国际物理林匹克竞赛试题Theoretical_Exam_Solution_2_English

IPHO2003TheoreticalQuestion2(Solution)(2003/07/17)p.1/5Solution-TheoreticalQuestion2APiezoelectricCrystalResonatorunderanAlternatingVoltagePartA(a)RefertoFigureA1.Theleftfaceoftherodmovesadistancevtwhilethepressurewavetravelsadistanceutwith/Yu.ThestrainattheleftfaceisuvtutvS(A1a)*1FromHooke’slaw,thepressureattheleftfaceisuvuvYYSp(A1b)*(b)Thevelocityvisrelatedtothedisplacementasinasimpleharmonicmotion(orauniformcircularmotion,asshowninFigureA2)ofangularfrequencyku.Therefore,if)(sin),(0tuxktx,then)(cos),(0tuxkkutxv.(A2)*ThestrainandpressurearerelatedtovelocityasinProblem(a).Hence,)(cos/),(),(0tuxkkutxvtxS(A3)*)(cos),()(cos),(),(002tuxkkYtxYStuxkuktxuvtxp(A4)*-------------------------------------------------------------------------------------------------Alternatively,theanswersmaybeobtainedbydifferentiations:)(cos),(0tuxkkuttxv,)(cos),(0tuxkkxtxS,)(cos),(0tuxkkYxYtxp.-------------------------------------------------------------------------1Anequationsmarkedwithanasteriskcontainsanswertotheproblem.putt=0t/2ppFigureA1vttppkxtvx0FigureA2IPHO2003TheoreticalQuestion2(Solution)(2003/07/17)p.2/5PartB(c)Sincetheangularfrequencyandspeedofpropagationuaregiven,thewavelengthisgivenby=2/kwithk=/u.Thespatialvariationofthedisplacementisthereforedescribedby)2(cos)2(sin)(21bxkBbxkBxg(B1)Sincethecentersoftheelectrodesareassumedtobestationary,g(b/2)=0.ThisleadstoB2=0.Giventhatthemaximumofg(x)is1,wehaveA=±1and)2(sin)(bxuxg(B2)*Thus,thedisplacementistbxutxcos)2(sin2),(0(B3)(d)Sincethepressurep(orstressT)mustvanishattheendfacesofthequartzslab(i.e.,x=0andx=b),theanswertothisproblemcanbeobtained,byanalogy,fromtheresonantfrequenciesofsoundwavesinanopenpipeoflengthb.However,giventhatthecentersoftheelectrodesarestationary,allevenharmonicsofthefundamentaltonemustbeexcludedbecausetheyhaveantinodes,ratherthannodes,ofdisplacementatthebisectionplaneoftheslab.Sincethefundamentaltonehasawavelength=2b,thefundamentalfrequencyisgivenby)2/(1buf.Thespeedofpropagationuisgivenby33101045.51065.21087.7Yum/s(B4)and,giventhatb=1.0010-2m,thetwoloweststandingwavefrequenciesare)kHz(27321buf,)kHz(81823313buff(B5)*-------------------------------------------------------------------------------------------------[AlternativesolutiontoProblems(c)and(d)]:Alongitudinalstandingwaveinthequartzslabhasadisplacementnodeatx=b/2.Itmayberegardedasconsistingoftwowavestravelinginoppositedirections.Thus,itsdisplacementandvelocitymusthavethefollowingform)]2(sin)2([sincos)2(sin2),(utbxkutbxktbxktxmm(B6)tbxkutbxkutbxkkutxvmmsin)2(sin2)]2(cos)2([cos),((B7)where=kuandthefirstandsecondfactorsinthesquarebracketsrepresentwavestravelingalongthe+xand–xdirections,respectively.NotethatEq.(B6)isidenticaltoEq.(B3)ifwesetm=±0.IPHO2003TheoreticalQuestion2(Solution)(2003/07/17)p.3/5Forawavetravelingalongthe–xdirection,thevelocityvmustbereplacedby–vinEqs.(A1a)and(A1b)sothatwehaveuvSanduvp(wavestravelingalong+x)(B8)uvSanduvp(wavestravelingalong–x)(B9)AsinProblem(b),thestrainandpressurearethereforegivenbytbxkkutbxkutbxkktxSmmcos)2(cos2)]2(cos)2(cos[),((B10)tbxkuutbxkutbxkutxpmmcos)2(cos2)]2(cos)2([cos),((B11)Notethatv,S,andpmayalsobeobtainedbydifferentiatingasinProblem(b).ThestressTorpressurepmustbezeroatbothends(x=0andx=b)oftheslabatalltimesbecausetheyarefree.FromEq.(B11),thisispossibleonlyif0)2/cos(kbor,5,3,1,2nnbffbukb(B12)Intermsofwavelength,Eq.(B12)maybewrittenas,5,3,1,2nnb.(B13)Thefrequencyisgivenby,5,3,1,22nYbnbnuuf.(B14)ThisisidenticalwiththeresultsgiveninEqs.(B4)and(B5).---------------------------------------------------------------------------------------------------(e)FromEqs.(5a)and(5b)intheQuestion,thepiezoelectriceffectleadstotheequations)(EdSYTp(B15)EdYSYdTpTp)1(2(B16)Becausex=b/2mustbeanodeofdisplacementforanylongitudinalstandingwaveintheslab,thedisplacementandstrainSmusthavetheformgiveninEqs.(B6)and(B10),i.e.,withku,)cos()2(sin),(tbxktxm(B17))cos()2(cos),(tbxkktxSm(B18)whereaphaseconstantisnowincludedinthetime-dependentfactors.Byassumption,theelectricfieldEbetweentheelectrodesisuniformandIPHO2003TheoreticalQuestion2(Solution)(2003/07/17)p.4/5dependsonlyontime:htVhtVtxEmcos)(),(.(B19)SubstitutingEqs.(B18)and(B19)intoEq.(B15),wehave]cos)cos()2(cos[tVhdtbxkkYTmpm(B20)ThestressTmustbezeroatbothends(x=0andx=b)oftheslabatalltimesbecausetheyarefree.Thisispossibleonlyif=0andhVdkbkmpm2cos(B21)Since=0,Eqs.(B16),(B18),and(B19)implythatthesurfacechargedensitymusthavethesamedependenceontimetandmaybeexpressedastxtxcos)(),((B22)withthedependenceonxgivenbyhVdYbxkkbdYhVdYbxkkYdxmTpTpmTpTmp)]1()2(cos2cos[)1()2(cos)(222(B23)*(f)Attimet,thetotalsurfacechargeQ(t)onthelowerelectrodeisobtainedbyintegrating),(txinEq.(B22)overthesurfaceoftheelectrode.Theresultis)]1()2tan2([)]1()2tan2()[()]1()2(cos2cos[)(1),()(1)()(2202202200