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PH214_2008SPRING_EXAM2_PROFSOLN_[0] - Ph 214 Spring 2008...

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Ph 214, Spring 2008 Exam # 2 Question #1 Name:SOLUTIONS 1. Short answer questions [20 pts] Answer all three questions and try to limit yourself to the space provided. Make minimal use of equations and explain clearly. If you need extra space, use page # 3 and clearly identify which part you are answering. (a) [5 pts] Spacetime events The spacetime coordinates (as observed in the ground frame) for three pairs of events ( A 1 , A 2) , ( B 1 , B 2) & ( C 1 , C 2) are shown in the figure at right. Is it possible to find a frame moving with velocity v relative to the ground in which one of these pairs of events is observed to take place at the same location? If so, for which pair and what is the velocity of this frame relative to the ground? If two events are timelike separated [1 pt] in frame S , it is pos- sible to find a frame S in which the events occur at the same location but at different times. Timelike intervals are defined as those for which ( c Δ t ) 2 > x ) 2 −→ s ) 2 > 0. Comput- ing the intervals for our events, we obtain the following results: x, c Δ t ) = ( 2 , 3) , (4 , 2) & (4 , 4) for pairs A , B and C , respec- tively. Therefore, only pair A is timelike separated . [2 pt] These two events occur at the same location if the ct axis of the new frame passes through them. The slope of this axis in the S frame is 1 β = 3 2 −→ the velocity of this frame relative to the ground is v = βc = 2 3 c . [2 pt] (b) [5 pts] Photon Decay Is it possible for a single photon to decay into a pair of identical massive particles? Explain. No, because it is not possible to conserve both energy and momentum simultaneously in this process. [2 pt] E M, v M, v θ θ (1) (2) (3) We can see this by writing the 4-momenta before and after the decay: vector P 1 = parenleftbigg E c , E c , 0 , 0 parenrightbigg , vector P 2 = ( γMc, γMv cos θ, γMv sin θ, 0) & vector P 3 = ( γMc, γMv cos θ, γMv sin θ, 0). Conservation of energy implies E c = 2 γMc and conservation of x-momentum leads to E c = 2 γMv cos θ . Substituting the first equation into the second gives us 2 γMc = 2 γMv cos θ −→ v = c cos θ c which is not physically allowed for particles with non-zero rest mass . [3 pt] (c) [10 pts] Modified Michelson Interferometer The upper figure at right shows the standard configuration of a Michelson interferometer with arms of equal length L . An engineer wants to use this device to measure the index of refraction n of an unknown liquid. The liquid completely fills a transparent container of length d ( L > d ). Using monochromatic light of wavelength λ , the engineer first measures the fringe pattern with the standard configuration. The liquid is then inserted along one arm of the interferometer (lower figure) and a new fringe pattern is measured. 1
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Ph 214, Spring 2008 Exam # 2 Question #1 Name:SOLUTIONS [5pts] What is the difference in light travel time along the arms for the new configuration? ( Assume that the container is of the same index as the liquid and of negligible thickness ) We will refer to the upper arm as #1 and the arm with the liquid as #2.
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