Problems for Final Exam. (General Physics II, PH142) (2016. 12. 16. AM 9:00 – 12:00) ● Answers can be written in English or Korean. ● DO NOT use any electronic devices (e.g., calculators, PDAs, computers, cell phones, dictionary etc.) 1. Ampere’s law is ∮ • . Prove that if you have a parallel plate capacitor in the circuit as shown in the figure, then it should have a displacement current term, , to get the • . (10pts) complete the law, �∮ 2. The energy densities of the electric and magnetic fields are and , respectively. Derive the intensity, ( Poynting vector), of a plane electromagnetic wave propagating through a cross-sectional area at right angle, or the rate of energy flow per unit area. (10pts) 3. Find an expression for the speed of light in a material with refractive index in terms of (light speed in air) and the critical angle (i.e., total internal reflection angle) at an interface between the material and air. (10pts) 4. By what factor is the image magnified for an object located at 0.5 focal length from a converging lens? Is the image upright or inverted? (10pts) 5. Galilean telescopes use a pair of convex (objective) and concave (eyepiece) lenses, with the arrangement of the eyepiece before the focus of the objective lens, often satisfying . (a) Using ray-tracing, show that this telescope gives upright images. (5pts) (b) Explain why the angular magnification in this telescope approaches for an object at infinity. (5pts) 6 (a) What is the meaning of the diffraction limit? (3pts) (b) Find an expression for the diffraction limit (the angular resolution in radians) for a given circular aperture of diameter D (Rayleigh criterion)? (4pts) (c) Using the result of (b), list practical approaches to achieve better resolution (sharper images) in optical instruments, and explain the reasons? (3pts) - 1 - 7. Our Milky Way and the Andromeda Galaxy are approximately at rest with respect to each other and are 3 million light years (Mly) apart. Supernova explosions occur simultaneously in both galaxies, as judged in the galaxies’reference frame. A spacecraft is traveling at 0.8c from the Milky Way toward Andromeda. (a) Find the time between the supernova events as measured in the spacecraft’s reference frame. (7pts.) (b) What’s the meaning of the calculated value? (3pts.) 8. The most energetic proton ever detected in the cosmic rays coming from the space had an astounding kinetic energy of × (a) Calculate the proton’s Lorentz factor and speed (4pts.) (b) Suppose that the proton travels along a diameter ( × ly) of the Milky way galaxy. Approximately how long does the proton take to travel that diameter as measured from the common reference frame of Earth and the galaxy? (3pts.) (c) How long does the trip take as measured in the rest frame of the proton? (3pts.) 9. Bohr Model (a) Following the suggestion of de Broglie, derive the quantization of Bohr’s atomic orbit. (3pts) (b) Using the Bohr model, find the allowed energy levels in a hydrogen atom. (4pts) (c) In the case of He+(hydrogen-like ion), describe the change in the allowed energy levels from those of a hydrogen atom.(3pts) 10. (a) Solve the time-independent Shrdinger equation to find the wave function and allowed energy levels of a particle confined in one dimensional infinite square well (U=0 when 0≤x≤L, U=∞ when x<0 or x>L). (4pts) Time-independent Shrdinger equation: (b) Find the quantized energy levels when a particle is confined in a two-dimensional square well. (U=0 when 0≤x≤L and 0≤y≤M , U=∞ otherwise). (4pts) (c) Using the result of (b), find the degeneracy of the first and second excited state when L=M (2pts). - 2 -
