In this question, you will investigate the dominant sources of error in measurin

Question

In this question, you will investigate the dominant sources of error in measuring the speed of light in the lab by assuming some measurement outcomes and uncertainties given for this problem. Throughout this question, assume that the uncertainty in each time cursor measurement on the scope is delta t = 1 ns and the uncertainty on the length measurement is delta x = 1 cm. Suppose that in the lab, you measured Delta t = 72 ns and Delta x = 21.6 m. Given this, and the fact that the speed of light is calculated as c = Delta x/Delta t, answer the following questions. (a) Write an expression for the speed of light in terms of the time cursor measurements and the length of the beam path. (b) What is the uncertainty on your measured speed of light, delta c? (c) Now assume that we have the means to measure the beam path lengths with infinite precision (zero uncertainty). What is the uncertainty on the measured speed of light now? (d) Now assume that the scope's time cursors are infinitely precise (and beam path error is now non-zero). What is the uncertainty on the measured speed of light now? Products and ratios (they follow the same rule): f = x times ... times z/u times ... times w delta f/|f_best|) = Squareroot (delta x/|x_best|) ^2 + ... + (delta z/z_best|) ^2 + (delta u/|u_best|) ^2 +... + (delta w/|w_best|) ^2

Explanation / Answer

a) given Dx = 21.6 m ( length of beam path)
   Dt = 72 ns = 72*10^-9 s ( time period of measurement)
   dx = 1 cm = 0.01 m ( error in length measurement)
   dt = 10^-9 s ( error in time measurement)

   so speed of light, c = Dx/Dt
   c = 21.6/72*10^-9 = 3*10^8 m/s
b) now, let uncertianity in c be dc
   then dc/c = sqroot([dx/Dx]^2 + [dt/Dt]^2)
   dc = 3*10^8*sqroot((0.01/21.6)^2 + (1/72)^2) = 4168980.838 m/s = 4.168*10^6 m/s
c) now, assuming dx = 0
   dc = c*dt/Dt = 3*10^8*1/72 = 4.166 *10^6 m/s
d) assuming dt = 0
   dc = c*dx/Dx = 3*10^8*0.01/21.6 = 1.388*10^5 m/s

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