Pearson Physics Level 30 Unit VII Electromagnetic Radiation: Chapter 14 Solutions

Transcription

1 Pearon Pyic Level 30 Unit VII Electromagnetic Radiation: Cater 14 Solution Student Book age 704 Concet Ceck Te colour of te tar are a firt indication of te temerature of teir urface. Te brigt red tar, Aldebaran (in Tauru), a a urface temerature of about 4000 K, werea te brigt blue tar, Vega (ig overead in te ummer ky), a a urface temerature of nearly K. Student Book age 706 Concet Ceck From te imle form of Planck' formula, E f, ubtitute f c equation reult in E. c. Combining tee Examle 14.1 Practice Problem 1. Given f = Hz oton energy (E) To find te energy of te oton, ubtitute into Planck formula. E nf E (1)( J )( ) J A oton of frequency Hz a an energy of J.. Given = 555 nm = m oton energy (E) To find te energy of te oton from it wavelengt, ubtitute into Planck formula. Pearon Pyic Solution Unit VII Cater 14 1 Coyrigt 009 Pearon Education Canada

2 E nf nc (1) J m J A oton of wavelengt 555 nm a an energy of J. 3. Given 15.0 ev 15.0 ev in joule If 1 ev = J, ten 19 J ev J ev 15.0 ev i te ame a J. Examle 14. Practice Problem 1. Given 8 Student Book age nm m frequency (f) To find frequency, ubtitute into te univeral wave equation. c f m Hz A 10-nm oton a a frequency of Hz. Ti oton i a ard or dee UV oton.. Given 10 nm 16 f = Hz (from quetion 1) energy (E) Ue E nf and te given value for f. Pearon Pyic Solution Unit VII Cater 14 Coyrigt 009 Pearon Education Canada

3 34 16 E (1) J Hz J A 10-nm oton a an energy of J. 3. Given nm m E = 10 J number of oton (n) Find f for te oton. c f m Hz Ue E nf to find te energy er oton, and divide te total energy (10 J) by ti number E = ( J )( ) 19 = J E nf E n f 10 J J You need oton of green ligt (550 nm) to deliver 10 J of energy. Examle 14.3 Practice Problem 1. Given n = λ = 400 nm = m energy (E) Subtitute into Planck formula. c E nf were f Student Book age 708 Pearon Pyic Solution Unit VII Cater 14 3 Coyrigt 009 Pearon Education Canada

4 nc E (1000) J m J Te beam deliver J of energy.. Given P = 10 W = 10 J/ nm m number of oton er econd (n/) c Ue E nf n, were E = 10 J. Solve for n uing te equation E n. c J 7 10 ( m ) n 34 ( J ) oton/ You need nm oton er econd to deliver 10 W of ower Ceck and Reflect Knowledge 1. Given 7 λ = 450 nm = m energy of te oton (E) Ue te equation E nf, were n = 1. Student Book age 710 Pearon Pyic Solution Unit VII Cater 14 4 Coyrigt 009 Pearon Education Canada

5 E nf c 34 ( J ) m J 19 Te energy of te 450-nm oton i J.. Given E = 15.0 ev wavelengt ( ) c Ue te equation E nf, were f. Recall tat 1 ev = J. c E n nc E 34 (1)( J ) (15.0 ev ) J ev m 8 Te wavelengt of a 15.0-eV oton i m. 3. Given 7 = 300 nm m 7 = 600 nm m oton energy (E) nc Ue te equation E nf, were n = 1. E J J m Pearon Pyic Solution Unit VII Cater 14 5 Coyrigt 009 Pearon Education Canada

6 J E m J 19 Te 300-nm oton a an energy of J and te 600-nm oton a an 19 energy of J. Te 300-nm oton i terefore twice a energetic a te 600- nm oton, or E 300 = E (a) Given 5 E = 100 kev = ev frequency (f) Convert energy from kev to J J kev ev 14 ev J Ue te equation E = nf, were n = 1. E f J 34 J Hz Te oton a a frequency of Hz. (b) From Figure 14.6, ti oton i a gamma ray. Alication 5. Given P = 100 W = 100 J/ 7 λ = 550 nm m t = 10.0 number of oton emitted (n) E c P and E nf n t, o nc E Pt. Solve for n. Pearon Pyic Solution Unit VII Cater 14 6 Coyrigt 009 Pearon Education Canada

7 Pt n c J m J In 10.0, a 100-W ligt bulb emit oton. 6. Given P Sun = 1400 W/m 7 ave = 700 nm m number of oton er econd er quare metre (n) P E t P 1400 W Area 1m W/m Let Δt = 1 and area = 1 m. 3 E J nf c n Solve for n. E n c 3 7 ( J )( m ) J A 1-m area receive aroximately oton eac econd. Extenion 7. Given n = oton/ d = 10d 1 number of oton (n) Te ligt will read out over a ere. A = 4πr = urface area of a ere. Since you are increaing r by a factor of 10, te area over wic te oton are read increae Pearon Pyic Solution Unit VII Cater 14 7 Coyrigt 009 Pearon Education Canada

8 by 100. Te number of oton emitted by te tar remain te ame but i read over a larger area. So, te number of oton received eac econd will dro by a correonding factor of 100 or oton eac econd. 100 You will receive 100 oton er econd from te more ditant tar. 8. Given P = 100 W n = 500 oton ditance (d) average wavelengt ( ave ) area of ligt bulb (A) Te coice for wavelengt will vary. Ue λ = 600 nm. A tyical ligt bulb a a radiu of m, o let r i = m. number emitted Te number of oton er quare metre 4ri To find te number of oton emitted by te ligt bulb, ue te equation E nf. E E n f c 7 (100 J )( m ) n J Ti value give te denity of oton, σ. n Te denity of te oton 4 r oton 4 (0.030 m).7 10 oton/m Te area of te uman eye = πr uil Te number of oton received by te eye = (denity of oton)(area of eye): N = σπr uil A quick Web earc reveal tat te radiu of te uil (r uil ) i 3.5 mm in te dark. Ue a value for a dark adated eye for a bulb tat i jut dicernible. N = σπr uil oton m m oton Pearon Pyic Solution Unit VII Cater 14 8 Coyrigt 009 Pearon Education Canada

9 Once you know ow many oton are entering te eye wen it i at te urface of te N bulb, move to a ditance r f, were 500 oton. Ue te ame reaoning ued 4rf in quetion 7 to determine te ditance at wic your eye would detect 500 oton. N varie directly a te quare of te ditance (ee roblem 7), o ri N rf Nri f 500 r r f 500 i Nr 500 ( oton)(0.030 m) 500 oton m m 1300 km Ti olution ugget tat you could ee a 100-W ligt bulb from more tan 1300 km away! Ti reult would only be oible in a comletely dark environment, wit no aborbing dut or aze. A more robable figure i in te ten of kilometre becaue te nigt ky i never comletely dark. Anoter comlication i te curvature of Eart, wic become a factor for ditance longer tan 10 km. Ti roblem raie many quetion. Student Book age 715 Concet Ceck (to) Potoemiion will occur only if te incident oton a an energy greater tan te work function for te metal. Ti relationi can be exreed a E oton f W. Concet Ceck (bottom) Te collector late i given a oitive carge o tat it will attract otoelectron tat ave been knocked free from te metal urface by incident oton. Even witout a carge on te collector late, ome otoelectron would migrate down te tube and reac te collector, etabliing a very weak electric current. Examle 14.4 Practice Problem 1. Given E = J Student Book age 716 Pearon Pyic Solution Unit VII Cater 14 9 Coyrigt 009 Pearon Education Canada

10 toing otential (V toing ) E k max = qv toing Ek V max toing q J C 3.3 J/C 3.3 V Te toing otential required i 3.3 V.. Given E = J J in ev 19 1eV J 19 1eV J 3.3 ev J J i equal to 3.3 ev. 3. Given V toing = 3.1 V maximum kinetic energy of electron E k max Ek qv max toing 19 ( C)(3.1 V) J or (1 e )(3.1 V) 3.1 ev 19 Te maximum kinetic energy of te electron i 3.1 ev or J. Student Book age 717 Concet Ceck Relate te general exreion f qvtoing W to Planck' contant,, by noting tat qv toing i zero wen f i te treold frequency, f 0. Ti mean tat Pearon Pyic Solution Unit VII Cater Coyrigt 009 Pearon Education Canada

11 qv qv f W f 0 toing f 0 toing W W f 0 0. Examle 14.5 Practice Problem 1. Given nm m work function (W) W f0 c J m 19 1 ev J J Student Book age ev and Verify Te metal require oton of at leat.59 ev for otoemiion to occur. Ti value i reaonable given te data in Table Given 7 = 410 nm m W =.10 ev kinetic energy of te otoelectron (E k ) Pearon Pyic Solution Unit VII Cater Coyrigt 009 Pearon Education Canada

12 Ek Eoton W f W c W J m J J 19 1 ev J J 19 J.10 ev ev 0.93 ev Te otoelectron leave wit a kinetic energy of J or 0.93 ev, wic rereent te energy left after liberating te otoelectron. Student Book age 719 Examle 14.6 Practice Problem 1. Given E k =.1 ev eed of te otoelectron (v) Convert energy unit from electron volt to joule. 19 J Ek.1 ev ev J E 1 k mv Ek v m An electron a a ma of kg. 19 ( J) v kg m/ Te electron i moving wit a eed of m/ or 860 km/.. Given W =.10 ev (from Table 14.1) 7 λ = 400 nm = m kinetic energy of te otoelectron (E k ) Pearon Pyic Solution Unit VII Cater 14 1 Coyrigt 009 Pearon Education Canada

13 Ek Eoton W c W J ev 7 19 J m ev ev.10 ev 1.01 ev Te otoelectron a a kinetic energy of 1.01 ev. 3. Given 19 E k = 1.01 ev = J ev maximum eed of te otoelectron (v) m e = kg Ek 1 mev Ek v m e J ( J) kg m/ Te otoelectron i moving wit a maximum eed of m/. 14. Ceck and Reflect Knowledge 1. Given 7 λ = 400 nm m energy of te oton, in ev (E) Student Book age 70 Pearon Pyic Solution Unit VII Cater Coyrigt 009 Pearon Education Canada

14 E nf, were n1 c J m J 19 J ev 3.11 ev Te energy of te 400-nm oton i 3.11 ev.. Te work function rereent te amount of energy required to remove an electron from te metal. Te frequency of te oton i directly related to it energy by Planck formula, E = nf. So, te oton frequency mut be great enoug to make te roduct of Planck contant and te oton frequency greater tan or equal to te work function, uc tat E = f W. 3. Given cadmium treold frequency (f 0 ) From Table 14.1, te work function for cadmium i 4.07 ev. W f 0 19 J W 4.07 ev ev J W f J f J Hz and Verify Te oton mut ave a frequency of Hz, wic mean tat it wavelengt i 305 nm in te UV range. Ti anwer make ene becaue te work function i quite large and ence require an energetic oton. 4. Given ceium W.10 ev 7 λ = 500 nm = m to determine weter otoemiion occur Pearon Pyic Solution Unit VII Cater Coyrigt 009 Pearon Education Canada

15 Te kinetic energy of a otoelectron i given by te equation E k = f W. Kinetic energy mut be oitive, o otoemiion will occur if f W > 0 or f > W. 19 J W.10 ev ev J Ek f c J m J Terefore, f W and otoemiion occur. and Verify Ceium a a relatively low work function, o it i reaonable tat a blue-green oton of wavelengt 500 nm can caue otoemiion. E qv 5. kmax toing Ek V max toing q ev J ev C 1.5 J/C 1.5 V 6. Te toing otential of te electron varie directly a it maximum kinetic energy. 7. Fale. Increaing te intenity of te ligt ource a no effect on te maximum kinetic energy of te electron and ence on te toing otential. Te kinetic energy of electron deend on teir frequency, not on te intenity of te ligt. Pearon Pyic Solution Unit VII Cater Coyrigt 009 Pearon Education Canada

16 Alication 8. Convert wavelengt to frequency uing te equation c f. Wavelengt (nm) Frequency (Hz) Kinetic Energy (ev) 500 c 0.36 f m Hz Hz Hz Hz Hz Hz Hz 3.05 Te gra ow a traigt-line relationi (y = mx + b) between f and E k. Ti relationi el ow tat E = f W. kmax 9. Given te gra contructed in quetion 8 loe (m) loe = rie run Wen comaring te equation Ek f W wit te general equation y = mx + b, te loe of Ek f W ould equal Planck contant. Ek m f ev ev 15 Te loe i ev. Pearon Pyic Solution Unit VII Cater Coyrigt 009 Pearon Education Canada

17 A exected, te loe of ti gra i cloe to te value for Planck contant. 10. To determine te metal ued, find te work function, W, by noting were te y- intercet of te E k f gra occur. From E k = f W, wen f 0, Ek W. A negative kinetic energy imly mean tat ti amount of energy i required to jut eject an electron from te metal. If you rearrange te equation to Ek W 0, ten you can argue tat te negative of te y-intercet i numerically equal to te work function. Te y-intercet of te gra i.1 ev. Te y-intercet i te ame a te work function. So, te metal in quetion a a work function of (.1 ev) =.1 ev. From Table 14.1, te metal wit ti work function value i ceium. Extenion 11. According to te oton model of ligt, te energy of te oton arrive a a acket and i aborbed intantly (or very nearly o) by te metal. If te incident oton doe not carry enoug energy to dilodge an electron, te incident energy i quickly tranferred by colliion between te electron and atom of te metal, tereby eating te metal. It i only wen te incident oton deliver ufficient energy for te electron to overcome te work function for te metal urface tat emiion will occur. Te energy of te incident oton deend only on te oton frequency, and not on te intenity of te incident ligt. 1. Te otoelectric effect i at te eart of many common alication. Even toug few of tee alication conit of electron being emitted from a metal urface into a vacuum camber (a reented in mot text), tey involve electron and te abortion or emiion of oton. One examle i otovoltaic cell, wic are made of emiconductor tat convert oton from te Sun into electric current (electron flow). Potovoltaic cell are ued in olar owered calculator and olar owered arking meter. Anoter common alication of te otoelectric effect i te carge couled device (CCD) tat form te eart of mot digital camera. Ligt trike te ixel of te CCD and caue te releae of electron tat accumulate on te ixel. Te electron create an electric otential tat i eventually recontructed a a digital icture. 13. Given P = W A beam = m E = W = 3.5 ev time taken for an atom to aborb enoug energy to emit an electron (Δt) Te energy in te beam i read over an area of m. An individual atom can only aborb a tiny ortion of ti energy. Firt calculate ow muc energy er econd an individual atom receive. Te atomic radiu i aroximately m. Te urface area of an atom i terefore aroximately 10 0 m. Tu, any given atom only aborb m.0 10 W.0 10 W 4 10 m or.0 10 J/. Determine ow long it take to aborb 3.5 ev. Pearon Pyic Solution Unit VII Cater Coyrigt 009 Pearon Education Canada

18 Since E = P atom Δt, E t P atom 19 J 3.5 ev ev J or 0.78 According to claical teory, otoemiion i a low roce: It would take minute to our for an atom to aborb enoug energy to releae an electron. Student Book age 71 Concet Ceck Momentum i inverely related to wavelengt, o te -nm oton a te greatet momentum J m N J 9 10 m N Concet Ceck Student Book age 7 1. Refer to te diagram above. Reolve f into x and y comonent: Pearon Pyic Solution Unit VII Cater Coyrigt 009 Pearon Education Canada

19 f f co co x f f f in in y f. From te original condition, te net momentum in te y direction wa zero. Hence, to conerve momentum in ti direction, it follow tat te electron mut move in te y direction wit equal and ooite momentum. To conerve momentum in te x direction, te um of te x-direction momentum of te oton and te electron mut equal te original momentum in te ytem. [Note: In a formal derivation of te Comton effect, you mut ue te relativitic concet of 4-momentum, wic i beyond te coe of te ig cool curriculum.] 3. Becaue te cattered X ray a a longer wavelengt, it a lot energy. Te miing energy a been given to te electron, according to te law of conervation of energy. Te electron final energy can be written a: 1 1 E ( ff fi) c f i Student Book age 73 Examle 14.7 Practice Problem 1. Given 9 = 10 nm = m energy (E) c E f J E m J 17 Te energy of te 10-nm X ray i.0 10 J.. Given 9 = 10 nm = m momentum () J m N Pearon Pyic Solution Unit VII Cater Coyrigt 009 Pearon Education Canada

20 6 Te momentum of te X ray i N. 3. Given 9 λ i = 10 nm = m 9 λ f = 11 nm = m cange in energy (ΔE) c 1 1 E, o E c i f 34 E J m J 18 Te electron gain J of energy m Examle 14.8 Practice Problem 1. Given 11 i = nm = m θ = 90 wavelengt of cattered oton f m e = kg f i (1 co ) mc f i (1 co ) mc Student Book age J f m (1 0) kg m 0.01 nm Te wavelengt of te cattered oton i 0.01 nm. Pearon Pyic Solution Unit VII Cater 14 0 Coyrigt 009 Pearon Education Canada

21 Student Book age Ceck and Reflect Knowledge 1. Given 7 λ = 500 nm = m momentum () J m N A 500-nm oton a a momentum of N.. Given 3 A B greatet momentum () Te momentum of a oton i inverely related to wavelengt:. A A B B A B But A 3 B. A B B 3 B 1 A B 3 Poton A a one-tird te momentum of oton B. 3. Given = kg m/ wavelengt (λ) energy (E) Pearon Pyic Solution Unit VII Cater 14 1 Coyrigt 009 Pearon Education Canada

22 J kg m/ m c E c 1 8 ( kgm/)( m/) J Te oton a a wavelengt of m and an energy of J. 4. Te oton a a wavelengt of m or nm. From Figure 14.6, ti wavelengt i in te ard X ray or gamma ray art of te electromagnetic ectrum. 5. Ti tatement i fale. Pyicit believe tat te law of conervation of momentum and of energy old in all cae! Tey would rater give u a teory tan eiter of tee two fundamental law. Alication 6. Given E = 100 kev wavelengt (λ) 19 Convert 100 kev to joule: 1eV J ev J, o ev J c Ten olve for wavelengt uing te equation E. c E J J m nm An X ray of energy 100 kev a a wavelengt of nm. 7. Given 11 λ = nm = m θ catter = 180 Pearon Pyic Solution Unit VII Cater 14 Coyrigt 009 Pearon Education Canada

23 eed of elium nucleu (v) Te ma of a elium nucleu i kg. Ue Comton cattering equation, f i (1 co ), to calculate te final mc wavelengt of te X-ray oton. f i (1 co ) mc f i (1 co ) mc J m (1 co 180 ) 7 8 ( kg)( m/) m Ue Planck formula, E = nf, were n = 1, to find te energy lot by te X ray. E ( ff fi) 1 1 c f i J m/ m m J From te law of conervation of energy, te energy lot by te X ray i te energy gained by te elium nucleu. Ue te equation for kinetic energy, E 1 mv, to find te final velocity of te elium nucleu. 1 E mv E v m He 18 ( J) kg m/ Te X ray catter off te elium nucleu and te elium nucleu recoil wit a velocity of m/. Pearon Pyic Solution Unit VII Cater 14 3 Coyrigt 009 Pearon Education Canada

24 Extenion 8. In order to ee a mall article, you mut cooe a mall wavelengt. However, te maller te wavelengt, te greater i te momentum of te oton and, becaue of Comton cattering, te greater i te interaction between te oton and te article. For ti reaon, te maller te article, te more difficult it i to ee. Student Book age 78 Examle 14.9 Practice Problem 1. Given 11 λ = nm = m momentum () J m kg m/ 3 An X-ray oton of wavelengt nm a a momentum of kg m/. Examle Practice Problem 1. Given v = m/ wavelengt Te ma of a roton i kg. Auming non-relativitic eed, mv J 7 5 ( kg)( m/) m Te de Broglie wavelengt of te roton i m or nm.. Given 7 = 40 nm = m Pearon Pyic Solution Unit VII Cater 14 4 Coyrigt 009 Pearon Education Canada

25 eed (v) 31 Te ma of an electron i kg. Auming non-relativitic eed, mv v m J 31 7 ( kg)( m) m/ An electron tat a a de Broglie wavelengt of 40 nm i moving wit a eed of m/. Student Book age 73 Examle 14.1 Practice Problem 1. Given l = 1.0 nm maximum wavelengt l Te wavelengt and lengt of te box are related by te equation n. n For maximum wavelengt, n = 1. l 1 1 (1.0 nm) 1.0 nm Te maximum (lowet) wavelengt for te electron i.0 nm.. Given 9 =.0 nm =.0 10 m momentum () J m kg m/ Pearon Pyic Solution Unit VII Cater 14 5 Coyrigt 009 Pearon Education Canada

26 5 Te momentum of te electron in te box i kg m/. 3. Given l = 1.0 nm minimum energy (E min ) For minimum kinetic energy, ue te equation E. m From quetion, te momentum of an electron in a box of lengt 1.0 nm i kg m/. 31 Te ma of an electron i kg. E m 5 (3.310 kg m/) 31 ( kg) J Te minimum energy for an electron confined to a box 1.0 nm in lengt i J. Student Book age 733 Concet Ceck Te maller te region, te maller i te oible wavelengt of te confined article. If you reduce te ize of te region from m acro to m acro, te wavelengt of te article would decreae by a factor of Momentum i inverely related to wavelengt, o te momentum would increae by te ame factor. Since energy varie directly a momentum quared, te energy would increae by a factor of Tyical energie on te atomic cale are meaured between electron volt and a few tent of electron volt, o energie on a nuclear cale are meaured from MeV to GeV Ceck and Reflect Knowledge 1. Given v = m/ wavelengt 31 m e = kg Student Book age 736 Pearon Pyic Solution Unit VII Cater 14 6 Coyrigt 009 Pearon Education Canada

27 mv J 31 ( kg)(0 000 m/) m 36.4 nm Te electron de Broglie wavelengt i 36.4 nm.. Given 7 = 500 nm = m momentum () J m kg m/ 7 A 500-nm oton a a momentum of kg m/. 3. Given 9 x 1.0 nm m te uncertainty in te momentum of te article x x J m kg m/ 6 Te uncertainty in te momentum of te article i kg m/. 4. Given 1 x.5 10 m te uncertainty in te electron momentum Pearon Pyic Solution Unit VII Cater 14 7 Coyrigt 009 Pearon Education Canada

28 Treat te ere diameter a Δx. x x J m kg m/ Te electron momentum i no le tan kg m/. You could ubtitute ti anwer into te equation E to find te minimum energy of ti electron. m Alication 5. (a) Given V = V energy (E) Te electric field doe work on te electron (W = qv), wic increae te electron kinetic energy. E 1 k mv qv 19 q C 19 Ek ( C)(1 000 V) J An electron accelerated troug a V otential acquire an energy of J. (b) Given 15 E k J wavelengt Ek m me k Pearon Pyic Solution Unit VII Cater 14 8 Coyrigt 009 Pearon Education Canada

29 Terefore, mek J ( kg)( J) m An electron of ti energy a a wavelengt of m. 6. From and = mv,. Te ratio i mv m v e me v me m Te wavelengt varie inverely a te ma. Subtitute m e = kg and m = kg kg kg e e 3 or Te wavelengt of an electron moving wit te ame velocity a a roton i aroximately 000 time larger tan te correonding wavelengt of te roton. Extenion 7. No, a tate of no motion at abolute zero i imoible in quantum yic. No motion imlie tat te article a exactly zero momentum and ence no uncertainty in momentum. But, if you know were a article i, ten, by definition, you know tat te uncertainty in it oition mut be no bigger tan te ize of te region it occuie. Te article cannot ave zero uncertainty in it momentum becaue x. 4 No motion i terefore an imoible tate in quantum yic. 8. Given n En, n 1,,3, 8ml Derive te given exreion. Pearon Pyic Solution Unit VII Cater 14 9 Coyrigt 009 Pearon Education Canada

30 Ue te concet of (i) wavelengt and tanding wave in a one-dimenional box, (ii) te connection between wavelengt and momentum,, and (iii) te connection between E k and, were Ek. m l n n l n n l Ek m n l m n 8ml Te energy of a article of ma m traed in a box of lengt l i quantized according n to te exreion En, were n 1,, 3,... 8ml 14.5 Ceck and Reflect Student Book age 740 Knowledge 1. Te bet coice i (a): Te double-lit exeriment illutrate te formation of node and antinode an interference attern wic demontrate tat ligt a wave-like roertie.. (a) Fale: Electron can alo form node and antinode wen beamed troug a double lit. (b) True: Te Davion-Germer exeriment owed tat electron can form a erie of node and antinode wen reflected off a nickel urface. Alication 3. (a) Ti examle illutrate te article nature of a quantum becaue te electron tat it te oor creen are localized in ace and time. (b) Ti examle illutrate te wave nature of a quantum becaue te effect i tat of wave interference. (c) Ti examle illutrate te article nature of a quantum becaue ligt i localized in a mall oint at an intant of time. Pearon Pyic Solution Unit VII Cater Coyrigt 009 Pearon Education Canada

31 Extenion 4. If Planck contant canged to 6.63 J, ten you would ave a muc larger wavelengt! For examle, if you ave a ma of 66.3 kg and walk at a eed of 1.00 m/, ten your momentum i 66.3 kg m/. Since, your wavelengt would 6.63 J be 6.63 kg m/ m. You would diffract into a erie of node and antinode a you aed troug te doorway! Student Book age Cater 14 Review Knowledge 1. Claical yic redicted tat a truck matc would emit increaing amount of ligt at orter wavelengt. Ti effect would mean tat a flame from a matc would emit more blue ligt tan red ligt, more UV ligt tan blue ligt, more X ray tan UV ray, and o on. Ti increaing amount of orter-wavelengt, and ence igerenergy, ligt would lead to te aburd rediction tat a ingle matc would incinerate te univere!. Planck formula i E = nf, were n i te number of oton. According to ti equation, a oton of ligt a an energy tat i equal to Planck contant time te frequency of te ligt. Wen n = 1, Planck formula become E = f. 3. Given 7 = 450 nm = m energy (E) c E f J m J 19 1 ev J J.76 ev 19 J ev A 450-nm oton a an energy of J or.76 ev. 4. Energy varie directly a frequency and inverely a wavelengt. If an X-ray oton a a wavelengt 100 time maller tan a viible ligt oton, ten it frequency, and ence it energy, will be 100 time greater tan tat of te viible ligt oton. 5. Heinric Hertz i credited wit dicovering te otoelectric effect in Pearon Pyic Solution Unit VII Cater Coyrigt 009 Pearon Education Canada

32 6. Eintein rovided te exlanation for te otoelectric effect by making te radical uggetion tat ligt wa beaving like a article. 7. Given 14 f Hz work function (W) Ek max f W Wen te metal i illuminated wit oton at te treold frequency, electron will jut be emitted. Ek 0 f W 0 W f W ( J )( ) J J 19 J ev.5 ev If te treold frequency for otoemiion from a metal urface i Hz, ten te work function for te metal i J or.5 ev. 8. Te Comton effect i te cattering of X ray wen tey interact wit electron or oter article. Te article model of ligt wa able to redict and exlain te mall cange in wavelengt in te cattered X ray and te relationi between te cange in wavelengt and te cattering angle (not redicted by te claical teory). Te momentum and energy of te cattered X ray canged exactly a if te X ray were a mall article undergoing an elatic colliion wit an electron. Te Comton effect tereby rovided critical evidence for te article model of ligt. 9. Given nm = 90 cange in wavelengt Ue te equation (1 co ). mc 31 m e = kg J (1 co 90 ) 31 8 ( kg)( m/) m nm Pearon Pyic Solution Unit VII Cater 14 3 Coyrigt 009 Pearon Education Canada

33 Te oton undergoe a wavelengt increae of nm wen it catter 90 from an electron. 10. Given = N wavelengt J N m 73 nm An electron wit momentum N a a wavelengt of 73 nm. 11. Given 7 = 100 nm = m momentum () J m N A 100-nm UV oton a a momentum of N. 1. No, te article cannot be at ret. Becaue it i confined to a mall region, you know it oition wit a relatively mall uncertainty. According to Heienberg uncertainty rincile, owever, tere mut be a correonding very large uncertainty in te article momentum: Te article could be moving very fat and ave coniderable kinetic energy. Alication 13. Given P = 1.0 W = 1.0 J/ 7 = 600 nm = m t = 1 number of oton (n) Pearon Pyic Solution Unit VII Cater Coyrigt 009 Pearon Education Canada

34 E P t nf t c n t For t = 1, nc P P n c Te number of oton emitted eac econd i J m n J oton/ A 1.0-W flaligt emit oton of 600-nm ligt in one econd. 14. Given 7 = 300 nm = m W = 1.88 ev maximum kinetic energy of te otoelectron E k max Ek max f W c W J J 1.88 ev m ev J J J ev.6 ev Te maximum energy of te electron emitted by te metal urface i J or.6 ev. 15. Given 5 V = 100 kv = V Pearon Pyic Solution Unit VII Cater Coyrigt 009 Pearon Education Canada

35 electron wavelengt Ue te equation. Relate to E k by noting tat te electron acquire kinetic energy from work done by te accelerating electric field, defined by te equation E k = qv. Ek m mek mqv Terefore, mqv J ( kg)( C)( V) m nm and Verify Te wavelengt of a 100-keV electron in te microcoe i nm, wic i more tan time maller tan viible ligt. For ti reaon, an electron microcoe a a muc iger magnification, and i conequently caable of owing muc greater detail, tan a ligt microcoe. 16. (a) Given m = 0.15 kg v = 40 m/ wavelengt mv J (0.15 kg)(40 m/) m 34 Te ball wavelengt i m. Pearon Pyic Solution Unit VII Cater Coyrigt 009 Pearon Education Canada

36 (b) You can afely ignore quantum effect becaue any uncertainty in te oition of te ball due to quantum effect will be inignificant te wavelengt of te ball i only in te order of m! 17. Given d = 100 m P = 100 W = 100 J/ d uil = mm number of oton entering your eye eac econd (n) E P t nf t c n t Etimate tat te oton ave an average wavelengt of 500 nm. Ten, determine te number of oton emitted eac econd. For t 1, nc P P n c J m J oton/ From a ditance of 100 m, tee oton will be read over an area A = 4πr. So, te denity of oton er quare metre i 0 n.5110 oton/ 4r 4 (100m) oton//m Your eye a an area of πr = π(1 mm) = π(0.001 m) = π 10 6 m. Te number of oton entering your uil eac econd i oton//m 10 m.0010 oton/ oton/ Tere are about ix billion oton entering your eye eac econd. 18. Given P = 00 kw = W = J/ f = 90.9 MHz = Hz 9 Pearon Pyic Solution Unit VII Cater Coyrigt 009 Pearon Education Canada

37 number of oton emitted eac econd (n) E P t nf t For t 1, P nf P n f n J J oton/ Te FM radio tation emit oton eac econd. 19. Given 9 l = 0.85 nm = m tree lowet oible energie for te electron (E n ) n Ue te equation En. Subtitute n = 1,, 3, and olve. Alternatively, olve for 8ml E 1 and ten ue E n = n 31 E 1. Te ma of an electron i kg. 34 (1) ( J ) E ( kg)( m) J J 19 J ev 0.5 ev E = () E 1 =.1 ev E 3 = (3) E 1 = 4.7 ev and Verify Te lowet tree energie for te article in te 0.85-nm box are 0.5 ev,.1 ev, and 4.7 ev. Te de Broglie wavelengt for te article mut be a tanding wave, o te only oible value of n are wole-number integer. 0. Given 5 E = 100 kev = ev momentum () Pearon Pyic Solution Unit VII Cater Coyrigt 009 Pearon Education Canada

38 c Ue te equation E f and. c E c c E c 5 19 J ev ev m/ kg m/ A 100-keV X-ray oton a a momentum of kg m/. Extenion 1. Poton carry momentum. Wen a urface aborb or reflect oton, te momentum of te oton cange. Becaue ti cange occur over time, te oton mut exert a force, uc tat F. If you calculate te average force over t F an area, ten te oton mut exert reure, becaue P. A. (a) Given A = 1.0 km F = 10 N E = 1.4 kw/m number of oton er quare metre (n) Cooe 550 nm a te average wavelengt of te oton. c n E P t t For t 1, nc P P n c n 3 J m /m J oton//m 1 Te number of oton from te Sun i oton//m. Pearon Pyic Solution Unit VII Cater Coyrigt 009 Pearon Education Canada

39 (b) Given A = 1.0 km F = 10 N momentum of eac oton () Eac oton a a wavelengt of 550 nm and carrie a momentum of J m kg m/ Te oton are reflected, o kgm/.4110 kg m/ 7 Te momentum of eac oton i.4110 kg m/. (c) Given A = 1.0 km F = 10 N te force tat unligt roduce on te ail (F) From oton momentum, calculate te reure te oton exert on te ail, er quare metre. F P A t A For t = 1, n P, were n i te number of oton er econd calculated in (a). A 7 1 (.4110 kg m/)( ) P 1m N/m Ue your anwer for reure to calculate te force on te ail, uing te equation F P. A Pearon Pyic Solution Unit VII Cater Coyrigt 009 Pearon Education Canada

40 F PA 6 N m m 9.3 N and Verify Te yicit atent claim i valid! Your calculation ow tat te ail get a net u of 9.3 N from te Sun. Ti value i in good agreement wit te claim tat te trut ould be about 10 N. 3. Given 8 f = 100 MHz = Hz 6 I = 1.0 A = A V = 10 mv = V number of oton er econd (n) Ue te connection between electrical energy, current, and otential difference, E = VI. Ti energy come from te oton interacting wit te antenna. E nf VI VI n f 6 ( V)(1.010 A) ( J )( ) J J Te minimum number of oton tat your radio receiver can detect i in te order of (a) Eintein wa objecting to te central claim of Heienberg uncertainty rincile, wic tate tat you cannot know bot te oition and momentum of a article at te ame time wit unlimited reciion. Heienberg uncertainty rincile imlie tat quantitie uc a oition and momentum ave a robabilitic nature. Eintein could not accet tat, at te quantum level, we can, at bet, determine te robability tat a article will be at a articular location or ave a articular momentum. (b) Eintein tatement i ironic becaue e wa te one wo uggeted te quantum (article) model of ligt te firt examle of te wave-article duality and one of te tenet of quantum indeterminacy. Pearon Pyic Solution Unit VII Cater Coyrigt 009 Pearon Education Canada