (a) the 130 cm side, (b) the 50.0
6 across
tude of the magnetic force on
down
the
from
rolling
keeps
theVcylinder
12.0between
V. Resistance
R isis given
the dees
80 kV.by
resistance,
and
more
the motor
no
than
(e) plane?
themass
rate
at
across
the resistor,
andwire
ence
pIn
..44
:the
is the magnitude
of
cm side, and (c) the 120 cm side? (d) Whattance,
Fig. 0.200
28-46,
aC)
metal
of
nl which
24.1 thermal
mg can
R : 0.36
Vli, where
Vtheisloop?
the
voltin the
resistor.
<-o cable
is produced
energy
In aoncertain
in amore
movesno
circle than
of sec.
Moment
cyclotron a proton
the net force
Dipole
The
Magnetic
The
mechanic
turns
28-10
0.040
f,)
resistance.
parallel rails
slide
with
negligible
friction
on
two horizontal
z
meter reading and i is the curradiusFigure
magnetic
field
is
1.20 T. .54fo65
0.500 m.
Theshows
magnitude
ofring
the of
carries
a
current
radius
15.0
cm
loop
of
separated
by
distance
d
2.56
The
wire
cm.
track
lies
in
a
A
circular
Fig.
27-64.,
l.zkv,
In
the
circuit
of
Z
C
6.5 p.F,
..49
1.8
radius
a
cm
a
wire
28-48
V
and
motor
across
the
battery,3.0
g
on
and measures
- vertical
rent(a)inWhat
resistance R. However.
FrG" the
27-&2 angle : 30o to the direction of
is the kinetic of 2.60
is the oscillator
frequency? (b) What
: Rz:
plane
its
an / - 0,
normalCtocompletely
placed
sofield
that
themagnitude
It ismagnetic
uniform
mT.makes
of
56.3
At uncharged,
time
R: : magnetic
MC).
With
Rr
0.73
to ithe
is perpendicular
that
the
ammeter
reading is not
butgeneral direction
Problemof56.a radially
field
of field magnitude
aofA.uniform
proton,
in
magnetic
of
energy
of
the
electron-volts?
41.0"
uniform
with
a
angle
device
G
is
connected
to
(at
(a)
is
closed
r
0).
At
t:0,
what
are
suddenly
switch
S
diverging
symmetric,
rather
i', which
is i plusmagnetic
the cur- fleld. The magnetic field at the
magnitude 0.50 T. In unit-vectorthe 1,magnitude
I2.0current
T. (a)
Calculate
rails,
the
a(b) cur- of the magnetic dipole
..37
resistor
3.4two
mT, meter
and
its
i1 inproducing
of
the same
is everywhere
ring
A proton
inmagnitude
approxicirculates
a
cyclotron,
rent
through
the voltmeter.
Thus,
the
ratio B
of-beginning
the
is the torque
notation,
what (b)
is theactmagnitude of the torque
What
loop.
moment
constant
current
i : and
9.13
(c)
everywhere
direction
at
rent
i2ofinthe
it passes
rest
at the
Whenever
mately
readings
is at
notthe
R ring
resistance
R' : through
Vli'.If the acting
but
onlycenter.
an apparent
thecurhinge
ing
on resistor
the coil2,about
loop?
oninthe
:
mA
the
wire
and
rails
20"
with
a
an
angle
makes
0
in
resistor
At
t: @
rent
3?
^Rl
electric potential
between
dees,
h ssM
(a) the
R -gap
O,, what are
reading,(b)difference
the voltmeter
85.0between
the ammeter
line?
(even
the
plane
to the
normal
many
timeturns
is, as
aftercoil
con- has
160
A circular
ofwire
(a)
is 200
Byring.
much does
its kinetic
energy .55(that
the dees
+ a radius of 1.90 cm.
(d)of
(c)
increased,
reading,
and
R'? V.
Ifthe
Rv
ishow
does the
difference
o48
current
A
single-turn
moves).
At
t:6L1
ms,
lead
wires
have
no
The
twisted
in a magnetic dipole
(d) that
current
the are
what
ir, (e)results
ir,
Calculate
gap?
(b) What is its (a) stants),
increase
with
each passage
through
the the
between
R' and
R increase,
decrease,
or remain
same?
of 4.00
loop,
carrying
current
(u)
what
are
the
wire's
Find the 100 passes through the gap? Let moment
effect
on energy
the problem.
(b) Find the maximum
A'm2.
magnitude
2.30
(f)of
potential
What
isa the
and
t3?
as it completes
kinetic
sec.
27-g RC
a right
in
shape
of that
A, is and
of Circuits
the force the field
magnitude
(b)
speed
direction
thetriantorque
magnitude
ofthe
the
resistor
V)
across
2coil, carrying this current,
difference
r16s be the radius of the proton's circular path as it completes
Itrlc. ?8-46 Problem 44.
.57
gle
120,
the
ring if the
ring
car- aconstant
50.0,
exerts
on100
What
multiple
of
the
time
r gives
motion
of
right)?
mT
magnetic field. ssM
35.0
insides
a or
uniform
experience
(h)
(i)and
(g) with
t-q?
/:0 (left
and
passes
let rrctthe
betime
its next can at
those
and
enters
dee,
and
FlG. 28-47 Problem 47.
FlG. 28-48
49.
riesacurrenti
loop is/ inbetween
a uniform FlG. 27-64 Problem 65.
in anProblem
l-30 cm.V2
taken
by an
initially
uncharged capacitor
RC series
- 4.6mA.
ooo{$
versus
.56Sketch
radius,
as it enters
a dee the next time. (c) By what percentage
AThe
1.0
kg
copper
rod rests
horizontal rails
has magnitude
on two
moment
of Earth
The magnetic
dipole
ssM
parallel
circuit
to
its
final
be
charged
to
99.0"h
of
charge?
magnitude
mT whose direction
magnetic
fleld
of
75.0
..50
www
ssm
lies in
28-49,a
rectangular
carrying
extreme
times.
In Fig.
That 8.00these
frorlcurrent
r"roo to r1e1?
does the
radius
increase
when it loop
changes
x 1022
is produced
charges
1.0
mtwo
that athis
apart
and carries
current
of 50 Abyfrom
oneflowrailisto the
JlT.Assume
is the magniWhat
in
the
130 cmIfside
of
the loop.
to
the
current
.58
magnitude
0.040
T.
magnetic
of
plane
the
A capacitor
isof a uniform
with initial
chargefield
is
discharged
through
is, what
their
circular
the
radius
of
core.
in
Earth's
molten
outer
ing ..66
other. The
coefficient
of staticthe
friction between rod
and rails is
4o
27-65 shows
Figure
(b)
(a)
force
cm side,
the 50.0
magnetic
on andthe
of thecalculate
wire
of a single
flexible
loopWhat
consists
gives the time
produce.
a The
resistor.
multiple
of the turn
timeof
const
ant
rconducting
current
rrct
pathcircuit
istude
3500
(b)130
magnitude
0.60.
What
the (a)the
angle (relative to the
- \oo
flashing
like they
ofkm,aare
percentage
rco%?
(c) thelamp,
120 cm side? (d) What is the magnitude of
- one-third
cm side,ofand
is wrapped
that
(a)atheflexible
loseincrease
first
the
capacitor
takes toaround
of
its charge
vertical)
the
smallest
magnetic
field of
that
theinrod
ftoo
.57those
barrels
theon the
attached
tocarrying
current
5.0puts
A, is
a at
loop,
A current
the net
force on the loop?
theitsdimensions
mount
such that of
(b) two-thirds
and
charge? of the
verge
of
sliding?
30, 40, and 50 cm. The loop
sides
triangle
sites.
The
..38
of a right
highway
construction
shape
@with
with dee
radius
cyclotron
(The
total 53.0 cm is operated at an
rectangleAcan
be changed.
ring 80
radius
a - 1.8acm
wire along
of
28-48
shows
.59
whose
oooQ,$
field
of amagnitude
magnetic
A 15.0thekO
is in
a..49
uniform
resistor and a12.0
capacitor
are connected in senegligilamp
L (of
fluorescent
AFigure
long,
rigid
conductor,
anmT
x axis,
carries
lying
MHz
oscillator
frequency
wire is not of
changed.)
As to accelerate protons.
of
length
perpendicular
a radially
to theingeneral
that
50 direction
cm A
side
ofofthe
parallel
current
isisof
to
the
direction
ries,(a)
is is
suddenly
applied
andWhat
thenmagnitude
a 12.0 V potential
difference
is
connected
capacitance)
ble
current
5.0
A
in
negative
field
the
x the
direction.
magnetic
field
required
to
achieve
B
of
magnetic
edge length x is varied from approximagnetic
field
at the
magnetic
fleld.
The
diverging
symmetric,
(a)
moment
magnetic
dipole
of
the
Find
the
magnitude
loop.
potential
the
The
difference
across
capacitor
across
them. (b) At that field magnitude, what is the kinetic
in
across
giventhe
is present,
F parallel
bV capacitor
F : 3.0i + 8.0x2j, with x in meters and
resonance?
mately
zero to its maximum value of
magnitude B - 3.4 mT, and its
of
the
same
isand
ring
everywhere
(b)
loop.
on
the
loop
the
torque
of
the
(a) Calculate
Va
constant
of
the
risesenergy
to 5.00of
in proton
1.30 ps. emerging
the
time
RC
There
is
of
an
circuit.
a
C
F
in
milliteslas.
Find,
in
unit-vector FlG.
notation,
theProblem
forcc on66.
the
fromr the cyclotron? Suppose,,
approximately 4.0 cm, the magnitude
27-65
rLW
ring
everywhere
at the
(b) Find
of the
capacitor.
circuit.
the capacitance
lamp
through
2.0direction
segment
moment
of magnitude
ofthe
thewith
conductor
lies between
aonly
dipole that
1.57 changes.
T. (.)
What
x - 1.0 m
Ammagnetic
dipole
that
The oscillator frequency is .58current
the torque
on B
the- loop
ofinstead,
20"inwith
a it reaches
an angle
makes
0 :rest
FtG.that
28-49
potential
when
the
difference
thefield
breakdown
magnetic
of
aacross
uniform
is released
from
andx-3.0m.
J/T
frequency, 0.020
required value
to achieve
now? (d) At
is 4.80 x 10-8
maximum
of r resonance
plane
of the
Problem 50.
the
normal
voltage
then
the
capacitor
discharges
Vy;to
completely
through
to the magrotation
the
dipole due
mT.
The
of ring.
magnitude
52
anloop?
emerging proton?
whatWhat
is the iskinetic
energyinofthe
N.m.
the current
lead
wires
haveWhen
noLoop
The
twisted
sec.
28-g
Torque
on
a
Current
lamp
and
the
lamp
flashes
briefly.
For dipole
a lamp rotates
with breakthe
the
it
is
unimpeded.
netic
force
on
..51
(see Sample Probgalvanometer
The coil
of a certain
Wire
$ec. 28-8
Force
Magnetic
on a Current-Carrying
.47
problem.
Find
the
effect
on
the
Figure
of
28-47
rectangular
2O-turn
of wire,and
shows
a its
moment
iscoil
aligned
where
dipole
the
V,wired
V ideal
through
orientation
down
voltage
Vr:72.0
to a 95.0
battery
a full-by a
of 7 5.3L:62.0
O; its needle
lem
28-7)
o39
force
theresistance
field
A has
13.0agresistance
suspended
thebyits
cm is shows
magnitude
wire of length
cm.
Itenergy
A two
10of
cm
5.0
carriesis a0.80
current
of What
and
0.10
(a)
field,
kinetic
magnetic
withadimensions
the
p,F capacitor,
what
R
ismJ.
needed
0.150
for
current
of 1 .62 mA passes through the
scale
pairdeflection
leadsa in
a uniform magnetic field of magnitude
the
ring
if
the
ring
carof flexiblewhen
exerts
on
is
hinged
long
is
plane,
at
along
one
side.
It
mounted
in
the
xy
moment
and
the
the
dipole
is the
initial
between
per angle
flashes
second?
coil. (u) Determine the value of the auxiliary resistance
FlG. 28-48 Problem 49.
riesacurrenti
is the angle when the dipole is next
What
magnetic
field?
- 4.6mA.
..67
In Fig.(b)27-66,
Rr : 10.0 kC), Rz: 15.0 kC), C - 0.400 p.F,
required to convert the galvanometer to a voltmeter that
(momentarily)
rest?
..50 InatFig.
y
28-49,a rectangular loop carrying
current lies in
reads 1.00 V at full-scale deflection. (b) Should this resistance
.59 the
circularmagnetic field of magnitude 0.040 T.
Twoplane
concentric,
of a uniform
be connected in series or in parallel with the galvanometer?
:
radii
,,
20.0
cm
loops,
of
wire
wire
of
a
single
turn
of
flexible
conducting
The
loop
consists
(c) Determine the value of the auxiliary resistance required to
in
are located
and 12:
30.0
iscffi,
wrapped
around
a flexible
that
convert the galvanometer to an ammeter that reads 50.0 mA
77
lz
L
FtG. A7-s8
Problem 52.
Problem 53.
I
7A
What are the (a) size and (b) direction (up or down) of
in Fig.27-68,, where all resistances are 4.0 C) and all
The starting motor no
of a car is turning
0 too slowly, and current i are
(Hin r: This can be
ideal
have an emf
of 10 V? 82.
and 27-74
R,r'
FlG.
Problem
lRr
the mechanic has to decide whether to replace the motor, batteries
answered using only mental calculation.)
Problems
:
I2.0
V
In
Fig.27-75,
the
ideal
batteries
have
emfs
%1
83
the cable, or the battery.The car's manual says that the 12V
..53
4.00Figure
8z:
and
theshows
resistances
1I.4V
are each 4.00 ,0. What
a
28-50
V
battery should have no more than 0.020 0 internal resiswood cylinder of mass m FlG. 27-74 Problem 82.
0.250 kg and length L - 0.100
tance, the motor no more than 0.200 C) resistance, and the
m, with N - 10.0 turns of wire
it longitudi- the ideal batteries have emfs %1
wrapped
In Fig.27-75,
83 around
cable no more than 0.040 f,) resistance. The mechanic turns
nally, so that the plane of the
and
8z: the4.00
on the motor and measures 1I.4V across the battery,3.0 V
long V
cen- and the resistances are each 4.00
wire
coil contains
x__4
at full-scale deflection. (d) Should this resistance be connected
in series or in parallel?
.)52 An electron
moves in a circle of radius r:5.29 x
10-t1m with speed 2.I9 x 106 m/s. Treat the circular path as
current loop with a constant current equal to the ratio of the
electron's charge magnitude to the period of the motion.
If the circle lies in a uniform magnetic field of magnitude B 7.10 mT, what is the maximum possible magnitude of the
a
torque produced on the loop by the field?
plane;
a
each
an xymount
thatcarries
the dimensions
of the
such
7.00 A
clockwise
current
(The total
rectangle
can of
be changed.
(Fig. 28-5I).
Find
theismagwire
not changed.) As
of the
length (u)
net magnetic
nitudeedge
of the
length
x is varieddi-from approxipole moment
theitssystem.
of to
maximum value of
mately zero
(b) Repeat
for reversed
4.0current
cm, the magnitude r
approximately
f^,1
tral axis of the cylinder.
The
cylinder is released on a plane
inclined at an angle 0 to the
horizontal, with the plane of
FlG. 28-50 Problem 53.
the coil parallel to the incline
plane. If there is a vertical uniform magnetic field of magnitude 0.500 T, what is the least current i through the coil that
keeps the cylinder from rolling down the plane?
sec.
28-10 The Magnetic Dipole Moment
.54 A circular wire loop of radius
15.0 cm carries a current
FlG. 27-68
70. to its plane makes an
so thatProblem
the normal
of 2.60 A. It is placed
angle of 41.0" with a uniform magnetic field of magnitude
you
in
chair,
charge sep71I2.0
that,
while
are
sitting
Suppose
magnetic
dipole
(a)
magnitude
ofa the
T.
Calculate the
puts you
poaration
between
clothing
andisthe
(b) What
magnitude
of at
thea torque
thechair
loop.
moment
of theyour
you
200
capacitance
between
and
the
tential
of
with
the
V
acting on the loop?
chair at 150 pF. When you stand up, the increased separation
.55 A circular
has the
a radius
of 1.90tocm.
160 turns
coilthe
ofchair
decreases
capacitance
between your
body and
in body?
a magnetic
dipole
(a) Calculate
current
that results
the is
What then
of your
That po10 pF. (a)
the potential
(b)
Find
the
A'm2.
magnitude
2.30
moment
of
drains
tential is reduced over time, as the charge on you maximum
carrying this current,
torque
magnitude
of the
(youthe
your body
and
shoes that
are coil,
a capacitor discharging
through
ssM
field.
in a uniform
can experience
resistance
a resistance).
Assume 35.0
thatmT
themagnetic
along that
through
.56is 300
route
If you dipole
touch an
electrical
component
while
has magnitude
moment
of Earth
TheGO.
magnetic
8.00 x 1022 JlT.Assume that this is produced by charges flowing in Earth's molten outer core. If the radius of their circular
path is 3500 km, calculate the current they produce.
.57 A current
loop, carrying a current of 5.0 A, is in the
shape of a right triangle with sides 30, 40, and 50 cm. The loop
is in a uniform magnetic field of magnitude 80 mT whose
direction is parallel to the current in the 50 cm side of the
x__4
ssM
in the inner
torque on the loop changes. The
of theloop.
FtG. 28-49
.160maximum
gives
Figure 28-52
value
of the
r is 4.80 x 10-8
Problem 50.
potential
mag- in
of acurrent
is the
the28-51
loop? Problem 59.
N.m.energy
What U
FlG.
..51
The coil of a certain galvanometer (see Sample Problem 28-7) has a resistance of 7 5.3 O; its needle shows a fullscale deflection when a current of 1 .62 mA passes through the
coil. (u) Determine the value of the auxiliary resistance
required to convert the galvanometer to a voltmeter that
reads 1.00 V at full-scale deflection. (b) Should this resistance
be connected in series or in parallel with the galvanometer?
(c) Determine the value of the auxiliary resistance required to
convert the galvanometer to an ammeter that reads 50.0 mA
at full-scale deflection. (d) Should this resistance be connected
in series or in parallel?
loop. Find the magnitude of (a) the magnetic dipole moment
of the loop and (b) the torque on the loop.
.58 A magnetic dipole with a dipole moment of magnitude
0.020 J/T is released from rest in a uniform magnetic field of
magnitude 52 mT. The rotation of the dipole due to the magit is unimpeded. When the dipole rotates
through the orientation where its dipole moment is aligned
with the magnetic field, its kinetic energy is 0.80 mJ. (a) What
is the initial angle between the dipole moment and the
netic force on
magnetic field? (b) What is the angle when the dipole is next
(momentarily) at rest?
.59 Two
concentric, circular
wire loops, of radii ,, : 20.0 cm
and 12: 30.0 cffi, are located in
an xy plane; each carries a
clockwise current of 7.00 A
y
l^,1