May 2013 1 Measurement of Speed of Light Shawn Kann Department of Physics and Astronomy, San Francisco State University, San Francisco, California 94132 Abstract: Previous studies have shown that speed of light, c can be determined by a time-of-flight method. In this study, we measured the speed of light by using a laser driven by a function generator. The distance that light traveled was varied, and the change in the time delay in light generated by the function generator was compared with light emitted by the laser after it has traveled over distance. The experimentally measured value of c = 2.4x108 m/s is consistent in orders of magnitude with the current national institute of science and technology (NIST) value. INTRODUCTION/THEORY The speed of light, c is an important constant used in many physics formulas and can be derived from Maxwell’s equations. In regions of space where there is no charge or current, Maxwell’s equations read [1], B (i) E 0, (iii) E , t E (ii) B 0, (iv) B 0 0 . t Preceding equations constitute a set of coupled, first-order, partial differential equations for E and B. They can be decoupled by applying the curl to (iii) and (iv) [2], B 2 ( E ) ( E ) E t 2E ( B) 0 0 2 , t t E 2 ( B) ( B) B 0 0 t 2B 0 0 ( E ) 0 0 2 . t t Or, since E 0 and B 0, 2 2 E B 2 E 0 0 2 , 2 B 0 0 2 . t t We now have second order decoupled differential equations for E and B [2]. In vacuum, then, each Cartesian component of E and B satisfies the threedimensional wave equation, 2 f 1 2 f . v 2 t 2 So, Maxwell’s equations imply that empty space supports the propagation of electromagnetic waves, traveling at a speed v 1 0 0 3.00 10 8 m / s, which happens to be precisely the velocity of light, c [2]. Maxwell’s theory predicts the extremely important idea that electrical disturbances are transverse waves of electric and magnetic fields. The velocity of propagation depends upon vacuum permittivity 0 and permeability µ0, which can be determined purely by electrical measurements and should be equal to the velocity of light. May 2013 2 this experiment. The reflected output was amplified by sending the output signal thru TDS1002 Tektronix Digital Oscilloscope to determine the time delay in the received light signal. The optical components included a 10 cm lens to focus the laser beam and a front-surface mirror to reflect the beam. To generate a sine wave, we used 33120A Agilent Function Generator. In addition, a leveling base to hold laser, lens, and detector were also used. The optical setup of this experiment is shown in figure 2. Figure 1. Photograph of a typical display on the dual trace oscilloscope showing the sharp shift between the signal generator and the signal of the reflected beam. EXPERIMENTAL METHODS In this experiment, we used “time-of-flight” method to measure the speed of light. The original attempt to do this was by Galileo, who used flags and lights, with human operators doing the timing measurements [3]. In this experiment, we used a special laser whose intensity was modulated with the help of a signal generator. Using the oscilloscope we then measured the change in the time delay in light generated by the signal generator and compared it with the light emitted by the laser after it traveled some distance (figure 1.). The peaks of the signals in figure 1 are slightly shifted. by t . By repeating measurements for long distances and short distances, we arrived at a time t to travel a distance x , 2d x t 10ns. c c Plotting x vs. t and performing fit to a straight line allowed us to compute the speed of light constant as 1/c. A ML868 Metrologic Laser with intensity modulation capability was used as the source of light. Metrologic detector unit with photodiodes and amplifier was chosen to detect the reflected light for Figure 2: Optical setup used to determine c. The signal generator was configured to provide a sine wave at 800 kHz with peak-to-peak amplitude of 900mV, to modulate the laser intensity RESULTS By plotting time delay versus distance and using a least-square-fit program in MATLAB, we can determine a close fitting slope as, c 1 m (1) The graph in (figure 3.) has a slope with a value of about 0.10745. Plugging into equation 1 yields, c 1 1 inch m 2.40 108 m 0.10745 ns s May 2013 3 2.9979x108 m/s. Comparing the discrepancy and the accepted value the descrpency is, 2.9979 108 m / s 2.3639 108 m / s 100 27% 2.3639 108 m / s The large descrepancy could be attributed to sources of error as discussed shortly in conclusion. CONCLUSION Figure 3: Plot of timedelay vs. distance. ANALYSIS The experimental value of speed of light constant is obtained using equation (1) where the value of the slope yields the experimental value of the speed of light constant, c 1 1 inch m 2.40 108 m 0.10745 ns s which agree in order of magnitude with the currently accepted value of 299,792,458 m/s as given by NIST. In addition, the graph in figure shows Δm, the uncertainy in the measured slope, m 0.0098 ns inch m 0.0098 9% m 0.1075 c m 9% c m c 0.2 108 m / s c (2.4 0.2) 108 m / s The theoratical value of the speed of light constant according to NIST is 2.9979x108 m/s and the experimental value of the speed of light constant is This study has demonstrated that the time of flight method is a suitable method to determine speed of light, c. While the experimental result of 2.40x108 m/s is in good agreement with the accepted value of 2.9979x108 ms-1 in orders of magnitude, there was a relative discrepancy of 27%. This discrepancy can be attributed to the possible systematic source of error inside the oscilloscope. There is also random error associated with our subjective judgment of tmeasure. Moreover, when the distance between detector and reflected mirror is large, it is hard to focus the reflected laser beam into the signal detector, in order to get high enough signal to noise ratio. Acknowledgement The author thanks Dr. Weining Man for providing the Matlab Plot1 script. The author also thanks lab partner, Leung Mike. Figures one and two are courtesy of Department of Physics at San Francisco State University, CA. References [1] James Clerk Maxwell, “A Dynamical Theory of the Electromagnetic Field,” Philosophical Transactions of the Royal Society of London , 155 , 459-512 (1865) [2] David J. Griffiths, Introduction to Electrodynamics (3rd ed, Addison Welsley, 1999), Chap. 9, p. 375. May 2013 [3] Maurice A. Finocchiaro, Retrying Galileo (Universiy of California Press, 2007), Chap. 3, p.63. 4 State University, CA.