Data Sheet MEASUREMENT  OF THE INDEX OF REFRACTION OF WATER. 

Blank Data Sheet here 
FOR THE INDEX OF REFRACTION LAB. The purpose of the  experiment is to measure the index of refraction of water using a tank of shallow water and two slots. The lab is not formal  But include a diagram with key geometry and answers to all questions with all work under each question. An email was sent with a reminder of format conventions.    In the formatting below , you will be allowed to quote your percent error with the accepted value and whether or not nacc = 1.33 falls within a statistical  range that will be discussed below.   You should compute the percent error as well as perform the discrepancy test  discussed  next: 

| nbest - nacc  | < overall error, where "overall error" is the standard deviation of the mean or uncertainty due to propagation of errors in a single measurement, which ever is larger, as discussed below 

The accepted value of the index of refraction of water is nacc = 1.33

You are to measure the angle of incidence theta_i and refraction theta_f  4 OR 5  times each, compute the index of refraction for each trial, compute the average index of refraction . You will also compute the standard deviation of the mean and compare it with the  uncertainty due to propagation of  error  for a single measurement = Sp  = a + b,  where the formulas for  a and b are derived from calculus.  

Here, 

a = [cos (theta_fbest ) /sin(theta_ibest)]*Δtheta_f
b =  [nbest*cot(theta_ibest)* Δtheta_i
[REMEMBER TO CONVERT TO RADIANS IF THE FINAL  OR INITIAL ANGLE CHANGE Δtheta IS USED OUTSIDE THE ARGUMENT OF THE SINE OR COSINE FUNCTION.]
These expressions are obtained by taking partial derivatives as shown in class.
Remember,

theta_fbest and theta_ibest  are each the average of 4 measurements; see sample data sheet below.


Using  lc/2 = 0.125 = 0.1 for the instrumental uncertainty of the protractor, then you might have collected numbers like:

50.25   or 50.00.

Blank Data Sheet here  and all calculations with the report. Sample data is included in the table below.  Note only certain values of numbers are allowed given Least Count (LC) rules. 
Explore 
http://www.physics.upenn.edu/~uglabs/ and 
for tips on sig. figs. and other topics of interest.

 

 SAMPLE DATA 

 

 

 Δtheta_iinst = LC/2 = 0.25/2 = 0.125 = 0.1

 

theta_i

 

 

 trial 1.  36.50
trial 2.  36.75
trial 3.  36.00
trial 4.  36.25

theta_ibest    (average of the 4 above) = 36.375

Δtheta_iinst  = 0.125 = 0.1

(theta_imax – theta_imin)/2 = 0.375 = 0.4

Δtheta_i (larger of previous two.) = 0.4
Below,  compute nw =  sin(theta_f)/sin (theta_i) = for each trial
Δtheta_iinst   = LC/2 = 0.25/2 = 0.125 = 0.1


theta_f

 trial 1. 52.25 nw = 1.329285951

 
trial 2. 52.75 nw = 1.330384881
trial 3. 52.00 nw = 1.340643969
trial 4. 52.50 nw = 1.341688475

theta_fbest (average of the 4 above)=  52.375

Δtheta_finst  =0.125 = 0.1

(theta_fmax – theta_fmin)/2 = 0.375 = 0.4

Δtheta_f (larger of previous two.) = 0.4
average
nw of  4 trials 		1.335500819
standard deviation of the mean Sm of the mean = Standard Deviation/2 = 0.00328553

 nbest

  sin(theta_fbest) /sin (theta_ibest) = sin(52.375) /sin (36.375) = 1.335468846

 
See formula at the top of the page  Sp = propagation of error uncertainty= 
a + b 
= [cos (theta_fbest ) /sin(theta_ibest)]*Δtheta_f  +  [nbest*cot(theta_ibest)* Δtheta_i

= [cos (52.375) /sin(36.375)]*0.4*pi/180 + 
+[1.335468846*cot(36.375)* 0.4*pi/180 = 0.01984412 = 0.02 		 
  Percent error = |average nw   - nacc |*100%/nacc  =
 | 1.335500819 - 1.33 | *100%/1.33 = 0%

 

  
overall error = larger of Sm and Sp =  0.01984412 = 0.02
  Compare:  
|average nw   - nacc | =
| 1.335500819 - 1.33 | = |0.0055008| = 
0 < overall error = 0.02? YES