第23届全国部分地区大学生物理竞赛试题

**********************************************************************************************232006.12.10lPQPABl/4Ql/4PQSPCDABl/4PQSl/41lB600µ2M/3h1+h2v1S2-S1pV=Ah1h2v2S17.f(T)bcWp0V0Qp0V0cm8.6T052TAAT00Amcab2V0V0VO00p2ppc2**********************************************************************************************9.lABCD3kl/2A′B′C′A′B′ABl/4B′C′BCl/4A′B′C′RI=A′B′B′C′A′C′A′C′ε=SR10.APcArrtdtdrrdtdr=nnλ11.oeAB12.LABCDSxyvxBCxSyxl=LABL′BCLv=0.6cxLL′=4.l/4l/4ABDCA′B••••′••••C•••••••′•SRPArABCDvxOSy1352m1m2a1a1a2αm1m2ααa1a24**********************************************************************************************uT4pT5551Q2Q3Q4Q5Q1Q1Q2Q4Q5Q31Q5Q2QQ236**********************************************************************************************Nt0()tNtNeλ−=λN0/2τλτ/2ln=M=1g→⎯⎯⎯→⎯⎯⎯⎯→⎯→==RnRaTh2208666.32248891.12289021dyττTh22890Ra224881t=0NTh228901NRa2248822t0NRa224882t;3Th22890Ra22488*2N)()(xQyxpdxdy=+⎟⎠⎞⎜⎝⎛+∫∫=∫−CdxexQexydxxpdxxp)()()()(710B17A1718171917)1()(22202rnrnα−=0αxOx12max2rα0n0=r0φ0φxr~O0φxrxOφ08**********************************************************************************************AV011Vn+1VnVn-1nBAnnnVRRVVV⎟⎟⎠⎞⎜⎜⎝⎛=+−−+112RARARARA21Vnn1αnneVV−=0α32RBRA10µmV0=30mVV(x)3RARBVnVn+1V2VN-1V1VNV0RBRBRBRBRBRB11235RARBV0V32xx=0ln(V/V0)0-13-2-3x(µm)01002003004005006007008009001000919BKmqq0t=00vKvKvfKKγ−=γ(1)t0vs(2)2/πv*10231222481l/4,vl2130.1,0316.0103=−R04πε,R02πε63321=,2112gh,0C,A,2ln212ln2−3263,⎟⎠⎞⎜⎝⎛3263lnRkl82010cRrrRc22−11,)(4eonn−λ12)1(2ββ+−,31112=−−ββ,cv=β1313m1)(amTTm1cos:111=−α2⎩⎨⎧−==−)cos(cossinsin:2122222ααααaamTamTgmm)3()2(22m2m1v2=v142aaa==124123)cos1(cos,sinsin,)cos1(2221ααααα−==−=−amTamTgmamTTamam12)cos1(coscos1=−−ααα5ααααsinsincoscos1222amamgm=⋅−−65⎩⎨⎧==++−0/01cos)2(cos212mmkkαα[][]⎪⎪⎩⎪⎪⎨⎧+−++++=1)4()2(211)4()2(21coskkkkkkα[]{}⎪⎩⎪⎨⎧=+−+=0/)4()2(arccos2121mmkkkkα36[{⎪⎩⎪⎨⎧+−+===)4()2(arccoscot2121kkkgaaαα]}2m2aT1•α•Ta1a211413ν2chmνν=3nνννννννuhncmnp3131312===5νννhnu=νuupp3131===∑∑νννν3∑∑==νννννhnuu4Tu∝4Tp∝2151311Q12Q2021=+QQ,0,04332=+=+QQQQ054=+QQ[]52155432211))()(QQQQQQQQQQQ+++=++++++++……(…Q∑==+5151iiQQQ)=,(21QQ),(32QQ),(54QQ),(51QQ51QQ1Q2∑===515121iiQQQ72)(21154322QQQQQQ−+++=)(21215433QQQQQQ6−−++=Q4Q5Q321613dtteNtNλ−=0)(dttNdteNdNt)(0λλλ==−−1tThRaN1(t)N2(t)dtN2(t)2211221122ln,2ln,)()(τλτλλλ==−=dttNdttNdN202=dN19066.336591.1)()(211221=×===ττλλtNtN1221191NNNN=+=ThRaTh21NNg228=µ21231064.21002.62281×=××==ANMNµ1922111038.1191,1064.2×==×=≈NNNN22t=0t0Th1teNtN111)(λ−=RaN2(t)dtdttNdttNtdN)()()(22112λλ−=teNtNtNdttdN11111222)()()(λλλλ−==+2⎥⎦⎤⎢⎣⎡+−=⎥⎦⎤⎢⎣⎡+∫∫=−−−−∫CeeNeCdteeNetNtttdttdt2122121121112)(λλλλλλλλλλ22)0(NN=11212NNCλλλ−−=3()ttteNeeNtN221211212)(λλλλλλ−−−+−−=221/2//12122212121)(ττττττtttNNtN⎟⎠⎞⎜⎝⎛+⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛−⎟⎠⎞⎜⎝⎛−=3Th1τ=tRa2121/2/121212*2212121)(ττττττττ⎟⎠⎞⎜⎝⎛+⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛−−==NNNN21ττ21NN181112*21094.638012×===NNNττ231710r⎟⎠⎞⎜⎝⎛−=⎟⎠⎞⎜⎝⎛−φπφπ2sin)(2sin00rnn2φdrdxφφcos)(cos00rnn=φ0O0002202coscostanφφφnnndxdr−==1x)(sincoscos220220022022202002202rnnrnnnnαφφαφ−=−−=−02202cossinφαφrdxdr−=2x0220220222020222020222coscossinsincossincosφαφαφαφφααφφαrrrrdxdrrrdxrd−=−−−=−−=0cos02222=+φαrdxrd20=x0=r⎟⎟⎠⎞⎜⎜⎝⎛=xAr0cossinφα100tanφ==xdxdrαφ0sin=AOx01cosφαπ=xφ0απ=1x1Oφ0απ=1xxxx1x2x13x1…141810Vn-11++=nnnIiIAnnRVV1+BnAnnRVRVV1−−+=−⇒nBAnnnVRRVVV⎟⎟⎠⎞⎜⎜⎝⎛=+−−+11222VαnneV−=0BARRee+=+−2αα⎥⎥⎦⎤⎢⎢⎣⎡−⎟⎟⎠⎞⎜⎜⎝⎛+±⎟⎟⎠⎞⎜⎜⎝⎛+=⇒422212BABARRRReααe1α⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎥⎥⎦⎤⎢⎢⎣⎡−⎟⎟⎠⎞⎜⎜⎝⎛++⎟⎟⎠⎞⎜⎜⎝⎛+=42221ln2BABARRRRα331000lnln001)(VVxVx=xVeVxV⎟⎠⎞⎜⎝⎛−=1000ln00)(221nαnneVV−=010xn=nVnVnxneVeVxVV⎟⎠⎞⎜⎝⎛−⎟⎠⎞⎜⎝⎛−====100ln0101000ln01000)(034.010030ln100ln0===⇒VαBARRee+=+−2αα3102.12−−×=−+=ααeeRRBA3RARBRBVnRAVn+1In+1Ininin+1519101vfγ−=mvadtdvγ−==∫∫−=tvvdtmvdv00γtmevvγ−=⇒03∫∫−−=⇒==ttmstmdtevdsevvdtds0000γγ⎟⎟⎠⎞⎜⎜⎝⎛−=−tmevmsγγ1032tt+dtqBmv=ρdθvdtd=θρdtmqBdtvd==ρθtmqB=θ2πθ=⇒qBmt2*π=qBtmevevv20*0*πγγ−−==46