8(4 m0) ( θ ) ( ) Solutions for HW 8. Chapter 25. Conceptual Questions
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1 Solutios for HW 8 Captr 5 Cocptual Qustios 5.. θ dcrass. As t crystal is coprssd, t spacig d btw t plas of atos dcrass. For t first ordr diffractio =. T Bragg coditio is = d so as d dcrass, ust icras for t coditio to b satisfid. But icrass as θ dcrass. Hc tr will b a dcras i t agl of icidc. 5.. (a a > b > c bcaus t rgy pr poto dpds oly o t frqucy so = f = c /. T sallr wavlgts corrspod to igr frqucis. (b Nc > Nb > Na bcaus t powrs ar qual, tr ust b or potos w t rgy pr poto is lss T rgy of a poto is giv by = f = c /. Trfor t ratio of rgis is, c / = c / = = 5.5. Fast lctros will av a sortr wavlgt ladig to lss diffractio spradig ad bttr rsolutio Bcaus 8L 9 dcrasd by a factor of 4. So t w = 0 J. = w s tat for a giv, is ivrsly proportioal to L. If L is doubld t is It is t sa, or.0 0 J. = = = H H H 80L0 L0 8(4 0 xrciss ad Probls 5.4. Modl: T agls of icidc for wic diffractio fro paralll plas occurs satisfy t Bragg coditio. Solv: T Bragg coditio is d cos θ =, wr =, For first ad scod ordr diffractio, rspctivly Dividig ts two quatios, d = ( cos = ( d θ ( θ ( = θ = cos cos = cos cos68 = 4
2 5.7. Modl: T agls corrspodig to t various diffractio ordrs satisfy t Bragg coditio. Solv: T Bragg coditio is d =, wr =,, 3, givs t ordr of diffractio. T axiu possibl valu of is t ubr of possibl diffractio ordrs. T axiu valu of is. Tus, w td to fid t valu of for t liitig valu of. d ( 0.80 d ( W ca obsrv up to t fourt diffractio ordr. = = = = Modl: Us t poto odl of ligt. Solv: T rgy of a poto wit wavlgt is =. f = c Siilarly, = c. Sic is qual to, c c = 600 = = = 300 Assss: A poto wit = 300 as twic t rgy of a poto wit = 600. Tis is a xpctd rsult, bcaus rgy is ivrsly proportioal to t wavlgt. 5.. W d to kow t prcis otu to obtai t D Brogli wavlgt. Howvr tis probl oly asks t ordr of t valu of wavlgt of a uc assiv body wigig i kgs. Assuig a stiat of your ass big, say, 70 kg ad your vlocity is /s. Tus, your otu is p = v (70 kg( /s = 70 kg /s. Your d Brogli wavlgt is Js = = p 70 kg /s Wic is ifact too sall to sow ay wavlik proprtis lik diffractio tc Visualiz: W'll ploy quatios 5.8 ( = / p ad 5.9 rgy. Solv: First solv quatio 5.9 for p: p =. ( p / J s = = = =.0 p 9. 0 kg.4 0 J 3 9 ( ( = to xprss t wavlgt i trs of kitic Assss: T rgy giv is about.5 V, wic is a rasoabl aout of rgy. T rsultig wavlgt is a fw to a fw doz tis t siz of a ato Modl: Modl t 5.0-f-diatr uclus as a box of lgt L = 5.0 f. Solv: T proto s rgy is rstrictd to t discrt valus wr =,, 3, For =, 34 ( Js 7 5 ( ( = = = (.36 0 J 8L kg J.36 0 J J =, for =, ( = =, ad for = 3, 3 = 9 =. 0 J. W ca obsrv tat t rgy of t uclos i a ato is of t ordr of MV wic is uc largr tat tat of a lctro i t ato wic is of t ordr of V.
3 5.. Modl: Us t poto odl of ligt. Solv: (a T wavlgt is calculatd as follows: ( Js ( /s c gaa = f = = = J (b T rgy of a visibl-ligt poto of wavlgt 500 is T ubr of potos suc tat gaa = visibl is ( Js ( /s c visibl = = = J = = = 3 gaa visibl J J 5.. Modl: Us t poto odl. T itt M radiatio(potos av t sa frqucy as t trasittig ata. Trfor t rgy of a 000 khz poto is ( ( poto = f = Js Hz = J T rgy trasittd ac scod is 0 0 J. T ubr of potos trasittd ac scod is 0 0 J/ J =
4 5.5. Modl: Us t poto odl of ligt ad t Bragg coditio for diffractio. Solv: T Bragg coditio for t rflctio of x-rays fro a crystal is d cos θ =. To dtri t agls of icidc θ, w d to first calculat. T wavlgt is rlatd to t poto s rgy as = c. Tus, Fro t Bragg coditio, Likwis, θ ( θ ( Js ( /s c = = = J 0 ( ( = cos = cos = cos ( θ = cos ( = 7.6 d 0. 0 = cos = 50.8 ad θ = Not tat θ ( 4 = cos is ot allowd bcaus t caot b largr ta. Tus, t x-rays will b diffractd at agls of icidc qual to 8.7, 50.8, ad Modl: Particls av a d Brogli wavlgt giv by = p. T wav atur of t particls causs a itrfrc pattr i a doubl-slit apparatus. Solv: (a Sic t spd of t utro ad lctro ar t sa, t utro s otu is p = v = v = v = p wr ad ar t utro s ad lctro s asss. T d Brogli wavlgt for t utro is p = = = p p p Fro Sctio. o doubl-slit itrfrc, t frig spacig is y = L / d. Tus, t frig spacig for t lctro ad utro ar rlatd by y = y = y = = = kg ( µ.67 0 kg (b If t frig spacig as to b t sa for t utros ad t lctros, kg 3 y = y = = v = v = (.0 0 /s. 0 /s 7 = v v.67 0 kg Modl: lctros av a d Brogli wavlgt giv by = p. T wav atur of t lctros causs a diffractio pattr. Solv: T widt of t ctral axiu of a sigl-slit diffractio pattr is giv by quatio.: 34 (.0 ( Js (.0 0 ( 9. 0 kg (.5 0 /s L L L w = = = = = = a ap av
5 5.3. Modl: lctros av a d Brogli wavlgt giv by = p. Visualiz: Notic tat a scattrig agl φ = 60 corrspods to a agl of icidc θ = 30. Solv: quatio 5.6 dscribs t Davisso-Grr xprit: D ( θ Dsi θ =. Usig = v, w av Js D = = = = v si θ 9. 0 kg /s si ( ( ( si =. Assuig =, tis quatio siplifis to Modl: A cofid particl ca av oly discrt valus of rgy. Solv: (a quatio 5.4 siplifis to 34 ( Js 3 9 ( ( = = = 9 (.3 0 J 8L kg Tus, = (.3 0 J ( =. 0 J, ( ( J J, (b T rgy is = J. 0 J = J. 8 = = ad 3 =. 0 J. 9 (c Bcaus rgy is cosrvd, t poto will carry a rgy of = J. Tat is, ( Js ( /s c c = = f = = = = J poto Modl: A particl cofid i a o-disioal box as discrt rgy lvls. Solv: (a quatio 4.4 = = =,, 3, 4, L 8L p for t = stat is 34 ( Js ( ( = = = 3 8L kg J 64 T iiu rgy of t Pig-Pog ball is = J. (b T oly for of rgy tat t ball ca av is cosidrd to b t kitic rgy of t ball. Trfor t spd is calculatd as follows: 64 ( J = J = v = ( 0 0 kg v v = = /s kg
6 5.39. Modl: A particl cofid i a o-disioal box of lgt L as t discrt rgy lvls giv by quatio 4.4. Solv: (a Sic t rgy is tirly kitic rgy, = = =,, 3, 4, L 8L p = = = v v = =,, 3, K 8L L (b T first allowd vlocity is Js 3 9 v = = 9. 0 kg ( ( For = ad = 3, v = /s ad v 3 = /s /s p
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