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-- Main.JorgeRodriguez - 2012-01-09
---++ Lab Assignment: The Millikan Oil Drop

The goal in this lab is to determine the value of the electric charge q<sub>e</sub>. We will be using the method developed by Robert A. Millikan's who was awarded the Noble Prize for Physic in 1923 for this work. We have, these days, a Pasco AP-831A apparatus in the lab that will allow us to extract a relatively accurate result with about a 10 percent precision. Although the experiment itself is conceptually straightforward, utilizing a balance of forces on a charged oil drop to extract the parameter, the data analysis is a bit tricky. The analysis consists of measuring the total charge on a series of old drops with an undetermined number of charges per drop. The analysis technique used by Millikan uses clever inferences to determine the number of charges per drop by comparing data. The entire experimental procedure is described in the "manual" provided by Pasco for your benefit below. In this assignment, you will not only conduct the actual experiment with the Pasco apparatus but also generate a sample of data simulated on a computer using the Monte Carlo smearing technique as described in Bevington and Robinson, pages 85 and 86. The simulated data part of the assignment will allow you not only further your experience with Monte Carlo sampling techniques but also provide you with a set of data that is completely under your control. This control can be used to better understand the real data, which usually leads to improvements to your analysis algorithm. It can also provide you with a very useful method to explore experimental uncertainties given the control the experimenter has over the input parameters, sample size, and a variety of additional handles not available with experimental data alone. In fact, this is usually the way modern experiments are conducted, especially in particle and nuclear physics as the computing power available improves. In fact, by first doing the experiment on a computer you will have a much clearer understanding of what you will be doing in the lab so your data collection will be greatly enhanced. 
---++ PART A: Generate a sample of simulated Millikan data using Gaussian Smearing

The baseline here is to use the parameters and methods suggested by the equipment we will ultimately use to conduct the actual experiment. The experiment consists of determining the charge on a single drop <img alt="LaTeX: Q" data-equation-content="Q" src="https://fiu.instructure.com/equation_images/Q" title="Q" /> where,

<img alt="LaTeX: nq_e=Q=\frac{6\pi d}{V}\sqrt{\frac{\left\{9\eta^3\right\}}{2g\rho\left(1+\frac{b}{pa}\right)^{3\:}}}\left(v_{fall}+v_{rise}\right)\sqrt{v_{fall}}" data-equation-content="nq_e=Q=\frac{6\pi d}{V}\sqrt{\frac{\left\{9\eta^3\right\}}{2g\rho\left(1+\frac{b}{pa}\right)^{3\:}}}\left(v_{fall}+v_{rise}\right)\sqrt{v_{fall}}" src="https://fiu.instructure.com/equation_images/nq_e%3DQ%3D%5Cfrac%7B6%5Cpi%20d%7D%7BV%7D%5Csqrt%7B%5Cfrac%7B%5Cleft%5C%7B9%5Ceta%5E3%5Cright%5C%7D%7D%7B2g%5Crho%5Cleft(1%2B%5Cfrac%7Bb%7D%7Bpa%7D%5Cright)%5E%7B3%5C%3A%7D%7D%7D%5Cleft(v_%7Bfall%7D%2Bv_%7Brise%7D%5Cright)%5Csqrt%7Bv_%7Bfall%7D%7D" title="nq_e=Q=\frac{6\pi d}{V}\sqrt{\frac{\left\{9\eta^3\right\}}{2g\rho\left(1+\frac{b}{pa}\right)^{3\:}}}\left(v_{fall}+v_{rise}\right)\sqrt{v_{fall}}" />

The variables are all explained in the Pasco manual but include, the voltage <img alt="LaTeX: V" data-equation-content="V" src="https://fiu.instructure.com/equation_images/V" title="V" />, plate separation <img alt="LaTeX: d" data-equation-content="d" src="https://fiu.instructure.com/equation_images/d" title="d" /> , fall, and risel velocities, <img alt="LaTeX: v_{fall},\:v_{rise}" data-equation-content="v_{fall},\:v_{rise}" src="https://fiu.instructure.com/equation_images/v_%7Bfall%7D%2C%5C%3Av_%7Brise%7D" title="v_{fall},\:v_{rise}" /> etc., All of the parameters are determined experimentally. In particular, the rise and fall velocities are measured for a series of drops, repeatedly and will form the majority of your measurements. To simulate the data on a computer you smear the fall and rise velocities with Gaussian smearing and draw a set of integers from a random uniform integer distribution between 1 and say 9, typically for the Pasco equipment.

Here is the procedure summarized as a list: (Goal is to generate 100 total measurements of 10 separate drops, each drop is measured 10 times)
   1 Create 10 individual drops each defined by making up reasonable inputs for 1. the number of charges on the drop and 2. the time it takes said drop to fall through 0.5 mm, the reticle spacing etched on the microscope. You can pick times from 9.5 seconds to 20.5. You do not really need to pick 10 separate values since you can also need to pick the number of charges per drop. Remember that the fall time is only related to the actual size of the drop and not its charge which can be changed without affecting the size or radius <img alt="LaTeX: a" data-equation-content="a" src="https://fiu.instructure.com/equation_images/a" title="a" /> of the drop. 
   1 Pick the number of charges to assign to each drop. You can do this by just assigning it or you can draw randomly from a uniform integer distribution from 1 to say 10. 
   1 For each drop above you have its fall time and its number of charges you can now calculate the rise and fall velocities by inverting the equation above. This will provide you with 10 individual "raw" data which forms the "perfect" data you will now smear to get your 100 samples of simulated data next. 
   1 For each of the 10 perfect data points above Gaussian smear, each one 10 times to simulate the dataset you will ultimately collect from conducting the actual experiment with the Pasco apparatus. Use a relative uncertainty of 10% to smear your perfect data. Remember relative uncertainty is defined as the ratio of the uncertainty in the measurement to the measurement itself.  
Now upload your code to the Project assignment for this lab. Again make sure the code is commented and runs on spyder. 

---++ PART B: Develop your Millikan Analysis

Now that you have a sizable set of simulated data you can develop the analysis procedure to extract the value of the electric charge constant that should, within errors, yield the value you input originally. Since you have the data and have complete control you can change the value of e to any value whatsoever and test your analysis procedure. You can also test the error determination. You should see the power of using Monte Carlo simulation to do data analysis. 

Some useful information and hints
   * Read through the entire experimental procedures outlined in the manual and pay particular attention to how to analyze the data. It is a particularly nuanced process by which you can extract the value of e without knowing the number of charges deposited on oil drops.
   * Also, you may avail yourself of the excellent write-up in Mellisinos and Napolitano and of course, google can help you find many student papers online. If you do find useful sources on the web make sure to cite them in your bibliography.
   * Also the original Millikan paper outlines the procedure he used to do his data analysis. It's a good idea to look through that for hints on how to extract the electric charge. His method works even if the drop with the lowest total charge is not 1. A link to the original paper is provided. 
---++ PART C: Conduct the actual experiment

Now go to the lab and do the experiment. Because you've essentially figured out how to do the analysis already in PARTS A and B you should be well equipped to inform your data taking exercise and take data quickly and efficiently. Report the results in your write up. 

Some useful information and hints
   * This lab requires a lot of patience and careful observations of very small objects that appear through the microscope. The charged droplets are tiny and thus are influenced significantly by the ambient environment in ways you'll have to figure out how to control. Its good practice to have plan when you start your data taking. The manual provides you with useful hints but you will have to think up things to help on your own.
   * As with all labs in this course a proper account of your experimental uncertainties is a must. As you determine parameters in your formulas and measure the variables make sure you take into proper account the experimental uncertainties and that these uncertainties are correctly propagated to determine your best estimate of the uncertatinty in your measurement. In this case that will be the value of the electrical charge. Keep in mind thought that the formula that relates the value of the charge to the various quantities you'll measure is somewhat complicated so propagation of errors can get somewhat involved. However, determing the relative uncertainty of each of the variables in the expression and ignoring those that are small relative to the others will help simplify things considerably. The way to do this is to consider the sources of uncertainties carefully by determinig the relative errors (err_val/val) of each measurement that goes into the formula. Please make sure that indicated what assumptions you make in determining the overall uncertainties in your write up. Documenting exactly what you did is of paramount importance so make sure you clearly describe this in your paper.
   * The Pasco AP-8210A manual is: [[%ATTACHURL%/Millikan-Oil-Drop-Manual-AP-8210A-corrected.pdf][Millikan-Oil-Drop-Manual-AP-8210A-corrected.pdf]]
---++ GradingRuberic
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I	Attachment	History	Action	Size	Date	Who	Comment
pdf	Millikan-Oil-Drop-Manual-AP-8210A-corrected.pdf	r1	manage	1055.4 K	2016-02-20 - 21:44	UnknownUser	AP-8210A Instruction manual
pdf	Millikan-Oil-Drop-Manual-AP-8210A.pdf	r1	manage	1050.5 K	2013-09-03 - 15:43	UnknownUser	AP-8210A Instruction manual
Topic revision: r9 - 2021-01-05 - JorgeRodriguez