Outer water acket Inter temperature constant to ±0.001·C) As you can see, this i

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Outer water acket Inter temperature constant to ±0.001·C) As you can see, this is essentially an interferometer with arms of unequal length. Let's analyze this experiment via a series of problems: (i) Assume that one arm of the interferometer is longer than the other arm. Show that a flash of light entering the apparatus will take a time 2/e longer to complete the round trip along the longer arm than the along the shorter arm. The difference in length 1 used by Kennedy and Thorndike was ~ 16cm. What is the approximate difference in time for the round trip of a light flash along the alternative paths? (ii) Instead of a pulse of light, Kennedy and Thorndike used continuous monochromatic light of wave- length -546.1 nm from a mercury source. Light that traverses the longer arm of the interferometer will return n periods later than light that traverses the shorter arm. Write down an expression for n (in terms of 1, c, and the period T of the light) and evaluate numerically as well. If the number is an integer, the reunited light will from the two arms will constructively interfere and the observed light (at point " in the diagram) wil be bright. If n is a half integer, destructive interference results and the experimentalist sees darkness. (iii) Rearrange the expression from the previous part so that you have the speed of light in terms of other variables. As the Earth goes around the Sun, its velocity relative to distant stars is constantly changing. Put another way, we can think of the Earth as constantly switching from one inertial frame to another. As the Earth (and therefore the experiment) moves between different inertial frames2, we can monitor the brightness of the output light, which tells us if n changes. Which in Adapted from Spacetime Physics by Wheeler and Taylor This is actually awfully careless language. As we've emphasized in class, events are events are events. They do not "belong to any inertial frame. Rather, we can describe events from any frame that is convenient for us (including, incidentally, non-inertial frames). So to be more precise here, we should've said "as the Earth goes around the Sun, the experiment is at rest in different inertial frames 3 PHYS 260: Modern Physics and Relativity Problem Set 3 turn tells us whether c is different between inertial frames. By making observations on timescales of a month, Kennedy and Thorndike showed that n varies less by 3 x 10-3. Differentiate your expression to compute the largest fractional change c/c in c that is tolerable given this precision on n. Evaluate the result numerically

Explanation / Answer

1. given one arm of interferometer is longer by dl

then

total distacne covered = 4l + 2dl

lenght of longer arm = l + dl

length of shorter arm = l

hence

for round trip, extra distance = 2*dl

extra time taken by light = dt

c = 2*dl/dt

dt = 2*dl/c

now, dl = 0.16 m

for c = 3*10^8 m/s

dt = 1.066*10^-9 s

ii. lambda = 546.1 nm

time period = T

now

c = lambda/T

T = 1.82033*10^-15 s

hence

n = dt/T = 2*dl/cT = 585973.26496978

iii. c = 2*dl/nT

dn = 3*10^-3

hence

dc/c = (-dn/n)

hence

dc/c = 3*10^-3/585973.2649697 = 0.000000005119687

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