Download Magnetism Answers

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1. A particle of mass m and charge q moves with a velacity v in a magnetic field B. The velocity of the
particle is perpendicular to the field, and the particle is moving in a circle.
fI1
(a) Calculate the period of th",mR.t.j~I1,
the time j:t takes to complete one revolution. (T=27tm/qB)
f. ~~
':::«'lie
~
V -:::~'tl('
v::
'f
:t&l = <2J!£
T
(V\
T _ a--1i' rYI
-
'b'b
T
(b) The cyclotron frequency is the number of revolutions the particle makes in one second. Calculate
the cyclotron frequency for this particle. (f=qBI27tm)
JO?
=
I~",J
2. A wire 0.5 m lies along the y-axis and carries a current of..!9 A in the +y direction. The magnetic field
e
IS
. Un!
.f orm
'- and has components
"Bx = 0.3T
By=-1.2T and B,=0.5 T
(al-Find the components of the force on the w~~~,
2.5 N)
fSX::TLB)I
'>/ ~.~r
"
=-L'O\(.'5}(.3)-~
f!:,y:o
F"e,2=rl-'?>~
-
=
i
• - ,. z
-;@.S N I
::p.q~N\
QQ)L.SX
.5)
(b) What is the magnitude of the total force on the wire? (2.91 N)
rNJ-=lQ'J)~1 (Z.S)1-
3. The rectangular loop shown below is pivoted about the y-axis and carries a current of 10 A in the
direction indicated.
(a) If the loop is in a uniform magnetic field of magnitude 0.2 T, parallel to the x-axis, find the torque
needed to hold the loop in the position shown. (0.00831 Nm)
'i';:: 0
£
y
r'[ """Fe. X f
~
-
'fp" ~, \~I(.olo)Sl'n CtJO'J
8 cm
;( 0, (:083N,MI
x
z
"
f
3 /I E
f6 ~ . '"oL 'f!.r;;...
I
(b) Same as (a), except the field is parallel ta the-z-a~is.(0.0048 Nm)
ry:::
F.e,
><
r
= ~ Ib)(
I
I
010 ) 5 \ n
xl -:.
[ C)(Y.f~tV. /YI
)
(c) Far each of the above magnetic fields, what torque would be required if the loop were pivoted about an
axis through its center, parallel to the y-axis? (same 1/2 cancels 2F)
rr ~(rB Xf )~~~X f):;
r lFE 'If)
= SOJYI e,
I
4. The rectangular loop of wire shown below has a mass of 0.1 g per centimeter of length and is pivoted
about side ab as a frictionless axis. The current in the wire is 10 A in the direction shown. Find the
magnitude and the ~en~f
t.hemag,!et~c fiel~, parallel to the ~s,Jhatwill
cause the loop to swing up
_until its plane makes a 'Iangle of 30 degrees with the yz-plane. (O.OIRT)
(f\ ~~ • ~.
u.>P;+
Cl
I c.M
'::\
y
::
- -"-.; ;X
t,=o
z
zr(::o
f\m~=~E
~ '/..r ::. 'Fe, 'X 'f
(, ()b1..~Y~.2X.
O~S if\
,0\ 312.
-
/
.
It;
0
30° =. (ID)(' oro) (I))(..o~)Sln 0
. S' q lQ 'b
'~
10009
,o\¥-!j(t"I1
J. SA
5. A longstraight wire carries a current of 1.5 A. And electron travels with a speed of 5 x 104 m/s
parallel to the wire, 0.10 m from it and in the same direction as the current. What are the magnitude and
direction of the force thcit the magnetic field of the current exerts on the movingelectron? (2.4x1O'''N,
away)
f:'6v13 .
.
f:: I.
:: 'J
';2
u
(.s
S)<IOq)
I
l{ 'I. tr{/'O
K
rc
:0
~)<.
to- ~ ( I ' n
CD )
I
;; '3,OX/O
-
/P
I
N
./
6. Two long parallel wires are hung by cords 4.0 cm long from a commonaxis, as shown. The wires have a
mass per unit length of 0.050 kg/m. and carry the same current in the opposite directions. What is the
current if the cords hang at an angle of 6° with the vertical? (46.5A)
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4c
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I~
--=::
-r '"
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Name
&ut~@f ~a~~ie
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_
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~~d:~
1.
Wli"m@li\l ~~@~
The figure belaw is an end view af two long, parallel wires perpendicular
a current I, but in opposite directions.
(a) show by vectors on the diagram the jj field of each
wire and the resultant
jj field at point P.
(b) Derive the expression for the magnitude of jj at any
point on the x-axis, in terms of the coordinate x of the
point. (!L~ahr(a2+x2)
'B",~+~ 1'>x.l 'Y
1>:::) := 0
t
tp\~l-
to the xy-plane, each carrying
YQ
t
..
I, out
-~ ..•...
x
-.Mo:t
~'\T'("
€>lI ::()AoT.~e )~
=
)A.;Xq,
\
1i' (a.1+)Ct.-)
(c) Construct
~c-
j
a graph of the magnitude of jj at points on the x-axis.
ci
(d) At what vaYueof x is jj a maximum?
)(:::.0
(e) Suppose that a third long, straight wire, parallel to the other two, passes through point P, and that
each wire carries a current I=20 A. Let a = 0.30 m and x = 0.40 m. Find the magnitude and direction of
the force per unit length o~.the third wire if the current is directed into the a e (1.92xlO'4 N/m,
down) and'
'FB;; T.3 L:B1'd-
.fa- "
I3 (-!A 0 La._
L.
\.1T'{a1l-yl)
;:. ',Q2
(i)
(ii) if the current
is directed
out of the page. (same, up)
Fe:," l.ctL~/o-~1 ./
up
-=
4KtD1 (.1.0)' (,3)
'lC1(J-41J/
(,3Y 4-~t./)1.
rfl
I
B <E--
I
2. The long straight wire ABshown above carries a constant current I.
(a) What is the magnetic field in the shaded are.,a perpendicular distance x
from the wire? (~I/27[x)
I
';2.
_
lA,I:
_
Dlol(~ - ~
-
;""0]:.
,:).11'
i
i
I
L
A
j--i!r
(b) What is the magnetic flux d<l>through the small, darker shaded area? [(~ILl27[)ln x]
-r:. (L' vtx) ~ )Ao T.l
l
-'d-'\I
J~1TX
i )(
I,
I
_ ~o-r!::-( I V\ )<\
~1T:
)
I
, (c) What is the_total fir
~;:
J.A 0 r L
G}1TX
I
through the rectangle CDEF? [(~ILl27[)ln bla]
rb-"
I
)-f(
()...,
<A.,
D
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(
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.
--~)
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~-
F
1
A
C)
Hall Probe
and Meter
2005E3. A student performs an experiment to obtain the value of I'll, the magnetic permeability of vacuum. She measures the
magnetic field along the axis of the long, IDO-turn solenoid PQ shown above. She connects ends P and Q of the solenoid to a
variable power supply and an ammeter as shown. EnCl7'Orinesoienoid is taped at the 0 em mark of a meterstick. The solenoid
can be stretched so that the position of end Q can be varied. The student then positions a Hall probe' in the center of the solenoid
to measure the magnetic field along its axis. She measures the field for a fixed current of3.0 A and various ositions of the end Q.
The data she obtains are shown below.
--- --
I
Position of End Q
(em)
40
9.70
X
10-4
c9.50
2
50
7.70
X
10-4
2.00
3
60
6.80
X
10-4
I In':l-
4
80
4.90
X
10-4
lJ.'<
5
100
4.00
X
10-4
100
Trial
Measured Magnetic Field (T)
(directed from P to Q)
n
(rums/m)
a. Complete the last column of the table above by calculating the number of turns per meter.
'" A Hall Probe is a device used to measure the magnetic field at a point.
b. On the axes below, plot the measured magnetic field B versus n Draw a best-fit straioht
line for the data points.
0
-7
10.0
'.0
:u1
:t:t:!=t
... _~-~_
f=.
1,
,.0
:;
~O
~ ./..0
3.0
2.0
1."
0-_
.J ••
",-.,-.,--r-
,
•
.
.
I
,.
"'- .•..,- ....
_4 •.••..••..•
_ •• _4
__
-~_.•.._ .._~. ..•..•.......•..
••.••••••
,
'
I
, ,
:;("7-~.t.
-~,-~_-, ,.....
- -~-.-~,
• ..1 __
•.. _ •.••.
- ...
,
-
I
••
•
,
I
•
I
••
-"-0
...
••••••.- •.-r-
.J __ '-_"_"_
•
_L.
,
-~-~--:-~-
+-~.t-,-.,--:.+-1-- -+-,-+-:-- -f-!
... _,
.:-.t-t-~-.~.-:--:-.~.
:~:~:~:~::
, - ..-.-~..... - - .. -~. :~:~:~:~::
-i-f-, .. :-.
,-
I
-.- .•.... - ..•.• -
•
.. :~:t..:?:r:
:~:~:~:~::
.
.-;--;--i--r-
-}--
-.,--.-- •.. - •.. -
'
100
...
"0
,
•
..
,
,
I
200
300
n (lumvm)
From the graph, obtain the value of I'", the magnetic permeability of vacuum. (-1.3x IO.6TI11/A)
Sh:¥Je..~T.M.o Slopl-= ""'o-'l \ 8b -
d.
I...I..j.,...
-}-t-l-,:F~=~:~:=
.':--:"'i4- :ttj:i::
' , , , :t:l
"T-'-"-"'-"-"-"-'" -~~~-~' , ,
•••
"-""T'"-,.-r-
.
"
c.
.
_ .• ____
---~, ,
' , , ,. -~-:
-+-~-~--:--:r.....r.r-'- -~-++-r- -,-,-,--,-, , , ,
, ,
'
,
,
.
-~.~-':.:f-:, , , , •..•.....•.- .. -.- -~-.. - ..---- , ,
.
-"-"-T-"- -'--1""-1".,.- -+-+.,.r.',':-- -"-"-T-'. .,-..,--,.-,.- -~-~--:--:---1"-'
.~.~-~-~:j-1-~.~--~-+-:-.~--f-~-~-+'
-~-~-t-~--~.-:--~-~, ,
,
..., -.,-.,-..,.... .,-.,-..,-- -,.-~-.-,- ... -~.~-.,.-:- .;--}-. ~-.f-i-,-+ :~:i
+-t-t.i.
+.t-t-i_L_
..
_
•._~_
_ ..... _ .. _~.
- .. -.
,
7.0
.~-.:--~-}-
.~.~-~-~-, , ,
.. - ...•...•..•...•..
, ,
,
..•- ..•. - ... - .. -,..-r-"-'"
'r-r-t-~- .-.-.,--I"-r"
, , ,
, ,
,
-~-- ,,.. -~-.-., ,.. -,•... - ,.. ,
Using the theoretical value of po =
paI1(c).(1.06%)
U<'[,'-._I/
,1•.•
< 0
4TCX
140
Ia
flo ~ 5}(ID-
5~lo.\ 5y..,16~
'3
.
10.7 TM/A. determine the percent ,errOl;.}nthe experimental value of j10 c?mputed in
•....•xOf,.
''-'"
lJrr'l:.IO-r
~/.
X 100 -:~
.
@
.'
r--
R
'ire carries current I in
e plane of the page as shown above. Using Ampere's law,
develop an expression for the I gRel;
1I1tensityat a point M that is a distance R from the center of the wire.
On the diagram. above indicate your path of integration land indicate the direction of the field at point M.
j~ ,dt
I
~)ADTM.~
'b [z..'lT(")
::- Ao]:
'\2. .••
I
V
I
pO]:
-
'/1if '("
I
I
I
b.
Two long parallel wires that are a distance 2a apart carry equal currents I into the plane of the page as shown
above-:-,
.
i. Determine the resultant magnetic field intensity at the point a midway between the wires. (0)
'& ( ~
I
=- 0 ~nd-
B2
I
ii. Develop an expression for the resultant magnetic field intensity at the point N. which is a vertical distance ofy
above point O. On the diagram above indicate the direction of the resultant magnetic field at point N.
[Illa/(rr(a'+y')]
"E:>
l/)\'r~:::
~"1:
I
.9'\1("
'6"e+ :. 'D)( -:: 1/ I Ch4-e + 6t
:;. ,&.MoTC\.
,tlf
p-'Z-t j ~ )
I
1m ~ .".~
e
Cos, 6
-; floTo.
11( a'1.t'jt)
+ ~
~:JO
C1J:J
e -:_0._
1O-1t'j'L
\....--- Coble
200 IE3. The circuit shown above consists of a battery of emf E in series with a rod of length I, mass /11, and
resistance R. The rod is suspended by vertical connecting wires of length d, and the horizontal wires that connect
to the Datlery are fixed. All these wires have negligible mass and resistance. The rod is a distance r above a
conducting cable. The cable is very long and is located directly below and parallel to the rod. Earth's gravitational
pull is toward the bottom of the page. Express all algebraic answers in terms of the given quantities and
fundamental constants.
a.
What is the magnitude and direction of the current 1 in the rod?
(I C)M ""l ...h-f/
(ri
C.U WI !;( )
u
b. In which direction must ther~ be a current in the cable to exert an upward force o)l,the rod? Justifyyour
answer.
v \C;W
~ ~
~ H r< -fN,. 'O-.f1 Q.ld ~:1
P-t.
+v ~
a. o. p -to
c.
:tB
evv
"Pr"v~
~
iA v-p
oY\ ..A-Dcl
With the proper current in the cable, the rod can be lifted up such that there is no tension in the connecting
wires. Determine the minimum current Ie in the cable that satisfies this situation. (mgR2m/Elfl)
"*L
'Cc.=
Trz.;; e.
('(\3
~-e
(Mo:rC.)=
rZ
~rrf'
1Y13
'Pc.=
.M. 0 Ie..
e;t1T'("
T c. = rng R
;l.Tfr
G Q ).1.0
j
'"-:b d. Determine th." mallnitude of the magnetic flux through the circuit due to the minimum current Ie determined in
part c. [(mgRr/~).ln(d+r/r)]
\..S-I
c\y~~Bfl.-
);toTe..
~r
Q.clr
;: jADTc..~
dt f"
I
E
a-1f
'(
'(
)Ao Tc..!
In
Q-.1T
y!+f
r
),t.-pk (fI~!P1'( .Q,()
~
[l;-p.e'
h'I£1 ~ r
f:.
r'
£!+r
'(
(~;;- \
•
Figure I
2008£3 .. The circular loop of wire in Figure 1 above has a radius of R and carries a current 1. Point P is a distance of R/2
above the center of the loop. Express algebraic answers to parts (a) and (b) in terms of R, I, and fundamental constants.
(a)
I
I.
State the direction of the magnetic field 8, at point P due to the current in the loop. lAp
-='?
ii. Calculate the magnitude of the magnetic field 8, at pospintP .;111«.
clt:>~
- ..-MoX dQ
Td.-tWe
"411
,'2.
~
r~
~
L.f17'
(15:
t -15
_ ,VAoT
=:>
==>'1
I,
~i!
I
Axis
1>1
R
5' lTg'
J~ - .}AoJ: if
0
- 05 y( (2.
R
• p
-f--~-AI
t
!c
s
I :::Jc
R
R
'
.
:::>
I
I,
Figure 2
Figure 3
A second identical loop also carrying a current I is added at a distance of R above the first loop, as shown in
Figure 2 above.
(b) Determine the magnitude of the net magnetic field B"" at point P. d-)l
~~
8MoI
.
:>ls r<..
A small square loop of wire in which each side has a length s is now placed at point P with its plane parallel to the
plane of each loop, as shown in Figure 3 above. For parts (c) and (d), assume that tl1emagnetic field between the
two circular loops is uniform in the region of the square loop and has magnitude B,,,,.
(c) In terms of B,,", and s, determine the magnetic flux through the square loop.
8)\oI Sd-.
515R..
I
( (d) The square loop is now rotated about an axis in its plane at an angular speed ro. In terms of B"",, s, and w,
( calculate the induced emfin the loop as a function of time 1, assuming that the foop is horizontal at 1 = o.
[-----N~".
Jf)f1eAS;){jx)f)
Jf
tIt
Bnel
S(;l
d
C1JJ-()Jb
(Jl3
=
-:-IBY1efS~
~
,..
iJ.)
5 (1\ UJi \
q ,
W
~N~
e ;-lJJ :t;
Cnls.'-scctilmal
View
(curren! inw p<lgc)
20 II. 3. A section of a long conducting cylinder with inner radius a and outer radius b carries a current 10 that
has a uniform current density, as shown in the figure above.
(a) Using Ampere's law, derive an expression for the magnitude of the magnetic field in the following regions
as a function of the distance r from the central axis.
r <
I.
jg
cis; =)lOT0\
II.
iii.
r ~ 2b
e'J.rrr:;
PoIo
~ '
:;t: I
(b) On the cross-sectional view in the diagram above, indicate the direction of the field at point P, which is at a
distance r ~ 2b from the axis of the cylinder.
point P. Describe any electromagnetic forces acting on the electron. Justify your
(c) An electron is at~t
r,B '" 0
answer.
V'" 0
\0 / (..
-- f>
-=-
'tV "E>
Now consider a long, solid conducting cylinder of radius b carrying a CUITent10 . The magnitude of the magnetic
field inside this cylinder as a function of
r is given by S' ~ Alor / 2"b'.
An ~xperiment is conducted using a
particular solid cylinder of radius 0.010 m carrying a current of25 A. The magnetic field inside the cylinder is
measured as a function of r, andthe data is tabulated below.
Distance r m
0.002
0.004
Magnetic Field B (T)
0.006
0.008
0.010
-4
(d)
i. On the graph below, plot the data points for the magnetic field B as a function of the dista1J9;.L and label
the scale on both axes. Draw a straight line that best represents the data.
~)
,
~1
l-."
)40=
ii. Use the slope of your line to estimate a value of the penneabilit)}lo'
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,--
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-....
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f
.........:: ~7--. . , :,.':.\
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li'Q$Jlj\,~ : (lui
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1993E3. A mass spectrometer. constructed as shown in the diagram above, is to be used for detemlining the mass of
singly ionized positively charged ions. There is a uniform magnetic field B = 0.20 tesla perpendicular to the page in the
shaded region ofthe diagram. A potential difference V = I;500 volts is applied across the parallel plates Land K, which
are separated by a distance d = 0.012 meter and which act as a velocity selector.
a.
In which direction, relative to the coordinate system shown above on the right, should the magnetic fi~cLpoint in
~rder for positive ions to move along the path shown by the dashed line in the diagram above? (.z)
I vric
P c; c.;
b.
Should plate K have a positive or negative voltage polarity with respect to plate L? (+)
c.
Calculate the magnitude of the electric field between the plates. (125,000 Vim)
bV~U~
I~OO
=-E
E
d.
(,ul7-J'
1
1, l. S)<lh')" V
I
rv'\
---,
'")
Calculate the speed of a particle that can,pass between ~he parallel plate without being deflected. (625,000 m/s)
I.
--,
?""S;</O)
-'2.---
;
.2-5.<[ i)S'M Is.
I
e.
Calculate the mass ofa hypothetical,singly
(2.56(XI:2~g~
~
f.
E>
J
charged ion1that travels in a semicircle of radius R = 0.50 meter.
m::!,al?;;
Ii'
C:5)(\,~r'('},)-L--r/---""="
~ 0. 2-5 'f-I D 5"
=0'.;,'0 '(v0)
,7---:-
•....
- ----.::.....-
~
A doubly ionized positive ion of the same mass and velocity as the singly charged ion enters the mass spectrometer.
What is the radius of its path? (0.25 m)
2 1
f==- (~.Sfo~.'I)-~"')(4?lS)(J~)
5M
(J,' I.CD~D\4)(.l-)
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EIlIf3nCC
Examinalion
Board. All ri~hts reserved
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.1010 AP Advanced
Physics Magnetism
Name:
and Induction Test
Questions 1-2 relate to the two long parallel wires shown
below. Initially the wires are a distance d apart and each has
1 current i directed into the page. The force per unit length
on each wire has magnitude Fo
••.••••
..:~v:
The direction of the force on the right-hand wire due to
the current in the left.hand wire is
a. to the right
b. to the left
c. upward in the plane of the page
d. downward in the plane of the page
e. into the page
A negatively charged particle in a uniform magnetic
field B moves with constant speed v in a circular path of
radius r, as shown above. Which of of following graphs
best represents the radius r as a function of the
magnitude of B, if the speed v is constant?
(AI
(8)
ot=. oL. L
Ie)
2.
3.
The wires are moved apart to a separation 2d and the
current in each wire is increased to 2i. The new force per
unit length on each wire is
a. FoI4
b. FJ2
c. Fo
d. 2Fo
e. 4Fo
A charged particle is projected
parallel to a uniform magnetic
a
a. spiral
b. parabolic arc
c. circular arc
d. straight line parallel to the
e. straight line perpendicular
,
o~.
(OJ
,
lEJ
oLL.
r- ---------..,
(A)"(D)
(B)'(C)
_
(E)
/
/
,
pr~o;-il 1 I I 1
E
L
.J
A proton traveling with speed venters a uniform electric
field of magnitude E, directed parallel to the plane ofthe
page, as shown in the figure above. There is also a magnetic
force on the proton that is in the direction opposite to that of
the electric force.
Which of the following is a possible direction for the
magnetic field?
(A)
In a region of space there is a uniform B field in the
plane of the page but no E field. A positively charged
particle with velocity v directed into the page is subject
to a force F in the plane of the page as shown above.
Which of the following vectors best represents the
direction of B?
•
I
6.
4.
0
Questions 6-7
with its initial velocity
field. The resulting path is
field
to the field
1
.."U- ..
......
.....
5.
I.
_
(D)
7.
0 (directed out of the page)
(C)-
(E) 0(directed
into the pag
If e represents the magnitude of the proton charge, what
minimum magnitude of the magnetic field could balance
the electric force on the proton?
a. Elv
b. eElv
c. vE
d. eE
e. evE
20 10 AP Advanced Physics Magnetism and Induction Test
8. A solid cylindrical conductor of radius R carries a
current I uniformly distributed throughout its interior.
Which of the following graphs best represents the
magnetic field intensity as a function of r, the radial
distance from the axis of the cylinder
'A'tili# (B'0ili
Name:
"t-
(8)
-"t-
t-
t--
-+-
,",,:i;~",:,~ '"~
-+-~- ,
-+- -,
O-;-;R-'
O-R-;R-'
:0
II. In cac!! of the following situations, a bar ma~t is
•
align~along the axis of a conducting 1001The magnet
and tH loop move with the indicated v2ocities. In
'
which ituation will the bar magnet N'pT induce a
current theconducting loop?
I
Field
Fidd
/
,A'
Uo
I
7',,0
(/;i(
... ,~
,~(j)
~..
.-;.:;?=o
'co
,0'( ""
~i.
"'lA'''j6-'\''
O-R-IR-'
'~V
A."
\
>-----/.
T
2,
~
-.L
~
9. The cross section above shows a long solenoid oflength
1and radius r consisting ofN closely wound turns of
wire. When the current in the wire is I, the magnetic
field within this solenoid has magnitude Bo' A solenoid
with the same number of turns N, length I, and current I,
but with radius 2r, would have a magnetic field of
magnitude most nearly equal to
a. B.,/4
b. B.,I2
c. Bo
d. 2Bo
e. 4Bo
~
X
y
10. At point X a charged particle has a kinetic energy of 9
microjoules ( ~ ). It follows the path shown above from
X to Y through a region in which there is an electric
ficld and a magnetic field. At Y the particle has a
kinetic energy of II ~. What is the work done by the
magnetic field on the particle?
a.
II ~
b. 2~
c.
-2 ~
d. -I I ~
e. None of the above
\.
F'I
M.,neuc
u:d
Gl
Gl
0/19\
Gl~0\
Gl~0
'\
GlGlGlGl
12. A squit", loop of wire of side 0.5 mete~d resistar
10.2 ohm~ located in a uniform magrietic field of
intensity OJitesla directed out oL1hepage as shown
above. The ~~nitude of the fi{]d is decreased to zero at
a constant rate In<: seconds/ As the field is decreased,
what are the magn'i ude nd direction of the current in
the loop?
a. Zero
b. 5 A, coun~e lockwi
c. 5 A, clock Ise
d. 20 A, co, nterclockwise
e. 710CkWise