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)1012-3(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

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