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Unformatted text preview: So I MAC Lo V’L % Physics 8.02 Final Exam Some (possibly useful) Relations: 4? 17: dA = 0
timed
surface s d  
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,uo =4Irx10'7Tm/A k = =9.0xlO°N—m2/C2 e P"
H cT f=1/T co = 2791': 21t/T double slit: constructive: dsinﬂzml, m=0,il,i2,:3,... destructive: dsin6=[m+%]k, m:0,il!i2.i3,... single slit: TE EasinEl: ”111:, m = i1, i2, i3, Fall 2002 4 Egg 215%
closed dt
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It = 1—3 L () Rm} )
rzL/R
r=RC
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single slit sin[§asin9
1:1 111 EX EasinB
7L double slit I = 1m“ cos2 [9]
2 with (1) = 2%dsin6 )2 Problem 1:(10 points) Question 1: Which of the following statement(s) correctly identiﬁes possible sources of
magnetic ﬁelds. Time chan ing electric ﬁelds .f . ‘
oPointlike Sgtationary magnetic charges —— éOV“ t E/’( l S ‘27
a Moving pointlike electric charges
I. Time changing magnetic ﬁelds
E. Pointlike stationary electric charges I r ‘5'
F. Moving pointlike magnetic charges ._ 7 Cid“ T 3K L E Answer: Question 2: When a current ﬂows in a wire of length L and cross sectional area A, the
resistance of the wire is .r‘ proportional to the area A and inversely proportional to the length L aproportional to the length L and inversely proportional to the area A
c) inversely proportional to the product of the length L with the area A
d) proportional to the product of the length L with the area A Question 3: Suppose that at the moment shown in the ﬁgure the magnetic ﬂux through
the 100p is decreasing (in the direction indicated by the normal I": ). Is the ycomponent of the electric ﬁeld Ey(x) . —‘ ”‘5 _ ~J “‘4 \
@Ew‘» “it SEAR 7 O
a) greater than (agrarian = [Estamt} Ebmﬂm ‘> 0 the electric ﬁeld Ey (x + Ax). S Q ——
—. EOE M J ﬂ> Ed .90 Question 4: Suppose that at the moment shown in the ﬁgure the electric flux through the
loop is increasing (in the direction indicated by the normal ii). 15 the z componenj—l: of the __5 —' magnetic ﬁeld B (x)
Q greater than %_J B 3:21,!» \iL {GL3
b) equal to c) lessthan E”? l)_z(?(+i§9()‘l' gatlw) FLAZ‘ > O the magnetic ﬁeld 32 (x + Ax) ? Question 5: Coherent monochromatic plane waves impinge on two long narrow
apertures that are separated by a distance (1. Each aperture has width a, with d>>a. @he adjacent interference fringes due to the doubleaperture interference
maxima are closer together than the distance between the adjacent
diffraction minima due to the width of each aperture. b) The adjacent interference fringes due to the doubleaperture interference
maxima are further apart than the distance between the adjacent diffraction
minima due to the width of each aperture. 0) The adjacent interference fringes due to the double—aperture interference
maxima are equal to the distance between the adjacent diffraction minima
due to the width of each aperture. ® Suppose that in the absence of any charges (free space), a plane sinusoidal
traveling wave has an electric ﬁeld of the form Problem 2: (35 points) Traveling Plane Wave and Energy E(z,t) = E“, cos[2ﬂ (z +ct)]l. 7.— a) What direction is the wave traveling? Explain your reasoning, u n
in ”Like, —'i'— JLCV‘GC‘LLO‘R, P“ CVE‘g‘K ..
50th 5i i e S acwq+ C'chegx; '3 mm 3 LC» «1. I
8 o dzicmegc z ~ C. b) Write down a vector expression for the magnetic ﬁeld associated with this wave
in terms of the quantities given in the electric ﬁeld and any additional constants
that you may need. You may ﬁnd the following relationships useful éEx(2,t)=—§t—By(z,t) and %3y(z,t)=%§7Ex(zat)
a at "
BCZ,"L) "—1 .. E90 COS l‘ (3+ 593 h; ”.3 ”\ . x,
EXB tvi ‘ a dunes—item} as
Woluived» c) What direction is the magnetic ﬁeld pointing at time t = 0 on the plane 2 = 0 ? “A “ A
B(Zﬂ> :7 — BSD J
1 leOﬁre C“ x
A .
Recon/Tl WC 5 étVeClww e) Suppose this wave strikes a perfectly absorbin plane Surface of area A aligned in
the x— y plane. What is the time averagea ‘radiation pressure’ that the wave exerts on the surface? '3 Problem 3: (35 points) Electric and Magnetic Fields inside a Wire and the
Paynting Vector A long wire of length 5 and radius a is connected to a constant ideal voltage source AV
(no internal resistance). The long wire has resistance R . (Assume the thin wire in the
ﬁgure below has zero resistance). Throughout this problem, express all your answers in
terms of the given quantities. a) What is the magnitude and direction of the electric ﬁeld inside the wire? Assume
that the current ﬂows uniformly through the wire. E W did/c CW5!“ ES cox/YMQL/ iii—0.23%
H = r“ b) Use Ampere’s Law to ﬁnd the magnitude and direction of the magnetic ﬁeld at a
distance r S a and at a distance r 2 a from the center of the wire? Show all your work (speciﬁcally your choice of Amperian loops for each case). Answers
without work will not receive any credit. Foe (‘43:, “MM \Owgliiowey
éw‘éb 7' £1er : Molten.“ c) What is the direction and magnitude of the Poynting vector S = if) x E on tho
surface of the wire at r = a ? H” .7 :8 __a r F 7777777 3""
5 {,5 uhUJonS ‘5 .L E W7 WW: V f .4 ’ ”WM“:1
030 HA) ’8’: V033! W? d) What is the ﬂux of the Poynting vector, é § 12%, on the cylindrical surface of
I 0 mad
.mrface Kit; Q51 \uode we '3 QT'OJZ
Flux @S M 2 am Ma— ,Q ﬁg :7 M1“ H
__._ 19R (MalRd e) Explain how the ﬂux of the Poynting vector on the surface of the wire is related to
the time rate of change of energy inside the wire. \r ekeStraw Wail»; QWEl/ l/ 19R
‘ 5 . 3 [5 beM5 ccmrned QCCVOSS Suﬁare ‘55", ECU/f $93l2V07/CC/ at “am rate 3—9 we "EAR Problem 4: (35 points) Experiments Part 1: Microwave Experiment The transmitting antenna in the microwave experiment is shown in the ﬁgure below.
A reﬂecting conducting plate is placed on the left and a receiving antenna is ﬁxed in
place to the right of the transmitter (not shown in ﬁgure). Approximate the radiation
generated by the transmitting antenna in the microwave experiment as a plane sinusoidal wave of wavelength 7L =15cm. Assume that the amplitude of the wave is E0 = 2V] m.
The speed of light c=3.0x103m/s. An ideal (zero resistance) conducting plate is a placed behind the transmitting antenna. a) What is the fre uenc of the lane wave? &
1y, 2; _ 300 arm/Sec
(\ "” \, sx\0”‘m 5i
”if 7 Q X )Ooi/see: QXiO \‘iZ— b) What is the closest distance d that an ideal (zero resistance) conducting plate can be placed to the left of the transmitting antenna in order that the reﬂected wave from the conducting plate will interfere constructively with the transmitted wave
hich is ﬁxed in place to the right of c) What is the closest distance that an ideal (zero resistance) conducting plate can be
placed to the left of the transmitting antenna in order that the reﬂected wave from the conducting plate will interfere destructively with the transmitted wave to
produce a minimal signal at the receiver which is ﬁxed in place to the right of the 2&2 %i Hus ides or Thomsmfi/teé . BM N+i§gibﬁ
Lam/04% es lgo° ems: mg... W up» 1 r \ i ”ﬁn,” D ‘ t (“.1
(AT Du: T STA/Wu eLDCSVqQEA‘Jt
(1) Explain how you used the results of parts b) and c) to measure the wavelength of
the radiation. \iﬁeuse é k$QLQVlgi ”Qt/om
Max 1,0 wxlxq me we move velar: 7 Dd; AkstomcQ A“): lac‘54 7* ‘34) 50
(\3 ll‘mg Part II: Interference and Diffraction You are given two narrow long apertures that are separated by a distance a’ . Each
aperture has width a . A plane wave of monochromatic coherent laser light of wavelength
it falls on the apertures. We place the screen so that the distance D to the screen is much
greater than the distance d between the apertures, D >> d . In addition, we assume that
the distance between the apertures d, and the width 0 of each aperture are much greater
than the wavelength of the monochromatic light, 0’ >> l , a >> 7L. a) Let y be the distance between the point P and the point 0 on the screen. If you
ﬁrst ignore the width of each aperture, ﬁnd a relation between the distance y, the wavelength 7L, the distance between the slits d , and the distance to the screen D such that a constructive interference pattern (maximum intensity of light) will
occur at the point P . BAP: Eis‘he =W\(\ ‘
low 5mm 9) tome ewe: A; d( ['33 1‘ M/\ b) Explain qualitatively how the pattern of the intensity of the light on the screen
will change if you take into consideration the effects of the width 0 of each
aperture. Also explain what happens to the interference fringes if the distance
between the apertures is ﬁve times the width of the aperture, 0’ = 5a. TWXTW;:\;\f Wt\\ gGCVF‘US‘e WME’W “Lt/06(—
lzau LUCA/0 accouwi1wk\\ J/ See. cm Hewvtlcpeuiuvu H $2..
"in war»; (in = 8 $0pr Fa'etueg Problem 5: (35 points) Capacitance, Inductance, and the Speed of Light Consider a plane parallel capacitor of plate separation at and plate area A . The capacitor
is fully charged with charge Q on the positive plate. Ignore edge effects. Throughout this problem answers without work shown will receive no credit. Express
your answers in terms of the given quantities. a) Use Gauss’ 3 Law to ﬁnd the direction and magnitude of the electric ﬁeld b
the capacitor pla es. @y show your ewe of~ Gaussian surface. 31; as _,, W450 E“ 3753 ‘ {5 '0“ b) What is the voltage difference across the plates? A V g A c) Find the capacitance C . CT ’iiiv’: 9? A current I ﬂows through each turn of a long air core cylindrical shaped solenoid that
has N turns, length h, and radius a. C!) Use Ampere” 5 Law to ﬁnd the direction and magnitude of them mgnetic ﬁeld
inside the solenoid. ngngrgydg: effects gm WMK e) Calculate the selfinductance L of the solenoid. The capacitor is now connected to the solenoid to form a resonant circuit without any
resistance. The capacitor is initially charged with charge Q on the positive plate, and when the switch is closed the current in the circuit is found to undergo sinusoidal
oscillations with period T. is i) Find an expression for the speed of light in terms of the given quantities T , N , h , a,
d , and A . ...
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