2007年综合试试题

PanPearlRiverDeltaPhysicsOlympiad2007年泛珠三角及中华名校物理竞赛1PanPearlRiverDeltaPhysicsOlympiad20072007年泛珠三角及中华名校物理奥林匹克邀请赛Part-1(Total7Problems)卷-1（共7题）(9:30am–12:30pm,02-26-2007)Q.1(3points)题1（3分）Anairplaneisinitiallyrisingupatspeedv0atanangletothehorizon.Findthetrajectoryoftheplanesuchthatweightlessconditioncanbeachievedintheplane.一架飞机以与水平面成角的初速度v0上升。求飞机以什么样的轨迹飞行，能使飞机里的物体处于失重状态。Q.2(6points)题2（6分）Asshown,twoidenticalweightsarefixedonthetwoendsofauniformrigidrodoflengthL.Theupperweightisrestrictedtomoveonasmoothhorizontalrailandtherodisfreetoswingalongtherail.Themassesoftheweightsandtherodareequal.Findthesmallanglevibrationfrequencyofthesystem.如图所示，两个质量为m的重块分别固定在一根长度为L质量为m的均匀杆两端。上面的重块可以沿光滑的水平轨道滑行，杆可沿轨道方向自由摆动。求整个系统的小角度振动频率。Q.3(6points)题3（6分）(a)AdiscshapedmediumblockofradiusRandthicknessd(R)isuniformlymagnetizedwithmagnetizationMperpendiculartothediscplane.Findthemagneticfieldatpoint-Oonthecentralaxisofthediskandatadistancehfromthecavitycenter.一半径为R，厚度为d(R)的圆盘形均匀磁化介质，磁化强度为M。盘的表面垂直于M。求圆盘中心轴上到圆盘中心距离为h的点O的磁场。(b)AlongandthincylindricalmediumisuniformlymagnetizedwithmagnetizationMalongthecylinderlongaxis.Findthemagneticfieldinsideandoutsidethemedium.一细长圆柱型介质沿柱轴方向均匀磁化，磁化强度为M。求介质里、外的磁场。mmmmmmMO(a)(b)MO(a)(b)PanPearlRiverDeltaPhysicsOlympiad2007年泛珠三角及中华名校物理竞赛2Q.4(5points)题4（5分）Alargeflatdielectricslabofthicknessdanddielectricconstantismovingalongthex-directionatspeedv.Itslargesurfaceplaneisperpendiculartothey-axis.AmagneticfieldofstrengthBisappliedalongthez-direction.Findthesurfaceboundchargedensityonthetwolargesurfacesoftheslab,andtheelectricfieldintheslab.一个厚度为d，介电常数为的大平板以速度v沿X-方向运动。它的表面与Y-轴垂直。Z-方向加有磁场B。求平板两表面上的束缚电荷密度，以及平板中的电场。Q.5(10points)题5（10分）ThespacebetweentwoconcentricconductorsphericalshellsofradiiR1andR3isfilledwithtwotypesofmedia.Thedielectricconstantandtheconductivityofmedium-1andmedium-2are1,1and2,2,respectively.ThevoltagedifferencebetweenthetwoshellsisV0.(a)Incase-A,themediaformtwoconcentricshellswiththeconductorshells,andtheradiusoftheboundarybetweenthetwomediaisR2.Findthefollowing:(i)totalcurrentfromtheinnershelltotheoutershell;(ii)totalfreechargeonthetwoconductorshellsandontheboundarybetweenthetwomedia.(b)Incase-B,medium-1fillstheupperhemisphereandmedium-2fillstheotherhalf.Findthefollowing:(i)totalcurrentfromtheinnershelltotheoutershell;(ii)totalfreechargeontheupperandlowerhalvesofthetwoconductorshells.Case-ACase-B如图所示，两个半径分别为R1和R3的同心导电球壳之间充满了两种介质。球壳之间电势差为V0。介质1和2的介电常数和导电率分别为1,1和2,2。(a)两种介质为与导电球壳同心的球壳，其界面为半径为R2的球面。(i)求两导电球壳间的总电流；(ii)求两导电球壳以及两介质之间界面上的电荷。(b)介质-1填充上半部分，介质-2填充下半部分。(i)求两导电球壳间的总电流；(ii)求两导电球壳上、下部分的电荷。R1R312R1R312R1R2R312R1R2R312xyzxyzPanPearlRiverDeltaPhysicsOlympiad2007年泛珠三角及中华名校物理竞赛3Q.6(12points)题6（12分）(a)Assumethatatmosphereismadeofdiatomidealgasinadiabaticequilibrium.DetermineairpressureP,temperatureTanddensityasafunctionofaltitudeh,providedthattheirvaluesath=0areknown.(Hint:Setupadifferentialequationforathinlayerofairatsomealtitude.111xdxx,where1isaconstant.)(6points)(a)大气可看成绝热平衡下的双原子理想气体。求空气压强P、温度T和密度作为高度h的函数，假定它们在h=0处的值为已知。（提示：对某高度的一薄层气体建立微分方程。111xdxx，1）(6分)(b)Whenthepartialpressureofwatervaporinairexceedsthesaturatedwatervaporpressure(Ps)atagiventemperature,thewatervaporwillcondenseintodropletswhichfalldownasrain.Ps=55.35mmHgat40C,andPs=6.50mmHgat5C.Theair/vapormixturecanbeconsideredasdiatomidealgasandthemassofawatermoleculeisapproximatelythesameasan‘air’molecule.Inthehumidairatsealevelat40Cthewatervaporpartialpressureis90%ofPs.Thedensityofairisρ0=1.18kgm-3at20Cand1.0atm.Thehumidairthenrisesadiabaticallytoanaltitudewherethetemperatureis5C.Ignoreairpressurechangeduetothereductionofwatervapor.(b1)Howmuchraincanonecubicmeterofthehumidairatsealevelgenerate?(5points)(b2)Usetheresultsin(a),findthealtitudewherethetemperatureis5C.(1point)(b)当空气中水蒸汽的分压强超过该温度下的饱和水蒸汽压(Ps)时，水蒸汽将凝聚成滴导致下雨。已知40C时Ps=55.35mmHg，5C时Ps=6.50mmHg。空气/水蒸汽的混合物可当作是双原子理想气体，水分子的质量近似等于‘空气’分子的质量。40C时海平面上的潮湿空气中，水蒸汽分压是Ps的90%。已知20C时，1个大气压下的空气密度ρ0=1.18kgm-3。忽略由于水蒸汽的减少导致的气压改变。该潮湿空气绝热上升到某一高度，该处温度为5C。（b1）一立方米海平面上的潮湿空气能够产生多少雨？(5分)（b2）用（a）的结果，求温度为5C处的高度。(1分)PanPearlRiverDeltaPhysicsOlympiad2007年泛珠三角及中华名校物理竞赛4Q7(8points)题7（8分）(i)FindthetorqueonanelectricdipolepinauniformelectricfieldE.(1point)(ii)AmediumisuniformedpolarizedwithpolarizationPbyanelectricfieldE.Findthetorquepervolumeonthemediumexertedbytheelectricfield.(1point)(iii)Anelectromagneticwave()000()ikztEExyeispropagatingalongthez-axisinanisotropicmedium.InsuchmediumtherelationbetweentheelectricdisplacementDandEisgivenby0DE,soDandEarealwayspointinginthesamedirection.Findthetorquepervolumeonthemediumexertedbytheelectromagneticwave.(1point)(iv)Anelectromagneticwave12000()ikzikzitEExeyeeispropagatingalongthez-axisinananisotropicmedium.Insuchmediumtheelectricdisplacementis120000()ikzikzitxyDExeyee,soDisnotparalleltoE.Notethat1xkcand2ykc,wherecisthespeedoflightinvacuum.Findthetime-averaged(overoneperiod)torquepervolumeonthemediumexertedbytheelectromagneticwave.(3points)(v)Following(iv),findthetime-averagedtotaltorqueonasecti