Download Gemcra! Frimciples

Summary 903
The goatr of Chapter 29 has been to caiculate and use the efectric potential and ele€tri€ potentia! energy.
Gemcra! Frimciples
$cayaes e€
Vf
:
C@BnseqiieKiees ffi€
The ele*r;c Fot€lrtial, like the etectric field, is.created by
charges.
.
A charged particle has poieEtial ereigy
.
Two major tools for calculating Vare
U=qV
at a point wheie source charges have created an electric
.
The potential of
.
The principle of superposirion
a
p
ont cnuse V =
potenlial Y
,l!
The elecnic forie is a conseryative fotce, so the mechdnical
energy is conserved for a charged particle iII an electric potential
Kr+ 14= Ki+
iviuliipl-^ po;ni charges
Use superposition: V=V1+V2+V,+ -.
.
The potential energy of two
distance r is
Coriinuou: distributicn of cbarge
.
.
.
Kr
point charges separated by
Kqtqz I QB:
4t<o r
,.
Divide tle charge into pointlike A0.
Ul
-t +q1 r
Find the potential of e ch A.Q.
Find y by summing the potentials of all AO_
The summation usually becomes an integral. A critical step is
replacing A@ with an exprcssion involving a charge (ensity and
an-integratio,l_coordinate. Calculating yis usually easier than
calculating E because the potential is a scalar.
The zero point ofpoteDtial and potential energy is chosen to be
convenient. For poitt charges, we let U = Q when r --+ co.
The potential en€irgl in an electric field of an electric dipole with
dipole moment is
f
IJa61"
-
-pEcos?
- -fi.
E
ApplieatEores
Craphical representations of the porential
s?heie sa charge Q
Same as
v
ifr>R
a
o
point charge
Parallel-Flaie capa.itot
V : Es, where s is measured
fiom the negative plate. The
electric field inside is
Potentialgraph Equipotenfialsurfaces
Lv.
d
Contourmap
Un;ts
Elevatiolgraph
l.V 1 lrc
=
field: 1 V/m.= I N/C
Electric potential:
Electric
Ternrs auld Notatiom
electric potential energy, a/
escape
velocity
electric potential, Y
volL,
V
potential difference, AV
voltage, AY
equipotential surface
contour map
elevation graph
9O4
cHAprER
29
The Electric Potential
.;
{64d For home*ork assigned on MasteringPhysics, 8o to
T*. wwwmasteringphysics.com
integrate significant material from earlier
Problems labeled
chapters.
Problem ditliculty is labeled as i (straishtforward) to ill (challenging)
L a. Charge qr is distance
positive Point charge O Charge
q1l3 is distance 2r from I. What is tl\e ftl1o UtlU1of
theirpotential energies due to their interactions with 0?
b. Charge q' is distance d from the negative plate of a parallelblate capacitor. Charge q2: ql3ls distance 2d from the negative plate. What is the ratio UrlU2 of their potential energies?
2. I,t/,l?) is the potential energy of two opposite charges a negative
number? (Saying that the fo.mula gives a negative number is not
:
42
r from
a
6. Rank in order, from most positive to most negative. the potential
energies 4 lo Ur ofthe six electric dipoles in the uniform electric field of
FTGURE
e29.6. Explain.
E
an explanation.)
3.
FtcuRE e29-5 shows lhe potential energy
of
a
proton (4
and a lead nucleus (s = +82?). The horizontal
ls
offetntometers,where I fm : i0 m.
a.
b.
c.
u( rn
= +e)
scale is in units
FIGURE Q29.6
A proton is fired toward a lead nucleus from very far away
How much initial kinetic energy does the proton need to
reach a tuming point 10 fm from the nucleus? Explain.
How much kinetjc energy does the proton of part a have
when it is 20 fm from the nucleus and moving toward it,
before the collision?
How much kinetic energy does thc proton of part a have
when it is 20 fm from the nucleus and moving away from it,
after the collision?
7.
tric potentjal along ther-axis.
a. Dmw a graph ofthe poten-
tial energy of a 0.1
FrcuRE Q29.4
the trajectory of FIcURE Q?9.4 from
i
to
f.
Does the electric potential energy increase. decrease, or stay
the same? Explain.
b. Is the electron's speed at f greater than, less Lhan. or equal to
its speed at i? Explain.
5. Two protons are launched with the same speed from point 1
inside the parallel-plate capacitor of FICURE Q29.5. Points 2 and 3
are the same distance from the negative plate.
a. Is A{l-2, the change in potential energy along the
b.
--+ 2. larger than, smalle. than, or equal to
2
FtcuRE q29-7
Im
the charged particle is shot toward the right from "r
with 1.0 J of kinetic energy, where is its tuming point? Use
your graph to explain.
Will the charged particle ever reach ir 0 m? so. how
much kinetic energy will it have at that point? If not, why not?
axes
:
:
lf
If
0
change as the seParation
If
not. why not?
Does tbe electric field strength E change as the separation
increases? Ifso. by what factor? Ifnot. why not?
c. Does the potential difference AVc change as the separation
increases? Ifso, by what factor? If not, why not?
9. Rank in order, from largest to smallest, the electric potentials y,
to % at points a to e in FIGURE Q29.9. Explain.
increases?
so, by what factor?
b.
path
AUr-2?
Is the proton's speed r,: at point 2 larger than, smalleithan. or
equal to v]? Explain.
2
I
2
FiGURE
q29.5
a
All wires
a. Does the capacitor charge
a.
I
0l
and batteries are disconnected. then the two plates are pulled apart (with jnsulated handles) to a new separalion ofdistance 2/.
/(lm)
4. An elechon moves along
0
potential difference AV6.
40
FIGURE Q29,3
t0
8. A capacitor with plates separated by distance d is charged to
@;
20
C
in
b. If
c.
0
20
this
region of space. Provide a
numerical scale for both
charged particle
rJ)
0
v(v)
FtcuRE e29.7 shows the elec-
FICURE Q29.9
FIGURE Q29.i
O
Problems 905
Exercises and
q?9.10 on the previous page shows two points inside
a
capacitor Let y = 0 V at the negative plate.
a. What is the ratio y,/yr of the elect.ic potentials at these two
10.
FTGURE
points? Explair.
What is the ratio E /Er of the elect c field srrengths at rhese
tNo points? Explain.
11. Rank in order, from largest to smallest. the electric potentials yo
b.
to % at points a to e
in
GURE
e?g.i
t.
Explain.
tz
FtcuRE e29.r2 shows
two points near a positive point charge.
What is the rado y?/Vr of the electric potentials at these two
points? Explain.
b. What is the ratio E2lEr of the electric field strengths at these
two points? Explain.
l3 FTGURE e29.r3 shows three points i[ the vicinity of two point
charges. The charges have equal magnitudes. Rank in order.
ftom most positive to most negative, the potentials
% to y".
a.
e
(a)
,:.
a
ib)
b
FrGUR! q29.i3
{}l
FrcuRE Q29-i
:t
i
2
14 Reproduce FtcUR€
ez9.t4 on your paper. Then draw a dot (or
dots) on the figure to show the position (or positions) at which
the electdc potential is zero.
FrcrJRE Q29.i 2
@
\'!:
i
FtcuRE Q29.14
Exercises
Section 29.1 Electric Potential Energy
1.
L
3.0 nC
?.0 nC
o
2.0 nC
3.0 cm
@)
@
il The electric field srrengrh is 50,000 N/C inside a parallel_
plate capacitor with a 2.0 mm spacing. A proton is released from
3.0cm /
rest at the positive plate. What is the proton's speed when it
reaches the negative plate?
2. lt The electric field strengrh is 20,000 N/C insjde a parallel_
plate capacitor with a 1.0 mm spacing. An electrcn is released
from rest at the negative plate. What is the electron's speed when
it.eaches the positive plate?
3. il A.proton is released from rest at the positive plate of a
parallel-plate capacitor It crosses the capacitor and reaches the
4.
negative plate with a speed of 50.000 m/s. Whar will be the proton's final speed if the experiment is repeated with double the
amount of charge on each capacitor plate?
il A proton is released from resr at the positive plate of a prrallel_
plate capacitor. lt crosses the captcitor and reacher the negarive
f
.
8.
What is the electric potential energy of the group ofcharges in
I
A water molecule perpendicular to an electric field bas 1.0 X
J more potential energy than a water molecule aligned
with the field. The dipole moment of a water molecule is
6.2 x 10 30 C m. What is the strength ofthe electric field?
I
The graph shows the potential energy ofan electric dipole_
u
(d)
Consider a dipole that oscil-
What is the electrjc potential ene.gy of the electron jn FIc
aae
il
Flc!_iRE FXz9_7
10_21
Section 29.2 The Potential Energy ofpoint Charges
The protons
?.0 nC
Section 29.3 The Potential Energy ofa Dipole
9.
URE Exzg.g?
1.0 nC
FIGURE EX29.7?
negative plate?
il
-
FI6URE EX29.6
plate with a speed of 50,000 m/s. The experiment is repeated
with a He+ ion (charge e, mass 4 u). Wlat is the ion's speed at the
5.
a)
'C
1.0 nC
lates between
a. What
fixed and cannot move.
b.
:!60'.
is the
dipole's
mechanical energy?
What is rhe dipole's kineric
energy when it i$ aligned
with the electric field?
050 nd
-
r80.
0
130"
-2
F!6URS EX29.9
0.50 nm
6.
Section 29.4 The Electric Potential
FiGURE EX29.5
10.
ll
i What is the speed of an electron that has been accelerated
from rest through a potential difference of 1000 V?
11.
What is the speed of a p.oton that has been accelerated from
rest through a potential difference of - 1000 V?
Whatis the electric potential energy ofthe group ofcharges in
FICURE EX29.6?
I
906
12.
cHAprER
29
The Electric Potential
ll
What potential difference is needed to accelerate a He+ ion
(charge +e. mass 4 u) from rest to a speed of 2.0
I 06 m/s?
What potential difference is needed to accelerate an electron
106 m/s?
from rest to a speed of2.0
An electron'r!,ith an initia] speed of500,000 m/s is brought to
13.
X
24.
ll A 1.0-mm-diameter ball bearing has 2.0
trons. what is the ball beaing's potential?
25-
i In a semiclassical model of the hydrogen atom. the electron
orbits the prcton at a distance of0.053 nm.
a- What is the electric potential of the proton at the position of
x
x
14.
lOe excess elec-
the electron?
rest by an electric field.
a. Did the electron move into a region of higher potential or
lower potential?
b. Wtat was the potential difference that stopped the electron?
15. i A proton with an inilial speed of 800,000 m/s is brought to
rest by an electric field.
a. Did the proton move into a region of higher potential or
b.
What is the electron's potential energy?
Section 29.7 The Electric Potential of Many Charges
26.
i
What is the clectric potenlial at the point indicated with the
dot in F[cuRE Ex29.?6?
lower potential?
What was the potentiai difference thai stopped the proton?
16. r Whatis the ratioAVr/A% of the potential differenies thatwill
accelerate a proton and an electron from rest to (a) the same final
speed and (b) the same hnai kinetic energy?
b-
20nC
2.0 nC
{l$
20nC
o
30c
l.0cm
Section 29.5 The Electric Potential Inside a Parallel-Plate
1.0 cm
Capacitor
17.
18.
I
i
Show that
1
l V/m : I N/C.
l
a. What is the potential of an ordinary AA or AAA banery?
(Ifyou're not sure, find one and look at the label.)
b. An AA battery is connected to a parallel-plate capacitor
having 4.0-cm-diameter plates spaced 2.0 mm apafi. How
much charge does the battery supply to each plate?
19. i A 2.0cm X 2.0cm parallel-plate capacitor has a 2.0 mm
spacing. The electric field strength inside the capacitor is
10
x
105
@
l.0cm
-
FIEIJRE EXZ9.V6
F!GURE EX29.27
2'1.
f,,)
i.0 nC
?0nC
1.0 nC
What is the electric potential at the point indicated with the
dot in
FIGURE EX?9.27?
28. li The electric potential at the dot in FlctiRE Ex2e.28 is 3140 V
What is charge g?
V/m.
a. Wlat
is the potential difference across the capacitor?
b. How nuch charge is on each plate?
20. Two 2.00 cm x 2.00 cm plates that form a parallel-plate
capacitor are charged to :10.708 nC. What are the electric field
strength inside and the potential difference actoss lhe capacitor
if the spacing between the plates is (a) 1.00 mm and (b) 2.00 mm?
I
21.
I
Two 2.0-cin-diameter disks spaced 2.0 mm apan fo.m aparallel-
plate capacitor. The electric field between the disks is 5.0
105
a.
b.
c.
23
4.
v/m.
What is the voltage across the capacitor?
How much charge is on each disk?
F'6URE EX29.?8
An electron is launched from the negative plate. It st kes the
po\iti\e plate at a speed of 2.0 \ IOTm/s. what wa. the
29.
nC
is at jr
=
0 cm and a
-
1.0 nC charge is at
point or points on the ,r-axis is lhe electric
potential zero?
charge is on the r-axis at ir: -9cm and a
+4.0 nC charge is on the r-axis at r = 16 cm. At what point or
points on the)-axis is the electric potential zero?
31. il Two point charges qa and 4b
are located on ther-axis at,r : rl
30.
i
A +25 nC chatge is at the origin.
a. what are the radii ofthe 1000 V 2000 V. 3000 V. and 4000 V
equipotential surfaces?
Draw a contour map in the i)-plane showing the charge and
these four surfaces.
a. What is the electric potenlial at
pointsA. B- and C in FI6URE EX?9-?3?
b. What are the potential differ-
il A -3.0nC
t:
and
b. FlcuRE Ex?9.1t is a
graph of E", the jr-component of
the elect.ic
field.
a. What are the signs of 4, and
ences AVo6 and AVsq?
B
?0nC ,/
FIGUiIF EX29.2]
-5.0
ii A +3.0 nC charge
r = 4 cm. At what
Section 29.6 The Electric Potential of a Point Charge
b.
nC
x
electron's speed as it left the negative plate?
22.
-50
(21
c
411
b.
c.
What is the ratio q,/qb ?
Draw a graph ofY. theelectric
potentia[. as a fi]nction ofjt.
FI6URE EX?9,31
Exercises aad
Problecrp W,'
,r
Tli o Foinl ciiarges % and qb are iocated on the r-axis at x : a
ailC.r = i. ncuR! rtzs.l? is a graph of y, the electric potential.
a lvhat are rhe s;gns of q^aJtd q6?
b- lfirel ;s $e ratio lg,/gbl?
c, Dieq.a glaph offr. ther-compoDent ofthe electric field. as a
tinction
l0 nC
l0 nC
6.r
_li,
l
I
0
ofr
B
{}
l0nC
Ifl nC
Ft6URE P29-4]
t0 Y
30
0
FlGr_.,R8
F29.42
ij A proton's speed as it passes pointA is 50,000 m/s. It follo\r,s
lhe tEjectory sho$n in FTCURE p29.42. What is the proton's speed
at pojnt B?
43; ll An arrangement of source charges prcduces the elecdc
potential y : 5000-rl aiong the r-axis, where yis in Yolts ard.t
is in meters.
a. Graph the potential between r : - l0 cm and r = + l0 cm_
b. Describe the motion of a positively charged particle in this
potential.
c. What is the mechanical energy ofa 1.0 g, 10 nC charged particle if its tuming points ale at a8.0 cm?
d. What is the particle's maximum speed?
44. ii A proton moves along the x-axis, where an anangement ol
source charges has created the electric potential V = 6000-r:.
where V is in volrs and,:r is in meters
a. Graph the potential between r : -5.0 cm and -r : -5-0 crlb. Describe the motion of the proron.
c. By exploiting the analogy Nith the porsnfal e*e+i-of,a Eie.t
on a spring- determine the "efiecii\'g sFrieg crnr$#t*- {.',ifu
elect ic porentiai.
d. \\hal is the protoB s o<ciiiatior 5ta.:e x^r ,: .,:
45. ri In Ft6uF€ p?s-e:, a prot+a is *se$ t?l
.
$l4':a
42.
FIGURE EX29.32
FiGURE eX29,33
33.
The two halves of the rod in FTGURE Ex?9.1j are uniformly
charged to :10. What is the electric potential at the point indicated by the dot?
Problems
34.
35.
36.
li
-
Two point charges 2.0 cm apart have an electric potential energy
180 /,J. The total charge is 30 nC- Wlar are rhe two charges?
i Two positive point charges are 5.0 cm apart. If the elect c
potential energy is'72 pI,'r,that is the magnitude of the force
between the two charges?
i, A -2.0 nC charge and a + 2.0 nC charge are located on the raxis at -r
1.0 cm and "{ = + 1.0 cm, respectively.
a. At *hat posilion or positions on the i-axis is the electric field
:
zf'fo.'
b. Ai $har posirion
{.,
3l
L',
or positions on the J.-axis is the elect c
Fotential zeio?
Dre1" ,sraphs of the electric
iole&tial along the.r,ax j s.
.q
-
i$.S hC painr charse and a +20_0 nC poinr charge are
irl] epa!? 9a ?h€ i,aiis.
B. &}r;t ;s t&e ekefic porential
tr
field strength and the electric
*irh a rp-ed rti i{F.Olfi cnls**d,*€:.
iriri;,r';l; +r {te i".;u;rt-,: t.,r{f g;sr'
5"*
ei.{xrir fi*id is zerc?
h- l&*nr is *l*, sljl8:r};tsde oi
at the point on the
i-a)iis $.he.e
gSE{.;1*tit6.,
*;
the
.{-eRi-s" li}$nsa{:}
3..,.F.i
rl
fieid at &e p*irlr e,r
a4}
*1,*
rxtr is
if-
,'*'"6r" aaii. Si.
are
?l:f!s.s$*{a.&:
zi{*r$: bati,}?} '*nax@t '{tu'iiz"
The teaC, arr y,cts+- tiw :r-+ "\=r-u,," :5-tive beads. in cnrr's. 11htli rh,e sre }q4$. fi!6
,{':€C . }f "-..
of 15 g and a charge"<p*t:
ai -].ft aC- Ee&d i}
has a mass of 25 g and a charge of - 10.0 nC. The beads are heid
12 cm apaft (measured between their centen) and released.
39, r
** fl n*.. {i\4ri s:
eha;ges, rthere the elecr.i. pfiEr*i.:i? i$
i,*F iq-"**r"ix}{h{h}rg{drs +5.*
dis-.grl$
$Aee€'&€4.$#.
*:.gdtit*.lk{
?he
7*t{!?
T\e}>
l-* e l*ad
.
*e
rhe elecrric
Bead A has a mass
What maximum speed is achieved by each bead?
Hint: There are n{,o conserved quantities. Make use ofboth.
40. li Two small metal spheres with masses 2.0 g and 4.0 g are tied
together by a 5.0-cmlong massless string and are at rest on a
frictionless surface. Each is charged to +2.0 /r,C.
a. What is the energy ofthis system?
b. What is the tension in the string?
c. The string is cut. What is the speed of each sphere when they
ate far apan?
Hint: There are tuo conserved quantities. Make use ofboth.
41. The lour 1.0 g spheres shown in F-tcURE p29-4t rre released
simultaneously and allowed to move away from each other.
What is the speed ofeach sphere wherl they are very far apart?
i
5e*.4{rr
,
nr 'Y3'l ei,e{er!!t
ai'4rr'
;!ii.t,-l*
si$il!
l,*irgr r"3-c5
.
,4$.
I
rr.
i
* T1I Ficftlre tsi}e &e!.alerete3 el€${s1s
&rf{r.l*is 1R{! Ferutle; Flates i-: c.}r ap*r.: q.itli a f5 tt F{*Biial
diiferl:rce
SD.ll r!:,
betrq,een lhear. The elecurras e,r:tE &$ai€h a srila.1l
hole in the negative plate, accelerate, then exit through a smali
hole in the posjtive plate. Assume that the holes are small enough
nof to affect the elect.ic field orpotentiai.
a. What is the electric field strength bet$'een the plates?
b. With what speed does an electron exit the electlon gun ifits
entry speed is close to zero?
$oIE tr The exit speed is so fast that u,e really need to use the
theory ofrelativity to compute an accurate value. Your answer to
pa.t b is in the right range bur a lirde roo big. .d
47. ll A room with 3.O-m-high ceilings has a metal plate on the floor
with V : 0 V and a separate metal plate on the ceiling_ A LO g
glass ball charged to +4.9 nC is shot srraight up at 5.0 m./s. How
high does the ball go if the ceiling volrage is (a) +3_0 X 106 V
and (b) 3.0 x i06V?
90E
48.
il
cHAprER
2q
The Eleclric Potential
The hydrogen molecular ion
Hi.
with one electron and two
protons, is the simplest molecule. The equilibrium spacing
between the protons is 0.11 nm. Suppose the electron is at the
midpoint between the p.otons and moving at 1.5 x 106 m/s perpendicularto alinebet\teen theprotons. How far (in nm) does the
elecfon move before reaching a turning point? Because ofthei.
largermass, the protons remain fixed during this interval of time.
t,igrF s An accurate descriplion of Hl requires quantum
mechanics. Even so. a classical calculation like this provides
some insight jnto the molecule. {
i
What is the escape speed of an electron launched from the
surface of a 1.0-cm-diameter glass sphere that has been charged
to 10 nC?
50. Your lab assignment is to use positive charge p to launch a
prolon- starting fiom reil. so that it acquires lhe ma-{inium possible speed. You can launch the p.oton from the surface of a sphere
ofpositive charge 0 and radius R, or from the center ofa ring of
charge
and radius R, or from the center of a disk of charge O
and radius R. Wlicb will you choose?
51. ii Anelectric dipole consists of 1.0 g spheres charged to a2.0 nC
at the ends of a 10-cn-1ong massless rod. The dipole rotates on a
frictionless pivot at its center. The dipole is held perpendicular to
a uniform electric field with field strength 1000 v/m, then
released. What is the dipole's angular velocily at the instant it is
49.
i
I
aligned with the electric field?
52. ill Three electrons form an equilateral triangle 1.0 nm on each
side. A proton is at the center of the triangle. What is the potential energy ofthis group of charges?
53. lil A2.0-mm-diameter glass bead is positively charged. The potential difference between a point 2.0 mm from the bead and apoint
4.0 mm from the bead is 500 V What is the charge on the
54. A proton is fired from far away toward the nucleu s of an iron
atom. Iron is element number 26. and the diameter of the nucleus
is 9.0 fm. what initial speed does the proton need to just reach
the surface of the nucleus? Assume the nucleus remains at rest.
55. li A proton ii fired from far away toward the nucleus of a mercury atom. Mercury is element numbe. 80, and the diameter of
bead?
i
the nucleus is 14.0 fm. If the proton is fired at a speed of
4.0 X 10r m/s. what is its closest approach to the surface of the
56.
nucleus?Assume the nucleus remains at rest.
li In the form of radioactive decay known as alplu clecay, an
unstable nucleus emits a helium-atom nucleus, which is called
a\ alplv particle. An alpha particle contains two protons and
two neutrons, thus having mass z? : 4 u ati.d chatge q:2e.
Suppose a uranium nucleus with 92 protons decays into thodum,
Bith 90 protons. and an alpha particle. The alpha particle is initially at rest at the surface of the thorium nucleus. which is 15 fm
in diameter. what is the speed of the alpha particle when it is
detected in the laboratory? Assume the thorium nucleus remains
b.
c.
Why is the final product a helium atom? Explain.
The nuclei of both 3H and 3He have radii of 1.5 x l0-rsm.
With what minimum speed must the electron be ejected if it
is to escape from the nucleus and not fall back?
58. i The sun is powered by /rsior, with four protons fusing
together to form a helium nucleus (two of tbe protons tum into
neutrons) and, in the process. releasing a large amount of thermal energy. The process happens in several steps, not all at once.
In one step, two protons fuse together, with one proton then
becoming a neuton, to form the "heavy hydrogen" isotope
deuterium (2H). Aproton is essentially a 2.4-fm-diameter sphere
of charge, and fusion occurs only if two Protons come into contact with each other This requires extaordinarily high temPe.atures due to the strong repulsion between the protons. Recall that
lhe average kineric energy ola gas panicle ir'itul
a. Suppose two protons, each with exactly the average kinetic
energy, have a head-on collision. What is the minimum tem. perature for fusion to occur?
b. Your answer to part a is much hotter than the 15 million K in
the core of the sun. lf the temperarure were as high as you
calculated, every proton in the sun would fuse almost
instantly and the sun would explode. For the sun lo lasl for
billions ofyears, fusion can occur oDly in collisions between
two protons with kinetic energies much higher than average
Only a very tiny fraction ofthe protons have enough kinetic
energy to fuse when they collide, but that fraction is enough
to keep the sun going. Suppose two protons with the same
kinetic energy collide head-on and just barely manage to
fuse. By what factor does each proton's energy exceed the
average kinetic energy at l5 million K?
59 ll Two 1O-cm-diameter electrodes 0.50 cm apart form a parallelplate capacitor. The elecfodes are attached by metal wires to the
terminals of a 15 V batte.y. After a long time, the capacitor is
disconnected from the battery but is not discharged. What are the
charge on each electrode. the elecfic field strength inside the
capacitor, and the poterltial difference between the electodes
a. Right after the battery is disconnected?
b. After insulating handles 6re used to pull the electrodes away
from each other until they are 1.0 cm apart?
c. After the original electrodes (not the modified electrodes of
part b) are expanded until they are 20 cm in diameter?
60 il Two 10-cm diameter electrodes 0.50 cm apafi form a parallelplate capacitor. The electrodes are attached by metal wires to the
terminals of a 15 V battery. What are the charge on each electode, the electric field strength inside the capacitor. and the
potential difference between the electrodes
a. While the capacitor
b.
is attached to the battery?
After insulating handles are used to pull the electrodes away
from each other until they are 1.0 cm apart? The electodes
remain conn€cted to the battery during this process.
c.
After the original elechodes ([ot the modified electodes of
a
neu_
decd),
occurs
when
of
nu
clear
.adiation,
beta
One form
part b) are expanded until they are 20 cm in diameter while
tron changes into a proton, an electron, and a neutral particle
remairling connected to the battery?
called a neut rino: n' p 'e - v where z is the symbol lor a
a.
Find an algebmic exp.ession for the elect c field strength
61
il
neutrino. When this change happens to a neutron within the
E0 at the surface of a charged sphere in terms of the
nucleus of an atom. the proton remains behind in the nucleus
sphere's potentiai Yo and radius R.
while the electron and neutdno are ejected from the nucleus- The
b. What is the electric field strength at the surface of a l o-cmejected electron is called a beta particle. One nucleus that
diametermarble charged to 500V?
exhibit' beta decay is fie isolope ofhydrogen lH. called lrilirm.
whose nucleus consists of ooe proton (making it hydrogen) and 62 li Two spherical drops of mercury each have a charge of0.10 nC
and a potential of 300 V at the sudace. The two drops merge to
two neutrons (giving tritium an atomic mass ru : 3 u). Tritium
rH
lHe
form a single drop. What is the potential at the surface of the
--+
+ e + z.
is radioactive, and it decays to helium:
new droP?
process?
Explain.
a. Is charge conserved in the beta decay
at rcst.
57 . il
Exercises and
63.
li A Van de Graaff generator is a device fot generating a large
electric potential by building up charge on a hollow metal
FlcuRE pz9.7l shows a
0
of 30 cm.
for the electric potential at the center
c.
The generator is charged by placing charge on the inrd? surface oflhe metal sphere. What happens to the charge after it
is placed there? Explain.
How much charge is needed on the sphere for its potential to
be 500.000 V?
What is the electric field strength just ir? side and jrst outside
the surface ofthe sphere when ir is charged ro 500,000 V?
ll
A thin spherical shell of radius R has roral charge O. What is
the elect.ic potential at the center ofthe shell?
65. r FICURE P29.65 shows two
l00cm
uniformly chaiged spheres.
ence between points a and b?
Which point is at the higher
potential?
Hint: The potential at any
Cb seo
FI6URE P29-7]
72.
a
25 nC
r01m
60 cm
Fl6uRg P?9-6s
In Problems 73 through 75 you are given the equation(s) used to solve
a problem. For each of these.
a. Write a realistic problem for which this is the co ect
equatjon(s).
b. Finirh lhe sohrrion of lhe problem.
the potentials due to all charges.
An electric dipole with dipole momentp is oriented along the
(9.0
73
,y-axis.
a.
Find an expression for the electric potential on the r.--axis at a
point where y is much la.ger than the charge spacing ,r. Write
your expression in rerms of the dipole momentp.
b. The dipole moment of a water molecule is 6.2 X 10-30Cm.
What is the electric potential 1.0 nm from a water molecule
atong the axis ofthe dipole?
67. I Two positive point charges q are located on the ,--axis at
.\
=
lir.
14
Find an expression for the poterltia] along the r-axis.
b. Draw a graph of V versus r for - co < .r < 6. For compadson, use a dolted line to show the potential of a point charge
24 located at the origin.
68. i The arrangementof charges shown in F'CURE p?9.68 is called a
linear electric quar)ntpole. The positive charges are located at
) = :!-r. Noticc that the net charge is zero. Find an expression
forthe elect.ic potential on the)-axis aldistances l >> .r.
x
10e
Chrge o
(9.0
x l0,Nmr/cr)(2.0 x
(9.0
x
o:
10 ec)(1.60
x t0 rrc)
0.0010 m
'75
10,
Nmlcr)(3.O
x
10
x
10 ,
e
c)
0.030 m
(9.0
x
10e
Nm,/c,)(3.0
(0.030 m)
c)
+d
+
:
1200
V
Chatrlenge Problems
76. A L0 nC charge is at the origin. A -3.0 nC charge is on the raxis at r : 4.0 cm. Find al1 the points in the r)-plane at which
the polenlial is zero. Give your ans\,\er a5 a conlour map !how-
:
0 V equiporential line.
?7. A proton and an alpha particle (4 = +2e, m : 4 u) are fired
directly toward each other from far away, each with an initial
speed of 0.010c. Whai is their distance of closest approach, as
measured between their centers?
1,!
L
q
FI€URE P29.6A
ll
:90 X 10 6J
q + .1.:40nC
j1t.el x to ,7kg)(2.5 x to6m/s)r +
,?
i(1.67 x lo kg),,i +
ing the Y
PoinL on biseclinc linc
Nm,/c2)qr4l
0.030 m
a.
-
corect
b@
point is the supe.position of
il
ii A disk with a hole has inner radius Rin and outer radius Rou,.
the disk is uniformly charged with total cha.ge p. Find an
expression for the on-axis electaic potential at distance i from
the center of the disk. Verify that your expression has the
behavior when R,n --+ 0.
loo n
What is the potential differ-
69.
thin rod with
thar ha. been benr inro a semi-
circle of radiu\ R. Find an e\pression
b-
66.
il
charge
sphere. A typical classroom-demonstration model has a diameter
a.
64.
71.
Problems 909
78. The 2.0-mm-diameter spheres in FIGURE cp29.?B are released from
rest. Whal aro theil speeds rr. and r," when they are very far apart?
1.0 e
FIGURE F29-6S
2.0 nC
.Ae..
2.0c
I0
2.0 nC
1.0 nC
oflengrh L and charge O. Find
-c
an expression for the electric potential a distance _x away fiom
FiGURE CP29.7S
the center of the rod on the axis ofthe rod.
70. ll FrcuRE pz9-69 showed a thin rod of length L and charge O.
Find an expression for the electric potentiai a distance i away '19 The 2.0-mm-diameter spheres in
FtcuRE F2s.6e shows a thin rod
from the center of rod on the line that bisects the rod.
-l0nC
d
)D
l0 mm
FlcuR! cF29-?9
FTGURE cp29.79 are reieased ftom
rest. What are their speeds rc and rb ar rhe instant they collide?
9f0
cHAprER 29
.
The Electric Potential
80. An electric dipole has dipole momentp. If r >> s. where.r is the
separation between the charges, show that the elect c potential
of the dipole can be wdtten
I Fcoso
,4.,r" ,.
81
where / is the distance from the center of the dipole and d is the
angle from the dipole axis.
. Electrodes of area A are spaced distance d apart to form a parallelplate capacitor The elecfodes are charged to lq.
a. What is the infinitesimal increase in elect c potential energy
dUif an infinitesimal amount of charge /4 is moved from the
negative electrode to the positive electrode?
b. An uncharged capacitor can be charged to :!0 by transferring charge l4 over and over and over Use your answer to
part a to show that the potential energy ofa capacitor charged
to :!Ois 4"e
=
tSAVc-
82. A sphere of radius R has charge q.
a. What is the infinitesimal increase in electric potential energy
dU if an infinitesimal amount of charge 14 is brought from
infinity to the surface of the sphere?
b. An uncharged sphere can acquire total charge O by the transfer of charge lq over and over and over. Use your answer to
Stop to Think 29.1: Zero. The motion is always perpendicular to the
elecuic force
Stop to Think 29.2: U,
= Ua) a^ = U". The
potential energy
part a to find an expression for the potentiai energy of a
sphere ofradius R with total charge O.
c. Your arswer to pan b is the amount of energy needed to
assemble a charged sphere. It is often calledthe self-energy
ofthe sphere. What is the self-energy ofa proton, assuming it
to be a charged sphere with a diameter of 1.0 X 10 15 m?
83. The wire in FtcuRE cp?g.ai has linea. charge density l. What is
the electric potential at the center of the semicircle?
2R
FI6URE CP?9.83
84. A circular disk of radius R and total charge g has the charge distributed with surface charge density 4 : cr. where c is a consmnt. Find an expression for the electric potential at distance i
oD the axis of the disk. Your expression should include R and O.
bul not c.
85. A hollow cylindrical shell of length a and radius R has charge 0
uniformly distributed along its length. What is the electric potential at the center ofthe cylinder?
Stop to Think 29.4: V, =
Vt) V"> yd = 4.
The potential
decreases steadjly frgm the positive to the negative plate.
only on the distance from the positive plate.
It depends
depends inversely on ,: The effects of doubling the charge and doubling the distance cancel each other.
Stop to Think 29.5: AV"" = AV6" > AV"6. The potentiai depends
only on the distance from the charge, not the direction. AV$ : 0
Stop to Think 29-3: c. The proton gains speed by losing potential
energy. It loses potential energy by moving in the direction of
because lhese points
decreaiing elecnic porenrial.
aae
at the same distance.