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M~~6-:ed
Name:
Unid: _ _ _ _ _ _ _ _ _ _ __
1
Exam 3
Discussion TA: _ _ _ _ _ _ _ _ __
Show all work!! Report all numbers to three (3) significant figures.
A wire carrying current I has two long straight sections connected by an arc of radius R centered at the origin as shown. Using unit vectors (t, j, and k [z-axis is out of the pageD, radius R, angle e, infinitesimal angle element de, and current I, as needed, find the infinitesimal
element of magnetic field, dB at the origin as produced
by the current in the infinitesimal element of length, Rde.
A. )Jo~JSXR
df--~l fl-J~ K~ -Alole ~
tfrl RZ B. Arc
L{nf<2.
yl1R
Assuming that the straight section ofwire in the figure above extend to infmite distance from the origin, find the
magnitude ofthe total magnetic field B • at the origin.
to.
Bto•• = - - - - - - ­
Alt~.--n~!tv~ (/';J kMW.'~'t £'M'ft
l(
- S1'Ivli1rt
S hurt
vi
0'(""
B,'o t
dB' ~
(z-axis: out)
l{jI\~ th J~ -::: R~ (}
V ~,'v1" i h~ V'l (~h+ haV! Ct{ rule w{ h~l)f-
F
Vector dB =
J !Je
I...
tOdf-
ci
J.(o I W~
~ bo lie.
p ; tAAoL
C. 4
R
.).1
(J(' I f'\.Q.
fA
11
Wi re
V\J.{
ViP"JL is/( r
J 01
(j
),-0
W/'f'.f5
r"/~)
titS Kr -=- 0
= 0 , Us i.-,t
de -f',.o_____
t01 K~rr
8~ 'd ~ =:
An infinite slab of thickness L has its mid-plane in the x­
y plane. It contains a uniform current density J flowing in
the +x direction. Find the magnitude of the magnetic
field above the slab at a distance z>U2 away from the
midplane. 11
B= ---~~~=--------
two sfyq"4ht
VtCltV( V1J ols
~; t(
\\ loop;: ~ ,
"L
7-"kf-
t::
I
JJAoIde()
Ll
rr R.
-lAol rt
'-frr~
l~:= AoI
If if(
z
J
l
h
W~ V<,~ A"'fe('e~ LI7JIN q~ ~k~ t\ '1>f/~re f/lj-~
4. for ouv-- Irflr..,,,ier, {OcJf', T~(/s jr3,Jl -::::){iJ I.eI\(lo~()
tJ-l'(('
jf~. J..R =::: B·~, + f$, hi_fi t6· WIJuJ'/<,1,+B. ~f'1I1t -: : . e"VII ft>tis w+0 .= 2. II? W
r;o "" OV( (<<se If'll - :JA "- J twol ti)uS L~W::: J'W L.Mo
3:::.}=
Ph~
SJlrin~
... it·... 2220
Name: _ _---'\.tF-}J~~4-lt-l_ _ __
Vnid: _________________
20 J3 Ikn
2
Exam 3
Discussion TA: ______________
Show all work!! Report all numbers to three (3) significant figures.
A. y
A square loop (length a to a side) lies in the x-y plane and is bisected by the x axis.
SI
as shown. The loop is pushed in the +x direction at speed v into a region-the 1 quadrant of the x-y plane---containing a uniform magnetic field with strength B directed into the page. as shown. Find the magnitude of the induced EMF in the
loop and the direction in which induced current would flow (clockwise or t. f
=
~
.Jj )
counterclockwise).
1 r:) <::L V j
5 at
z...
a
-1 (n
.Jl
E=
Bn)lnlc~
17 curre-n";:;t-d-ir-ec-t-io-n-:----,C....-c.:='" W ------~--------
1 !3 ==-(-B~x)
z
L.eM..1;'>
B. Iu.w
t-ft
.:L
ClA.rr~1
tv
off'ASe
decr.e a 5.1A.jfjeld..
ow couvJ-ev- c )"ckf.Jt'Je An inductor (L = 0.003 H) and resistors (R1 = 1 Q and ~ = 2 Q) are connected
to a 12 V DC power supply as shown. The switch has been closed for a long
time. What is the current I across resistor R2 at time a time t = 0.001 s after the
switch is opened?
-t/r:
L
1= r((J~""I/~ G./(pAICt):::<Jo e
L = R-z. -::v."ol5"---"9
f\{U5
I
...Je:
'1 - ':J ;:: 12 ft
C. A series circuit has an AC power supply [V(t)= 180 cos (100 t) V]. a resistor (R = 3 Q). a
capacitor (C = 0.005 F) and an inductor (L = 0.05 H). Find the current I as a function of
time and determine the average power Pavg dissipated by the circuit.
1(t) 1.(2.1.("3 u.S- (,,,.d
Pavg = '2..7DOtJ
-1')
""
_
I
"f -
¢:::.
t
f{
Ll.trr-u1.7-
10..<;;-> fM-F
11
'2 -r~.x
l~jn"L.f-(XL'-Xtl.-)
II'I('V' -= I ~::
'-f.LI{~
=
L
o
l(t)=
-r~ ..'"'/' (o5(wt - ;:)
-I.,
f,.... ... Jt ;:::
R
II-;:' ~ L = 5.0 52
.x,,-:::fc=-7.. vSi..
J.t.... -Xc­
-l
--t-tt 10{
L
~
~\AAA~~-
R,
t> ­
,-
=-
41Z-4~5t.
L./ -z... <-t 3 A
by ~ r?
I ~V,
C
c
-= ZI.,...
~""'<I.k c"'I.ct.)I.
A.
{or,
Name: ~~-I:';h
Unid:
-----------------------------
3
Exam 3
Discussion TA: ____________________
Show all work!! Report all numbers to three (3) significant figures.
A. Unusually powerful lightning can develop from the accumulation of positive charges in a cloud during a
thunderstorm. In a simple model, the clouds and the surface of the Earth act like a pair of circular conductors
(radius R = 2000 m), forming a parallel plate capacitor with separation d=500 m. At time t=0 the charge on each
plate starts to accumulate and grows according Q(t) = 5t C, where t>0 is in seconds. What is the strength of the
electric field E as a function oft> 0, assuming that E is uniform (as when d«R)?
J:,,-t
E(t) =
~
R;2000
IV\.;O~
(' :=.
__..:...;:...:,=.u._______
Cyl;~Jt;(,.~1 (;,Qu'J~;C\1'1 s4{'-~q[e cOfl-tCiIV1:nd
,
~J
L\fP~r fl~N'
. Ci1t.-Met~cJ:
1,.... 2.;"
aQ(i:)
j 'dv E' dA =' E ~dA -=
E(t)
B. U<:x'
Mlii-IVC I;
Iqw~
C. = n£.. RZ
U
Q(t)
(t-e!o o~ id€O/ ti-pl Qtt
{iDO""
10'1
k -= 0.0000 ~t
aGli-lX)
b1
T
c.:::. 2. qq7~1.IH~ ~ fl - Ltf( 1 ~7 T'Wl '0
17 -
A
~
1L£"~
A storm-tracker helicopter hovers 250 m above the ground just outside the storm at a distance r = 2100 m from the central axis of the storm. What is the strength of the induced magnetic field that it reads as the charges accumulate (t>0)? [Hint: You may give your answer in units ofnT (nano-Teslas). If you do not solve part A, you may attempt this part symbolically-define your variables clearly.] T
c.:I'("u!qr AWlp'ti\' Qn p Q-ri,
B = 0,4-7('
(A... . f~d,­
h = 2 t;o
Eo. =O,Ol'f~ ~ w::::: '1.44(0
I",f
E. ::; To
~
d::
WI
:r..,:= 5 :t
-C-
-=::
f'\1
c
U.Se
ICjw)
( (J,qu'>s
2. 1DO
.c.
VI
O~E.' --....
J"I\ B'd~ :. fo ICc> ~
B~J~ =
tdCe
GDVlCev1
on ft.if
1
~t (~E'JA)
d (~\
)
B{r): /!!:!:::21C r ""Jf 1tf~RL j dA
fa reo
'yl"~ ~er
B( ) _ ~
d ( Tot
'(" - l1C'r eN: T(f:.. R~
7(
2.)
R
~~ "" if,7b ' l0- 1D T ;::jO,/.f7b~i1 qoo d IlAc. \K W1.~OSL{,(': ~ .J,4'1 )
B(.)
)
i
(
1'\
At after about a minute or so, enough charge builds up to trigger a lightning strike. The transfer of charge (and
associated production ofE and B fields) generates electromagnetic radiation with an electric field
E(x,t) = 0.0143 cos(0.008 x - 2.4 xl 06 t) V1m, where t is in seconds and x measures length in meters along the
path of the wave at some large distance from the lightning bolt .What is the wavelength A. of the radiation?
1..= 785
WI - c. 0 s (I~ x- (» t )
E ( 1<; ) t) :::. Eo
"Aw
W
G -=- AV = 'tIT == k
A=~~G =
-:::..
¥ =Ei&s-'~l
Ee> R~ ( e((l"x-wt)\
J