※ [本文轉錄自 NTU-Exam 看板] 作者: yaujack (大頭) 看板: NTU-Exam 標題: [試題] 94下 蘇德潤 普通物理甲下 期末考 時間: Fri Jun 23 20:46:34 2006 課程名稱︰普通物理學甲下 課程性質︰機械系必修 課程教師︰蘇德潤 開課系所︰物理系 考試時間︰94-06-23 是否需發放獎勵金： 是 （如未明確表示，則不予發放） 試題 : 1. (a) The energy function of a free particle is E = K + V = p^2/2m + V(x) What is the momentum p of a matter wave in terms of the wavelength λ? In turn, what is p as a function of the angular wave number k = 2π/λ? What is the angular frequency ω of this E so that the wave function is ψ is proportional to exp(-iωt). Thus the energy function or the Hami- ltonian function is in form of E = E (k,x) = (h^2 k^2 / 8 π^2 m) + V (x) = H (k,x) (b) What is then the Schrodinger equation Hψ = Eψ when we replace k^2 = -d^2 / dx^2. The Schrodinger equation is a differential equation. 2. For a long solenoid of cross-sectional area A, length l, carryling a current i, the energy stored per unit volume is u = U / Al Since U = 1/2 Li^2 for inductance L. What is (a) u in terms of n,i? (b) u in terms of B at any point where the magnetic field is B. Hint: for a solenoid L = (μ_o)Aln^2. 3. In Figure (in the problem of 28-7 in the textbook), an electron accelerated from rest through potential difference V_1 = 1.00kV enters the gap between two parallel plates having separation d = 20.0mm and potential difference V_2 = 100V. The lower plate is at the lower potential. Neglect fringing and assume that the electron's velocity vector is perpendicular to the electric field vector between the plates. In unit-vector notation, what uniform magnetic field allows the electron to travel in a straight line in the gap? 4. In Figure (in the problem 68 of Ch33), two light rays pass from air through five layers of transparent plastic and then back into air. The layers have parallel interfaces and unknown thicknesses; their indexes of refraction are n_1 = 1.7, n_2 = 1.6, n_3 = 1.5, n_4 = 1.4, and n_5 = 1.6. Ray b is in- cident at angle θ=20 defrees. Relative to a normal at the last interface, at what angle do (a) ray a and (b) ray b emerge? Hint: Solving the problem algebraically can save time.) If the air at the left and right sides in the figure were, instead, glass with index of refraction 1.5, at what angle would (c) ray a and (d) ray b emerge? (Use sin A = A, A～20 degrees in radian, 1 radian～57 degrees) 5. Figure 29-16a (in the textbook) shows the cross section of a long conduction cylinder with inner radius a = 2.0cm and outer radius b = 4.0cm. The cylinder carries a current out of the page, and the magnitude of the current density in the cross section is given by J = cr^2, with c = 3.0×10^6 A/m^4 and r in meters. What is the magnetic field B at a point that is 3.0cm from the central axis of the cylinder? μ_o = 4π×10^-7 Tm/A --

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