Physics307L F09:People/Muehlmeyer/Lightspeed

Introduction

^{SJK 01:17, 1 November 2008 (EDT)}

01:17, 1 November 2008 (EDT)
I think you took some careful and good data for this lab. However, you did not go far enough with the error analysis or fitting methods. In fact, I couldn't tell what you did for the uncertainty analysis. Further, your comments about the large distance changes would be a lot stronger with an explanation for why it would matter.

From the simple fact that velocity is the derivative of position, we can derive the velocity of light if we have a position vs time plot of the light. We will do a linear least squares fit to our x versus t curve. The least squares line gives us a best fit line to our data based on the fact that the sum of of the squared "residuals" has its least value. A "residual" is the value difference between the actual data point, and the best fit line model. It gives us a trend line for which to model our data. We will use this line as our position vs. time line, whose slope will be our value for the speed of light. ^{SJK 01:17, 1 November 2008 (EDT)}

01:17, 1 November 2008 (EDT)
You have a good description of some of the terminology here, but when I look at your analysis in excel, you did not implement the "LINEST" function, which is for linear regression or least squares fitting.

Our set up is such that we measure the time difference between two different points of light emmission. The light is emmitted from an LED that we can push in and out of a tube. At the other end of this tube is a photomultiplier that converts this pulse of light from the LED into a signal that is imputed into a Time-Amplitude-Converter (TAC). The TAC converts the time difference between the start and stop signals (the LED pulse is start, the light the PMT received is stop) and outputs this time difference as a voltage. We read the voltage with an oscilloscope with the understanding that the TAC is outputing a ratio of 10 V per 50 nS.

With this in mind we can plot position vs. time and find the slope of the best fit line.

Please refer to my lab notebook for in depth discussion of the set up, the time walk, the data taking method, and such.

Speed of Light Notes

Results

Accepted Value of the Speed of light in a vacuum:

[math]\displaystyle{ {c}=3\cdot 10^{8}\frac{m}{s} }[/math]

Of importance to the following results is a discussion on the method of each trial. Each of the four trials below was characeterized by different changes in position of the LED. In the fourth trial we did not use the time walk correction of the polarizers. This of course validated the importance of the time walk correction, which takes into account the changes of light intensity as the LED comes closer to our PMT. Please refer to the lab notebook for more detail.

Trial 1: Changes in distance increased rapidly from the initial point.

[math]\displaystyle{ {c}=2.9371\cdot 10^{8}\frac{m}{s} }[/math]

• c_up = (4.2302) 10⁸ m/s
• c_down = (1.5972)) 10⁸ m/s

Trial 2: Small changes in distance.

[math]\displaystyle{ {c}=1.4202\cdot 10^{8}\frac{m}{s} }[/math]

• c_up = (1.7924) 10⁸ m/s
• c_down = (1.0481)) 10⁸ m/s

Trial 3: Larger changes in distance, but not as large as trial 1.

[math]\displaystyle{ {c}=3.4750\cdot 10^{8}\frac{m}{s} }[/math]

• c_up = (4.9498) 10⁸ m/s
• c_down = (2.0002)) 10⁸ m/s

Trial 4: No time walk correction.

[math]\displaystyle{ {c}=4.2009\cdot 10^{9}\frac{m}{s} }[/math]

• c_up = (6.1817) 10⁸ m/s
• c_down = (2.2201)) 10⁸ m/s

Conclusion

What I notice with my data is the importance of large changes in distance of the LED, and the necessity of the time walk correction. Trial 1 is obviously the closest value, and this trial was characterized by changes of distance that were increasingly large.