101年 - [無官方正解]101 台灣聯合大學系統_碩士班招生考試試題：普通物理#139142

1. In free space, there are two point charges of Q₁= 2q and Q₂ = 6q located at (0, 2r, r) and (r, 0, 3r) in rectangular (Cartesian) coordinate system. What is the magnitude of the force experienced by Q₂?(A)(B)(C)(D)
(E)None of the above

2. A sphere with radius a contains a uniform volume charge density ofρ_vin the region: 0 ＜ r ＜ a. If the sphere is isolated by a conducting spherical shell with inner radius b and outer radius c placed concentrically, as shown in figure. What is the electric field intensity in the region b ＜ r ＜ c?(A)(B)(C)(D)
(E)None of the above

3. A straight wire carrying a current I in the z direction is located along the z axis in free space. If the wire is infinitely long and infinitely thin, what is the magnetic field intensity at the point (0, L, 0) in rectangular coordinate system?(A)(B)(C)(D)
(E)None of the above

4. In a perfect dielectric medium with the permeability of \(\mu\), there exists an electric field given as \(\mathbf{E} = E₀cos(ωt - kz)ax, where E₀ is the peak value, \(\omega\) is the frequency, k is a constant, and \(\hat{a}_x\) is the unit vector in x direction. What is the magnetic field (H) in the region?(A)(B)0(C)(D)
(E)None of the above

5. A uniform electromagnetic plane wave is traveling in free space and its electric field is given by E = 5cos(ωt-2z)ax V/m, what is the angular frequency, ω in (rad/s), of the plane wave?(A)3✖10⁸(B)6✖10⁸(C)6✖10⁹(D)6.28✖10⁸
(E)None of the above

6. A long core-shell structure is made to transmit an optical wave through total internal reflection within the core region. If the refractive indices for the core and the shell are respectively n₁ = 1.5 and n₂ = 3.0, as depicted in the figure. What is the minimum value of the angle \(\theta\) for a successful transmission?(A)30°(B)45°(C)60°(D)90°
(E)None of the above

7. An object is placed at the position being 20 cm from a convex lens with the focal length of 10 cm. What is the distance between the image and the lens?(A)20 cm(B)10 cm(C)40 cm(D)30 cm
(E)None of the above

8. The spherical waves of wavelength \(\lambda\) emitted from a single point source are passing through two small holes, S₁ and S₂, which are separated by the distance of a, as shown in the figure. If c >> a, what is the approximated condition for destructive interference at a point P on the screen?
(A)asinθ =1.5λ(B)asinθ =1.5λ(C)c sinθ=λ(D)b tanθ=λ
(E)a sinθ =λ

9. A laser beam of wavelength \(\lambda\) shined on a spectrometer with the grating spacing a is split into two beams of wavelengthsλ₁ and λ₂ due to the 2nd order diffraction. If λ₂ is slightly longer than λ₁ and θ  is the diffraction angle for \(\lambda_1\), what is the difference in diffraction angles of the two beams?(A)(B)(C)(D)
(E)None of the above

10. A light beam is traveling in a structure with three layers of different refractive indices: n₁ = 2,n₂ =, n₃ = 1. If the beam impinges at the interface between n₁ and n₂ with the incident angle of 60°, as shown in the figure, what is θwhen the light beam enters the layer with n₃ = 1?(A)30°(B)45°(C)60°(D)90°
(E)None of the above

11. A particle is strictly confined in a region of length L, but moving freely in the region. Two possible wavefunctions, φ1 andφ2, of the particle are those shown in the figure. When the particle is in the state corresponding to the wavefunction φ1，  its total energy is 1 eV. What is its total energy in the state corresponding to φ₂?(A)1 eV(B)2 eV(C)3 eV(D)4 eV
(E)None of the above

12. An electron is trapped in an infinitely deep potential well with the width of 5 nm. When the electron is transferred from energy level E\(_2\) to E\(_1\), a photon of the wavelength 800 nm is emitted. If the width of the infinite well is increased to 10 nm, and the trapped electron is still transferred from E\(_2\) to E\(_1\), what can be the emitted photon wavelength?(A)200 nm(B)400 nm(C)1600 nm(D)3200 nm
(E)None of the above

13. A copper ball of mass m and specific heat c is at a temperature T₁ = 86.85°C. The ball is thrown into a large lake at T₂ = 6.85°C, which stays constant. What is the total change in entropy of the copper ball and the lake?(A)(B)(C)(D)
(E)None of the above

14. A particle is trapped in a potential well with the depth of E\(_0\). If the wavefunction of the particle is \(\psi(x,t) = A\sin(\omega t + kx) + iA\cos(\omega t + kx)\), where A, \(\omega\) and k are constants and i is the unit imaginary number, what is the probability, per unit length of the x axis, of finding the particle near the coordinate x at time t?(A)E₀(B)A²(C)(D)A
(E)None of the above

15. Two metal plates are connected to a battery and a current meter. When a light beam with tunable energy E is shined on one of the plates, the values read by the current meter is plotted as a function of E. E0 corresponds to the photon density of 10¹²cm^-3and the photon wavelength of 200 nm. When the incident photon wavelength is increased to 400 nm, what is the photon density to obtain the current value of 5 μA?(A)2✖1012 cm^-3(B)4✖10¹² cm^-3(C)8✖10₂₂cm^-3(D)10^12-3
(E)None of the above

16. Train A travels at 0.8c and passes a station platform. An observer on the platform records a time interval of 2 μs as the front and the rear of the train pass the observer. What is the proper length of the train?(A)288 m(B)320 m(C)480 m(D)600 m
(E)None of the above

17. A container contains 2 moles of ideal monatomic gas with volume V₀ and temperature T₀. What is the change in entropy of the ideal gas if the container is compressed isothermally as its volume is reduced to exactly one half of the initial volume?(A)3R ln 2(B)-3Rln2(C)2Rln2(D)-2Rln2
(E)None of the above

18. A 200 g ball is thrown at 30 m/s. It is struck by a bat which gives it a velocity of 40 m/s in the opposite direction. If the time of contact is 10^-2s, what is the average force on the ball?(A)1000 N(B)1200 N(C)1400 N(D)1600 N
(E)None of the above

19. A quarter circle with a linear density \(\lambda\) kg/m and a radius R as shown in the figure. What is the distance from the center of the quarter circle O to the center of mass of the quarter circle?(A)(B)0(C)(D)
(E)None of the above

20. A particle of mass m1 = 2kg moving at velocity u (m/s) makes a one-dimensional elastic collision with a particle of mass m₂ at rest. What is the possible mass of m₂if it has one-third the initial kinetic energy of m₁ after collision?(A)(B)(C)(D)
(E)None of the above

21. A marble of radius r rolls without slipping down an incline and then up along a vertical circular track of radius R, as shown in the figure. What is the minimum height H from which the ball must start so that it barely stays in contact at the top of the circle? Assume r ＜＜ H and r ＜＜ R.
(A)2 R(B)2.3 R(C)2.7 R(D)3 R
(E)None of the above

22. Water emerges at speed \(v_0\) from the opening of a faucet of radius R. In laminar flow, the cross-sectional area of the vertical stream of water decreases as it falls. What is the relation between the radius r of the stream and the vertical drop y?(A)(B)(C)(D)
(E)None of the above

23. What is the period T of the spring-block system for two springs with spring constants \(k_1\) and \(k_2\) respectively connected in series as shown in the figure? The mass of the block is m.
(A)(B)(C)(D)
(E)None of the above

24. A string of length L and linear mass density \(\lambda\) is under a tension F and vibrates in its fundamental mode at f₁. What is the new fundamental frequency if the length L increases by 25%, the linear mass densityλdecreases by 20%, and the tension decreases by 20%?(A)1.2 f₁(B)0.8 f₁(C)0.83f₁(D)1.25 f₁
(E)None of the above

25. An ideal monatomic gas undergoes an adiabatic process that the pressure and volume change from (P₁, V₁) to (P₂, V₂). What is the work done in this adiabatic process?(A)(B)(C)(D)
(E)None of the above