Introduction to Reflection and Refraction

Introduction: 

The nature of refraction and reflection angles may be explored with the aid of a semicircular acrylic glass piece.  In this experiment, we will explore the how light propagates through two different mediums. Namely, air and acrylic glass.  Air is approximated as vacuum with an index of refraction of 1.  Alerting our apparatus setup will allows to observe the critical angles of incidence.

Steps:

The apparatus requires the acrylic glass piece, a light box, a voltage generator, and a protractor.
Setup 1:
The light beam is aimed at the center of the flat side and exits through a radial path of the glass piece. Since we approximate air as vacuum it has the lower density of the two mediums.  The acrylic has the higher density.  Hence we go from low-high-low densities (once the lights exits back to air).

The setup for our first experimental run.  Angle of incidence is 0 degrees here.

Varying the angle of incidence by 5 degrees on each measurement we obtain the refraction angle along with their sin values.



θ1 θ2 sinθ1 sinθ2
0   0 0.000 0.000
5 3.5   0.087 0.061
10 5      0.174 0.087
15 9.3   0.259 0.162
20 12.6 0.342 0.218
25 16 0.423 0.276
30 17.5 0.500 0.301
35 22.2 0.574 0.378
40 25.5 0.643 0.431
45 28.5 0.707 0.477
50 31.5 0.766 0.522
55 35 0.819 0.574
60 37.5 0.866 0.609


Uncertainties (+/-)
uθ1 uθ2 usinθ1 usinθ2
1 1 0.017 0.017
1 1 0.017 0.017
1  1 0.017 0.017
1 1 0.017 0.017
1 1 0.016 0.017
1 1 0.016 0.017
1 1 0.015 0.017
1 1 0.014 0.016
1  1 0.013 0.016
1 1 0.012 0.015
1 1 0.011 0.015
1  1 0.010 0.014
1  1 0.009 0.014

We obtain the following graphs:

θ1 vs θ2

sinθ1 vs sinθ2

θ1 vs θ2 
mθ ≈ .64
sinθ1 vs sinθ2
msinθ ≈ .66

sinθ2/sinθ1 = .66

Setup 2:
The second setup involves using the curved surface of the semicircular acrylic.  Aiming the light beam at the curved surface and to the center of circle, we obtain straight beams passing through the piece.  This happens because the light beam is normal to the surface and thus creates a 0 degree angle of incidence.
The data obtained is shown below.

Curved surface incidence.

θ1 θ2 sinθ1 sinθ2
0 0 0.000 0.000
5 9.5 0.087 0.165
10 16.5 0.174 0.284
15 24.5 0.259 0.415
20 34 0.342 0.559
25 44 0.423 0.695
30 51.5 0.500 0.783
35 64.5 0.574 0.903
40 80    0.643 0.985


Uncertainties (+/-)
uθ1 uθ2 usinθ1 usinθ2
1 1 0.017 0.017
1 1 0.017 0.017
1         1 0.017 0.017
1 1 0.017 0.016
1 1 0.016 0.014
1 1 0.016 0.013
1         1 0.015 0.011
1 1 0.014 0.008
1         1 0.013 0.003


Similarly, we produce the following graphs:

θ1 vs θ2

sinθ1 vs sinθ2

θ1 vs θ2 
mθ ≈ 1.6 (ignoring 20 degree angle)
sinθ1 vs sinθ2
msinθ ≈ 1.5 


Questions/Conclusions:

We notice that beyond 40 degree we run into an issue when refraction no longer occurs and, instead, the light beam is completely reflected back inside the acrylic.

Using the axis variable we may fit have the equation:
sinθ2/sinθ1 = 1.5

This relationship is exactly inversely proportional to the equation obtained obtained in setup 1.  This is expected as the light beam, in setup 1, goes from low to high index of refraction mediums.  Setup 2 represents the opposite pathway.

The slope for setup 2 represents the refractive index.  The actual values for the refractive index of acrylic glass vary usually around a value of 1.49.

Electromagnetic Radiation

Introduction:
Antennas are known to transmit electromagnetic radiation and at measurable quantities when provided an alternating voltage source.  This principal behind this is the fact that Maxwell's equations state that a varying electric field (the voltage provided) induces magnetic field lines.  The combination of the two create propagating EM waves.   The voltage signal sent in this wave can be detected and measured.  This lab will explore voltage signal provided by a short antenna and measure the voltage as a function of distance from the detector, an oscilloscope. Assumptions and simplifications made will be discussed in the error/uncertainty analysis of the experiment.

Steps:

A BNC adapter will be used to receive the signal at the oscilloscope.
Simple tests are conducted to verify that the signal at our oscilloscope is generated by the antenna.

1)Check signal at short and far distances as the voltage should fall off as 1/r^3 closely and 1/r and greater distances
2) Vary the frequency at a reasonable distance to detect change in frequency at the oscilloscope.
3) Check close distance signal by varying angles of deflection to verify (sintheta)^2/r^2 voltage variations.

Signal will be minimized at close distance by varying angle of deflection.

Close distance signals fall off as 1/r^3

Data is collected by varying distance from the adapter (measured around its base).

A meter stick is used to approximate 5cm increments.  Signal strength decreases as a function of distance.

Peak to Peak Voltages(Y axis -Volts) as a function of distance(X axis - meters)

A/r fit

A/r^3 fit

A/r^n fit (n=.6)

As we observe, the plotted data fits a more A/r fit.  As long term data collection shows, voltage falls off mostly as A/r.
In this case, are A/r^n resulted in n =.6
The results may differ here because we do no vary at great enough distances to really obtain the A/r relationship of distance and voltage.

Theoretically our voltage may be derived to be



Where L is the length of the antenna, z is the distance from the adapter, and the charge Q is calculated from experimental data.



Distnace (m) Vtheoretical P2P
0
0.05                 0.011631
0.1                   0.005687
0.15                 0.003616
0.2                   0.002704
0.25                 0.002234
0.3                   0.001965
0.35           0.001796

The theoretical peak to peak voltage plot is shown below:

Theoretical peak to peak voltage (V) vs distance (m)

Comparing the graphs above to our theoretical peak to peak voltage, it is clear that experimental data and theoretical agree on the inverse relationship of voltage sensed as a function of distance (z).  Experimental results are not smooth and continuous as theoretical results.  This is caused by the inconsistencies in the inverse power relationship among distance and voltage—experimentally, we will experience a 1/r^3 relationship at close distances and tend towards 1/r at large enough distances.

Measurement Uncertainty:

Peak to Peak Voltage
ΔV = ±1mV

Quantization uncertainty for z:
Δz = ±.01m

accounting for BNC adapter length:
Δz = ±.05m







Questions/Conclusions:

Comparing our experimental results to theoretical we see that there is plenty of room for error given the above equations.  If we compare our theoretical values with experimental, our results agree within the same orders of magnitudes(mostly).
In our model we made several simplifications. We assumed linear charge density. Also, we did not take into account any interference from nearby wires and lab equipment.  We ignored fringing effects of the antenna rod that lead to different electric field distributions.
Future experiments may actually measure current supplied by our voltage generator to improve charge calculations.  Although negligible,  our calculations may also include the energy lost due resistance in the copper antenna.

Introduction to Sound

Introduction:
Using a LabPro and a microphone this lab looks at the characteristics of waves created by the human voice and tuning forks. Although the wave patterns created by these types of waves will vary drastically we will compare and contrast.  Frequencies, periods, amplitudes, and wavelength will be observed.
Steps:

The voice of our first participant produces a periodic wave funciton.

Second Participants Produced Wave Function (.3s collection)

Tuning Forks Results

Questions/Conclusions:
1)

a. Would this be a periodic wave? Support answer with characteristics.b. How many waves are shown?c. Relate how long the probe collected data to something in everyday experience (?)d. What is the period of these waves?
e. What is the frequency?f. Calculate wavelength, assuming speed of sound to be 340 m/sg. What is the amplitude of these waves?h. What would be different about the graph if the sample were 10 times as long?

2)
 Have someone else to say "AAAAA..." Compare.
3)
 Collect data from the tuning fork. Compare it with the one made by human voice
4. If the same tuning fork was used to collect data for a sound that is not as loud, what would be different from the graph?

In this lab there was not much error analysis to take into account.  As an introductory assignment we merely took characteristics of the results and analyzed them.  The units from the amplitude of the graphs were arbitrary as we were only interested in the resulting wave patterns and characteristics.

Standing Waves

Introduction:
 Waves on a string are useful in the study of standing waves.  Given a well-controlled apparatus setup, we may look at the properties of such waves.  This experiment will study the number of nodes present, wavelength, frequency, and wave speed of waves after creating 10 harmonics for each of two case setups.

Steps:

The apparatus is depicted in the following two images:

A mass-pulley system is set up on one end of the string to create the tension needed .

Below the standing wave, meter sticks are aligned to measure distance between nodes and find wavelength. A frequency generator and wave driver attached to the string will produce oscillations necessary to obtain empirical data.

In the two cases studied data was collected by adjusting our frequency to create the first 10 harmonics.  Case 1 and Case two had different masses attached to vary the frequency.  The mass of the string was

mstring = .72 +/- .01g
lstring = 2.26 +/- .01m

μstring = mstring/lstring =   .319 +/- .006 g/m

CASE 1:
m1 = .200 +/- .001kg
T1 = 1.96 +/- .01N


CASE 2:
m2 = .050 +/- .001kg
T2 = .49 +/- .01N

CASE 1:

Harmonic n	frequency +/- 1Hz	fn = nf1 +/- n1Hz	λ  +/- .01m	Δx nodes +/- .005m
1	24	24	3.32	1.66
2	48	48	1.66	0.83
3	72	72	1.11	0.555
4	96	96	0.83	0.425
5	120	120	0.66	0.33
6	144	144	0.55	0.275
7	169	168	0.465	0.233
8	193	192	0.405	0.203
9	217	216	0.365	0.183
10	241	240	0.33	0.165

 CASE 2:

Harmonic n	frequency +/- 1Hz	fn = nf1 +/- n1Hz	λ  +/- .01m	Δx nodes +/- .005m
1	12	12	3.32	1.66
2	24	24	1.66	0.83
3	36	36	1.11	0.555
4	48	38	0.83	0.415
5	60	60	0.663	0.332
6	72	72	0.543	0.272
7	84	84	0.475	0.238
8	96	96	0.412	0.206
9	109	108	0.369	0.185
10	121	120	0.334	0.167

Plotting frequency vs 1/λ we obtain the following charts

Two equations may be used for calculation of speed of the wave, they are given by

and

The latter equation is seen by the slope of the plots above.
For theoretical calculations we may use our tension/density equation and obtain:


v1 = 78.4 +/- .9 m/s

v2 = 39.2 +/- .8 m/s

Questions/Conclusion:
The data table above shows the wavelength and the n values that describe the wave ( the harmonic).  It is clear that the wavelength follows the equation:

The plots of frequency versus 1/λ compare nicely with our theoretical wave speed obtained and fall within uncertainty of the results.  The sources of error associated with the variables required (T, μstring, λ, and f) are all instrumental and amounted to relatively small uncertainty as seen by the final result of our theoretical wavespeeds v1 and v2 .  Taking the percent error we find:
|v1 - slope1|/v1 * 100% = .60%
|v2 - slope2|/v2 * 100% = 2.1%

The ratio of the the predicted wave speed was
v1/v2 = 78.4/39.2 = 2.0
or we could use

Comparing this to our slope1 and slope2
slope1/slope2 = 78.867/40.02 = 1.97
A percent error of 1.5% on this ratio shows much agreement between the ratios

As seen by the the charts above our values for nf1 agree identically up until the last four frequency values for case 1 and up until the last 2 values for case 2.  Each disagreement is from these last couple of values is only +/- 1 Hz which is within the uncertainty provided (The smallest denomination on our frequency generator was 1Hz and so this was taken as the uncertainty value).

Taking a ratio of our frequencies from case 1 and case 2 we obtain

Harmonic	f1/f2
1	2
2	2
3	2
4	2
5	2

The obvious pattern is that the ratio is 2.  This makes sense since the velocity increased by a factor of 2 and the wavelengths for the corresponding harmonics were kept relatively constant.

Wavelength and Frequency

Introduction:
In this short laboratory, an experiment was devised to find some mathematical relationship between wavelength and frequency of a wave.  The theoretical result is known to be an inverse relationship as described by the equation:

The experiment, devised among 2 lab partners and I was outlined and data was tabulated on a white board (Displayed on the steps below).


Steps:

Steps needed in our experiment.

The task of finding data for frequency was simplified by counting the time for 30 oscillations, dividing this number by 30, then finding the inverse (averaging 30 oscillations gives a better approximation).

Put on an Excel sheet:

wavelength	trial 1	trial 2	trial 3		Trial 1 period	Trial 2 period	Trial 3 period	Ave period	frequency
2.2	9.8	10.6	8.53	9.6433333333	0.3266666667	0.3533333333	0.2843333333	0.3214444444	3.1109574836
3.2	10.8	11.6	9.38	10.5933333333	0.36	0.3866666667	0.3126666667	0.3531111111	2.8319697923
4.2	11.9	11.3	11.7	11.6333333333	0.3966666667	0.3766666667	0.39	0.3877777778	2.5787965616

Average Frequency vs Wavelength was graphed with the data above 

Questions/Conclusion:

The result obtained was close to linear, but we were nonetheless able to fit an inverse onto the data.  There is a clear relationship describing a decreasing wavelength as frequency is increased.  This was definitely the case in our experiment as our effort to create a standing wave was eased as we increased distance from one another (less up and down motion required).  The exponent of this graph is at -1.22 which is near the expected value of -1  since