Optics: Reflection and Refraction

          The purpose of this lab is to see how light is affected by passing through substance interfaces such as air to acrylic. When a light ray is perpendicular (normal) to a substance interface, the light ray passes through unchanged. When the light ray hits an interface with an incident angle other that a right angle, the light ray will bend if it passes into the substance. In this lab, an acrylic half circle was used to relate incident angle with the refracted angle. A light source was directed to the center of the half disk and measurements were taken over a range of angles. Two sets of data were collected. The first set uses the flat side of the half disk for the incident angle. The second set uses the curved side for the incident angle. The interface of interest in both scenarios is the flat side of the half disk.

Trial 1 (Incident angle on flat side)

Incident Angle (degrees)	Refracted Angle (degrees)	Sin of Incident Angle	Sin of Refracted Angle
60	144	0.886	0.587
55	148	0.819	0.53
50	152	0.766	0.5
45	152	0.707	0.47
40	156	0.643	0.407
35	158	0.574	0.375
30	161	0.5	0.326
25	163	0.423	0.292
20	167	0.342	0.292
15	172	0.299	0.139

Trial 2 (Incident angle on curved side)

Incident Angle (degrees)	Refracted Angle (degrees)	Sin of Incident Angle	Sin of Refracted Angle
30	124	0.5	0.829
25	135	0.423	0.707
20	149	0.342	0.515
15	152	0.259	0.469
10	167	0.174	0.225
5	174	0.087	0.105
0	180	0	0
-5	190	-0.087	-0.174
-10	203	-0.174	-0.391
-15	211	-0.259	-0.515

          The data sets for each trial provided two graphs which totaled in four graphs all together. The first graph was incident angle vs. refracted angle. The second graph was sine of incident angle vs. sine of refracted angle. The follow is the sets of graphs for each trial.


Trial 1:


Incident Angle vs. Refracted Angle





















Sine of Incident Angle vs.  Sine of Refracted Angle





Trial 2:


Incident Angle vs. Refracted Angle



 Sine of Incident Angle vs.  Sine of Refracted Angle


          When looking at the RMS for each fit, the relationship between sine of incident angle vs.  sine of refracted angle are more reliable with a much lower RMS than incident angle vs. refracted angle. This leads to the idea that the sine graphs show a true linear relationship where as the angle graphs might appear linear and yet they should not be. The slope of the sine graph for trial 1 should be one over the index of refraction for acrylic. The slope of the sine graph for trial two should be close to the value of index of refraction for acrylic. The actual value of acrylic index of refraction is 1.49. 

Trial 1:
 Slope of sine graph = 0.648
 Inverse of acrylic index of refraction = 0.671

Trial 2:
 Slope of sine graph = 1.76
 Acrylic index of refraction = 1.49


          These values are relatively close when considering the experiment setup and the inaccuracy of measuring a "beam" of light. In the picture of the setup above it is easily seen that the beam of light has an inherent thickness that changes over a distance. This cam make measurements inaccurate. Furthermore, a paper protractor can be fairly inaccurate as well if the light does not reach the hash marks on the protractor.


          These graphs lead to the relationship of sine of incident angle is proportional to refracted angle. This is also know as Snell's Law:

where n is index of refraction on either side of the interface and theta is either incident or refracted angle. It is simple to see that if one of the indices in the experiment is 1 (index of air), then the sine graph relationship is truly linear which was depicted in the sine graphs.

Mechanical Waves: Standing Waves

          The purpose of this lab is to explore standing waves, the properties of these waves, and how the properties relate. The experiment consist of a string that is fixed at two ends. The tension is known in the string. Then string then has a wave driver attached which in turn is attached to a function generator. The function generator is turned on and the frequency is adjusted. At certain frequencies, the string will oscillate at it's fundamental node or a multiple of that fundamental node. At these frequencies which cause harmonic oscillations, we will measure frequency, wavelength, and the number of loops. The following data was acquired during this process along with three other groups of data obtained from peers conducting the same lab.

Due to technical difficulties with the camera, this picture is not of my personal lab set up. This is a picture of a peer's lab. source

Linear density of the string used = 0.001839 kg/m

T = tension in string

L = total length of the string

Loops = number of loops visible at a given frequency

T(N)	L(m)	Loops	Frequency (hz)	Wavelength (m)
3.924	2	1	15	4
	2	2	29.4	2
	2	3	44	1.33
	2	4	58.4	1
	2	5	72.3	0.8
	2	6	100.3	0.666
1.959	1.33	1	16	2.66
	1.33	2	32	1.33
	1.33	3	46	0.887
	1.33	4	63	0.665
	1.33	5	76	0.532
0.981	1.31	3	34	0.87
	1.31	4	46	0.655
	1.31	5	56	0.524
	1.31	6	61	0.437
	1.31	7	77	0.374
	1.31	8	93	0.327
1.962	1.31	3	46	0.87
	1.31	4	64	0.655
	1.31	5	79	0.524
	1.31	6	96	0.437
	1.31	7	110	0.374
	1.31	8	126	0.327

            This data was then used to construct a plot of f vs. 1/wavelength. The resulting curve fits yielded 1/v which could be used to find v (the wave speed). The following is the curve fit and the wave speeds derived from the graph, experimentally from the data, and predicted from known parameters.

 

Wave Speed "Graph" (m/s)	Wave Speed Predicted (m/s)	Wave Speeds Experimental (m/s)
66.3	53.15123	60
		58.8
		58.52
		58.4
		57.84
		66.7998
38.2	37.55485	42.56
		42.56
		40.802
		41.895
		40.432
43.4	26.57561	29.58
		30.13
		29.344
		26.657
		28.798
		30.411
43.4	37.5836	40.02
		41.92
		41.396
		41.952
		41.14
		41.202

           After computing those values for wave speed, the experimental average, standard deviation, and the percent difference between experimental average and predicted values was computed. The table below shows these values.

Experimental Average (m/s)	Standard Deviation (m/s)	%Dif.
60.06	3.379	13
41.65	0.9898	10.9
29.15	1.35	9.7
41.27	0.7053	9.81

          The overall data from my group and other groups was relatively consistent with computing values for n (harmonic number) and wave speed. When using the equation that relates the harmonic number to wavelength, all the values computed for N matched the observed values of N loops on the string. The waves peed values for the graph, experimental, and predicted all fell within a reasonable range of each other considering the standard deviation and the limitations on the equipment. The limitations in the equipment can be seen in the consistency of the percent difference between predicted and average wave speed values. Things such as variances in frequency, fine tuning the harmonic frequency, correctly measured wavelength, non ideal string and tension slightly oscillating can cause the 10.8% average error seem in the experiment.