Name: Page 1 Physics 263: Electromagnetism and Mol1:ernPhysics Sections0101-0105 Final Exam Prof J.J. Kelly Instructions:. This closed book, closed notes exam contains 8 problems to be completed in 2 hours. You may use a basic scientific calculator, but no other aids are permitted. The final page provides a brief compilation of possibly useful information, including relevant physical constants.. Work each problem in the space provided. If additional space is needed, use the back of the previous page and indicate that you have done so.. Please write your name on each page, including this one. Do not use red ink.. Explain your reasoning and show your work. Partial credit will be awarded when the relevant physical principles are applied even if mistakes are made in execution of the steps. However, correct guesses without any explanation may be penalized.. Algebraic answers must have consistent dimensions and numerical answers must have consistent units.. Honor pledge: please copy and sign "I pledge on my honor that I have not given or received any unauthorized assistance on this examination." Pledge: Signature: Printed name: :;:(9 ItA+; orls Student ID number: Section: Problem Score Problem Score 1 5 2 6 3 7 4 8 total Physics 263 Final Exam Prof J.J. Kelly
Name: Page 2 1. A long cylindrical shell with inner radius a and outer radius b carries total current I along its length. a) Find expressions for the magnetic field in the regions (i) r < a, (ii) a < r < b, and (iii) r > b. Sketch the function B(r).,. ~ E :: r~-i1l ;;;ii r B~ = ~o :r: lej?~/os~) ~ /1o'T:. c$ b,2.-tt 2. Cl~ ~ b [ 1- b~r-- E9 G:. I b (J'A~ (4... ~ ~ tb- -fj.~ CffJ~ 61- v'"rt' l ~ 0 r~{j, yo- b) Evaluate the pressure assuming that the shell is thin, such that a ::::b ::::R. Is the pressure inward or outward? [Hint: consider the force upon a narrow strip of current running the length of the cylinder.].mor (5 pts) ~ is - - :J-rr1(B I> ;; AD r -/ 9- ~n-r I... - _I F;:: ~ 1: I L d e -==='/F:;:, A () 1:"t L 01.l :;l1rr 1r "d.'\tr A -:; :ltfr..l ::::.~ ~ I ~ ~ ~ p ~ /t-o ~;:;R) r / / L { c) Compute the pressure for R =0.2 cm and I =10 A. (5 pts) p :::: 0" 7q6 0!- yyl7- Physics 263 Final Exam Prof J.J. Kelly
Name: Page 3 2. Interference and diffraction a) A beam of monochromatic light of wavelength Ao in vacuum impinges upon two slits of width a and separation d and forms a diffraction pattern upon a distant screen. A thin sheet of transparent material, such as glass or plastic, with thickness b and refractive index n is then inserted behind the top slit only. Derive an expression for the angular shift of the resulting diffraction pattern. Does the pattern shift up or down? Explain your reasoning carefully with the aid of a clear diagram. (I 0pts) ~~cjt~ ~--rl~ ~ (h\..-+.l.~ ~ ct ~ A ~ r'1~ I ~ ~ ~ ~ ~ pc;~"t;{,1' ~ rj'? I) ~ ~ I ~:::b/c/q"q& Ar:I>~ a; L())-~)) J-'=~/4"O81, 0, n~6/-::: ~e 1 JAJ~ ~. I -, I" ' 'L~ v' --? ~ e -=,J I - n-"1.~.. e = y\ y\'l- ~ v) ~. c.. I ) - - u-<- /.1~ (\ J = ~ ), b (;:r \ (\,.,..~~. -, & ~e ik~~~ ~~~~. ~ A rj; ~ A rp - ~ e1 ~ t9 J ~, I ~o \~ ~ ~ ~~"t~~ ~ ~ ~ ~ \LJfil-t.+e 1. '2.. ~i\o n-. -I-~ ) - - ~ e0 ~ ~~~'tl ~~J ~~ ~~la-fl'!j1'1.~~ ~'",;l:rr b ( A~ ~ ~ a~ ~ ~o ~.,. 0 VI r1 J '2. ri'1.-1 I ~J i.,!- -- n.-i.\&0'-+ Qe + \-)'\-=-0 =-";> Be ::",+1 [ ~) ~ ~ ~ - b I->.'f -0 -,. all b 0 ~~~~~ &o~o {.r..n~l. b) Use diffraction theory to estimate the maximum distance at which you can resolve the two headlights of an approaching car. (I 0 pts) '" 't:!. IN --- ~ rr; -:0: ~ ~ ~ Ae -: J. ~a. D:: L ).. ~ 'U 1m"", =:. ~ ~~ L ~ WD - I. J..::( )., rv q~ '>- rv S- $""'0 1'\I"'V\ Physics 263 Final Exam Prof JJ Kelly
Name: Page4 3. A coil consisting of N circular turns with area A is connected to an external circuit with resistance R. A uniform magnetic field B is directed perpendicular to the coil. The coil is now turned over, thereby reversing the magnetic flux through it. a) Derive an expression for the total charge Q that flows through the external circuit as the coil is flipped. Measurement of this charge can be used to determine the magnetic field strength. t:=:. - J "(fb dt 1Y.~rs:: J-.NA8 '".. (Q := R dq dt ~~ ~NA~,I ~ Q='R A~ 8 R ~~ b) Compute Q for B=l T, N=100, A=5 crn2,and R=0.2 Q. Q ::. (), t(;; C.
Name: Page5 4. A parallel-plate capacitor consists of two circular plates with radius R separated by distance d. A steady current I charges the capacitor. Use idealized electric and magnetic fields, neglecting fringing effects near the edges. a) Find an expression for the Poynting vector at the edge of the capacitor. From what direction does energy enter the capacitor? :r t E~ & ~~E~> ~.() -- 1:.. ~R ~ Eo ~ -?~='> ~;rrb =jk,, o ~ E:::)<0I. S~E~B ~ ~~~~.,.); II C! - :Ao - - - - Afo ~~R - ~~~E)B ~~~ s::- Q. J: IC- I:- ~ 2. 3 ). rr R. Eo J;. E! t ~ s b) Find an expression for the rate at which the energy stored in the capacitor increases. Compare this result with the electromagnetic energy flux into the capacitor. ~ ~~~fu.o.. '2 U -::;. to E E 7':. R c1 = I? (Tt fa. ~ E 1:.'"I...t r R"-) eo lid E 2. V -=. d- '2. n.rd. ::: 't:1) ~ -<- 1::. co"2j T'" f"'" a. ;;J- J:"'-t d - i... - ~ R ED ~V;t ~.-j t/v~~ ~ ~~. I _2-1-.J l}. -:::: L "l "'1 :::- (AE. 7T R to i< :::- ~ ~n-r.d
Name: Page 6 5. Phase shift for a series RLC circuit. a) Use a phasor diagram to write an expression for the phase shift 8 between the current and voltage in a series RLC circuit as a function of the driving frequency w. 'tl:::: wl \ 'l-.c-.:::: cue 'A ~ ~::::. '!:: - y.. c.. - R 1..-!... ~ c.,.,0 -= Lc.. R r - L l.. <.. 0.) - CNc.) 'cv r {1,.iti ~ ~+== - :t \ 0 G...1 -"- w ~ == 2:: ~ 1: r l..j~ ;.! r +,.jr 1. 't- 'f C,:)b'l. - - ;;t IZL 4::J C-c..>+ - w- -:::::r ~ ~ ~~"'-<-c. Gu-:::-LU D ~ ~d==o ~~ '1-[ R ') ; "- ::;;:.1" '- "f- '" < - - ;;~ - -~ L. c b) Sketch the function 8(w) and determine the resonant frequency in terms of R, L, and C. Identify the high and low frequency limits and determine for each whether the current leads or lags the voltage. Explain.. ~~~~. fi~-. 1...1. /- " U-fY'~ ~ ~... ~kr~, \.Io\to.~e 1~I1c1s - - -,- - - - - - fill c u. v-r n. -r \ e. (A").~ - - \ ~t5, \ I \ -( - -- ii/-;,~ '. W -Ti'J - - - - r'-t +- I ~ --- -Ti" c...c>o L r ( ( I ~ I c.w '::> c...:>o ') ~.,. D c.u <. UJo ~ S I.. () R ~ t;..:z:. \ ~~~ ~~~k~ Physics 263 Final Exam Prof J.J. Kelly R
Name: Page7 6. A charged kaon is an unstable particle that has a rest mass of mc2 =493.6 MeV and a proper lifetime of 1.24x 10-8s. Suppose that kaons with kinetic energy of 800 MeV are created in a nuclear physics experiment. a) What is the momentum of the kaons in MeV/c? (5 pts) '2.. 1.."2- liy\c7-) -:::: (K+fYlc'-) - (pc.) ;:.pc= [K(K-+ ~h1g~)j/"2...~. F:= J/N, M~ t-/fc b) What is the average distance that these particles travel in the laboratory before decaying? v 2. 2... K (5 pts) E =,,+ /'YIC = cyme :; d-'= I + - = 2. (pl. h1c'l ~ J -'l~ h Y :::: [I - (t) ~ f::: (( - d'-'-) =- D. q 24 - -If,1 ~~ -') t =:~"'t' =.:l.6~>< J.2.t.f-xlD S'..::: ;;g."2-~~/o.s: " 01 = vi =: 9 D.~....,. or: 0::::l{\J t' =:: ~ 1: c) The dominant decay mode produces a muon with mass 105.7 MeV/c2 and a massless neutrino. Compute the neutrino energy and the muon kinetic energy in the kaon rest frame. \""Y'\ C.2.. -:::: E 4- E.u K ~ P := p~ ~ E-.)/e-./'..'2. -'- "2.. (, 2.. ~ 2- l~c') '" c? - ~ ~ ~ = \,jake - f:.cj - (P.,c) '2..... (Y"\'"... - \"Y\ /A "2- v - C - ";)."3 S" tv1(.. V p (. - ~ fy\~ - -. /3.v -:::::.<."3S- IV'<=.,V!(~ =- Is-~ Me--tJ Physics 263 Final Exam Prof J.J. Kelly
Name: Page 8 7. Suppose that a particle of mass m is found in a harmonic oscillator potential V (x) = t kx2 where k is a positive constant. a) Show that with an appropriate choice for b the wave function tfi(x) =exp( - X2/2b2) is a solution to Schrodinger's equation and express its energy in terms of the classical oscillation frequency (0. How do you know that this is the ground state? (10 pis) 11/ -~/~b"2. "- (//1- - >< II) I =- e = ) T - );,- r II I U.J 'f...if,- - 1. - '><2.. 1/" Ij.J= - I:~ r - /;7.-'r - b7..' ( :r ) r '" I L. \ If.! I -k 2 ;:::..E -J-2- II II, d. ~ - 1:\ - / 1 - \) r -r i 7- (~ - b:2-lb~ - --- '-( + d- m it h,11 V, ;;:.. -"I; \ k. ;(m ~ b -= -,< ~~~~~'I-~\;-n\ ~7.._: ~ E:;; - b '2-' Ij~ ~fy'\. :. 6 ~ ~ \ rc 1- L f." lv,-x;/.- W ::::J «-1m ~-tt. E= ~f'\'-v. ~~ ~tij..~ ~:r ~~~) fi1h ~ -cl. ~~. b) Sketch the wave functions for the lowest three states of the harmonic oscillator and explain their basic features. I~V - vi.=:2.. - n := I - - - - - n-=ij - )( Zlf.<.. ~~ c <- -- ~ ~ -/::L ~~.~ ~ ~ ~.,.V ~ ~ ~ ~ pi., ~ ~~~ wv7l E<V, ~< ~ ~J ~ ~~, i. ~ Y'-~~<. ~~~ w-oj..~~)i~e. ~ct.-~. Physics 263 Final Exam Prof J.J. Kelly
Name: Page 9 8. Describe two phenomena that classical physics failed to describe correctly near the end of the nineteenth century and the new ideas that explained those phenomena and contributed to the development of the early quantum theory. (10 pts each)
Name: Page 10! ~ Co dc'pb y&.e=-d( - ~ -- D = ce = foe + P. - 1 f r Possibly useful information! ~ Q ydli.e=- static: E=- 4nco dqr2 c E2 u=~+- B2 2 2/10 1 1 1 - = (n-1)(---) f RJ R2 fdli.b=o! - y&.b=/1o ( I+cod( - ~ ~ ~ B = I.1H= /1o(H +M). ~ /1 static: B f ldsxr =---2... - 4n r2 S = ExB//1o 1 1 1 -+-=s0 s. I f dc'pe J. 1(0)=10 ( sma smgle slit: a 7Z'a a =-sin 0 A X' = y(x - ut) t' = yet - ux) c2 Y = (1- (7 Y ) -x. J 2. 1(0) = I sm N[3 grating: J( sin[3 ) 7r.d f3 =-sin 0 A E = ymc2 P=ymv (mc2) =E2 - (pcy 2 112 --Ij/'(x) + V(X)If/(X) = EIf/(x) 2m
Name: Page 11 2 1 c =- col1o Co= 8.8542x1O-12F m e = 1.6022 x 10-19C me = 9. 109x 10-31kg = 0.511MeV I c2 m c =2.9979x1O8 - S 110 =4nx1O-7 ~ A N A =6.022x1O23 mole-i mp =1.672 X 10-27 kg =938.27 MeV I c2 118= 9.27x1O-24 { T I1N =5.05x1O-27 { T h = 6.626 X10-34J. S nc =197.3eV run n = 1.055 X 10-34 J e2 1 =- 4nconc 137. s Dimensions and SI Unitsfor Basic Electromagnetic Quantities Quantitv Unit Abbreviation Conversions Dimensions Charge Coulomb C Q Electric potential Volt V JIC ML2T-2Q-I Electric field Voltlmeter VIm NIC MLT-2Q-I Capacitance Farad F CN M -I L-2T2Q2 Current Ampere A Cis QT-1 Magnetic field Tesla T N I(A m) MT-1Q-I Magnetic flux Weber Wb JIA ML2T-1Q-t Inductance Henry H J/AL ML2Q-2 Resistance ohm Q VIA ML2T-1Q-2