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Unformatted text preview: 41 PROBLELI (l) (17' points} A solid noneonducting spherical shell, of inner radius R and Wu
outer radius 35'. carries a uniformlyr distributed positive charge. /
The volume charge density of the shell is p. lClive your ensurers . , . . ., interrnsofso.R,p.msndq. [s16 pts.) ﬁnd the magnitude of the electric field at point P
which is at a distance 2!? from the center of the shell [point 0). Gmee' leeN": al1": pu n spherical Seattle'st Wf‘fﬂu of rallies Y": 19.: If
*11(t“5—P\3 t... t
(menum 3 ’i _+ as" W
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subsidisedv.19 'r ': ZR.
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{bl{7 pts.) Find the electric potential difference {Vp— Val between points P and G. r P t r u.
org =  IE. I; e _ fessmo'=Jeas=;j£ulr=;fctr;, are” ﬁr 4'5"“
o '5 G I... I I
Easement,“ hunt. em: If an =%,(r%) ﬁr egress. “gr!
I
"til" “alertstern}
g}. «lesseeease; [c)(5 pts.) A particle of mass tn and positive charge or is projected from a large distance toward the
shell with an initial speed v directed radially inward. What value of v is required for the particle to
just reach the shell before reversing direction? a a . I I ‘ I _ I L l
Closacrum“ {mmml we” .ﬂLdKL— Lill + qiMII omega3: {3: gm BaajL Hills = in [H941]; I “maﬁa .5
t 1 t “I weak"; :2; t! __a U. 5:. R = P. {51.1
EM“. 1' unit 3% FT it t to... '5' m Phys I06 Final Eran Page 2!? Mrs} 3!. 2m} 4 2
PROBLEM {2) {15 points) Two metoi spheres in the ﬁgure are separated by a distance that is much greater than their radii.
They are to be connected by a conducting wire of total resistance R. The sphere of radius a is
uncharged and the sphere of radius 2:: has a total charge Q on it. Give your answers in terms of Q, R, a, and k= NM.
3 2 Q
[a)[4 pts.) Find the current through the wire immediater after the switch 3 is closed. (h) Find the electrostatic potential energy of the twosphere system: (i]{5 pts.) Before 5' is closed.
u=u1+sf°'=..;:.cu‘=%__i when Cetiﬂiaﬂxllﬂﬁlet]: saga
' 1. t.
Lil. I: a = i9. = k 5‘!
mm _’ ” mm W (ii)(6 pts.) After Sis closed and siecrrnsratic equilibrium is reached. Compare (i) and (ii) and
discuss. (Jﬁrrt: Treat the spheres as capacitors.) Wm “ha. emmgﬁﬁc aﬁuﬂhwbuet Pia.0144, his arm MM‘M. 51M Wﬁﬂl»
.ﬂ. 6'" ...—'p 3:; = 51 (aged atoms, ﬁmldmgu.mp¢d~'vtly.) L Gwaeﬁecenﬁﬂrﬂd: q.stl+st _..;. (95331 _._, %.qqi.sg'=¥
13 I 1* 1 Q:—
Mﬁmlwzu:ﬂt_+§h l. 3 ae,’ 1;,
fat 1 w “3" 5 mi #39:.
= faminer 1%: W* “‘"ﬁ “‘“V '72 a II b at. = 6,2.
u" "surm k '6‘: I; we compare (£1 Ms fit),we set. int uyu'. ‘l'lww due/cu} a nurset5
WM E’sLU = gal:— L‘g; :Eﬁ?‘ ThiLNI‘ESI'“q W to lostm Luci“
Wm via M mmﬁnﬂ “Jim 1+5. f'EWMLL Phys 1455 FindEmm Friars3i?"r Mayifjﬂﬂf 4 3
PROBLEM {3) (19 points) Consider the currentcarrying loop shown in the ﬁgure. The
curved segments are use of circles of radii R and ER with the
common center 0. The straight segments are along radii. Give
your answers in terms of 1!, yo, i, R. and 3.. (given in part b}. sinso°=ﬁaanacosec°=ua (aNS pts.) Find the magnetic ﬁeld (in unit vector notation} at
point 0. E=E+K+E§+ﬂfa=ﬁ+ga I
difltheuiercfs citsubar ML: Br: 43%
* éﬂiq—t) = east) El " I 1 '
tints.“ This it a.“ at In“ a e In”. A
I I A = —— —— = —II—'
E  NHEEL (4.3;): it"; ['k}+lzﬂt )—) El 2"”. k
3 ' Lina IZR (h) Suppose the loop is subjected to an external magnetic ﬁeld §=Bn{2?+?) where ﬁn is a
positive constant. (iKS pts.) Find the forces (in unit vector notation) acting on the Wm the loop.
.1; F15: t:th —) 513M 2. : §=ikfce5£n'€+hhio'j)n BuiltH.)
E
a. l out [Jr (at) —— (jot +1591
—I. 3‘ i  ‘21 +8
5 LIL(viii: oust.
Steam Li '. E; = {ﬁt—'3]; ﬁoila‘i'a)”. ——+ EH: {5.35.3
(ii}(5 pts.) What torque (in unit vector notation} acts on the loop?
—s _.. 'iI ,  t l ' l‘. A ' L A
_ .._ ._ l1 a mi  Tit—R gﬂ ..
“CH/Lao —i/u Med ii}{1j[_‘Ti_l_TJT—i/a at J
*_ TIL L 4' a 3,113+li 1 NULL“ ﬂats: (iiiJIEfl pts.) How much work is required to turn the loop from the position shown in the ﬁgure to the
position in which the magnetic dipole moment vector ii is at right angles to the external magnetic
ﬁeld E? W=bUa off—U; = (5763; _ (:HE); = (FEM: ga‘{_tﬂ.[altzi+1}] W=_ Trial!»
.3. Phys MISme Pageﬂ? MSIJOW . ' " 4 4
PROBLEM (4} {16 points}
The figure shows a cross section cf a long cylindrical conductor of radius R {= 2 cm}. carrying a uniformly distributed current
i{=2A) directed into the page. (than 104 Tmi‘A}
[m4 pts.) What is the value of the integral ﬂittn along the
straight path KL shown in the ﬁgure? Amp42's. imtr fr "M. eluted squeeze inept §Ev dg=ﬂe inttr.
M44 Magmwdh‘lf wt. We: Jag. at 5m = MN , tannairixtm K It (hi6 pts.) Find the distance r of an external point P from the axis of the cylinder at 1which the
magnitude of the magnetic ﬁeld is the same as that at r = M, within the conductor. gr point P; M gluedvice! mnduchr Adan; ct lit«:3 wire . Sc, 3: % .
At MM rung : 393wg= rim—4 [Bltzn §)=h%n[§f=& {c}(6 pts.) Calculate the magnetic energy per unit length stored within the conductor. . '8‘
2mm chumh] , uh: —— 29,14.
* ‘ ' t'nr'
ég. Reﬁt,“ —g {mumh!” ﬁt utl __’ Eli('1': g?— mechmﬂnlwwilcfhtwnhw:* ‘ 1
uh ‘ jaut .. Su‘tﬂew .t J'JL. (artKarim = J;E(%ifi) furl4r) t. 1/“ t
m. ﬁt ‘u lg gti“ cl" n t‘.‘ rmt‘Twctm‘
a t; — e. 1' :__..————"'
L ‘ HarkIr r me “t c 1511 '1”
u". it??? s
1 "I
/ Phys lime Fag£53? MEL 21301 4 5
PROBLEM (5} (16 points} Consider the IR circuit shown in the ﬁgure.
(:06 pts.) What is the current through the hatter a
long time after the switch Sis closed + A lav3 nfir ‘fIniL 5wi+cle .5 Cloud 1 Mthducicretehem a. shirt circuit. so] inﬁll: 33..” —“*"' =34+zA I“; 16 (MG pts.) When the current through the inductor reaches its maximum, steel:ler value, the switch S
is opened at time t = ﬂ. Find the initial voltages across each resistor and lite inductor just after t = 0. was... Mswihh ,5 is tome nitinnit. tie, at current etsz MW“
414:. mildew m4 ellw WM 11. were. 114.1 “ﬁrmf a m Wu salty.
its) The mere«t will deeruf ffm 1,: an in: am Where. jM'i “Hf fto,‘ih.n. colic5M m:
Acme in. the. wear: u. = [1&1C‘IJL)—) Aﬁt‘atBM'EJL regainhr ., clanking—l
rim‘s‘n M Messier . a; [wt .4. efvsv io=zﬁt (c)(ﬁ pts.) Howloug after t = D will the magnetic energy,' stored by the inductor reach half of its
initial value at r = ﬂ? _ I hir— i"
%LLL=JTF L». we a — a "" F'z.
. 1:I'E._ L 5H 1H s
tum“. ufmm. rte? when. 'Z...= R‘Jfgt‘HJHIBJL L 1'7
‘f{' ‘H:. ‘'t
4:14: __, TL.” qhtﬁyhk )
t L
_,iiu,ﬁ,=—'~tt _, “91:15 Phys 106 ﬁne! Eran Page 6:"? May .H'. 200! 46 PROBLEM (6} (11r points) A long (ideal) solenoid with 21100 turnsim and radius
2 cm carries a current i. A single conducting loo of
radius 3 cm is held perpendicular to the axis 0 the
solenoid. Give your answers in terms of 11:. [3)(5 pts.) What is the mutual inductance M for the
loopsolenoid combination? Suziemid L949
n..= 1m “if«t N “l (1. 8 1m {adios} h = 3 m frail“) m % cﬁmnd"
l. l. M = fqnx1€?ﬁufn'}fmo wﬁytjnlﬁlﬂlt
i M = ain‘me' H [h)[5 pts.) If the solenoid's current i increases linearly from 2.4 A to 7.2 A in 113 s, what is the
induced emf in the hop? What is the direction of the induced current in the loop? :11: 2mm Mmajnl‘l'udl, DEMII‘ldﬂﬂlm‘filﬂl (l ‘1' ‘  A ' Tied(Bret _ 1.2”.” '1"! 1:¥'1 a=——v————J'11QT
I t 151'. At = 93 5 E: THEK” '“l' 0.3 5 3‘31”? 2,; a?” a = so. “LEWi V N: Mﬂqn'hw lﬁgruﬁuﬁ  ﬂhglﬂ EM :0 tlﬂdLWIH
errMd lep 4:: weatherman Mflmt. (elf? ptsJ While the solenoid's current is changing in the manner ruentioned in part {b}, what is the force (magnitude and direction) acting on a stationary positive point charge q = 1.5 x 10“ C placed
at point P that is 1 cm from the axis of the solenoid {see ﬁgure}?
I . a I 1
"TL: 5mm Merge1%; flux mint“ Mn. dWC‘Fidtl E d pain1+ P.
we arse. s2, ffedsmuh 2364:” Earl“J
cue . . 
ﬂax Dmehud mufﬁn 1‘ I I I.  i I = I 5:? i “#4:— ” minim”) ’5“ % 'diP ’ . . ' :5:
50mm“ Etiﬂru‘ﬂdnp’raﬁ_ ﬂ m% 5% 19% E  (ll11x15 Wan utm lfHlfilililml in mg)
n 1.. E a Music?“ or; ﬂuidi FWE efismigcxtrrmﬁltit) —; F: tarma‘luiemnrt Phys iﬂdFicciEmu Phys?” “0331.200! ...
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 Spring '12
 OnurFen
 Magnetic Field, pts, Electric charge

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