4/17 Relativity of time and length

Purpose
The purpose of this experiment was to gain knowledge of relativity and understand how choosing different reference fram can cause the time dilation and length contraction by running the simulation.

Time dilution

     1. The distance traveled by the light pulse on the moving light clock is longer than the distance traveled by the light pulse on the stationary light clock.

     2. The time interval for the light pulse to travel to the top mirror and back on the moving light clock is longer than the interval on the stationary light clock because the distance is longer in the moving clock and speed of light is constant.

 

     3. The light pulse travel a larger distance when the clock is moving, and hence require a larger time interval to complete a single round trip. The minimum time interval between events is referred to as the proper time.



     4. As the speed of the light clock is reduced, the difference between the distance traveled by the light pulse and the distance between the mirrors decreases. As this distance difference decreases, the time difference also decreases. Comparing the images below and above, the time difference decreases as the gamma value decreases. When gamma value decreases, the velocity of the traveling light clock decreases, and the time difference decreases.

     5. Using the time dilation formula, predict how long it will take for the light pulse to travel back and forth between mirrors, as measured by an earth-bound observer, when the light clock has a Lorentz factor (γ) of 1.2.

Δt = γΔt_{proper =} 1.2(6.67 µs) = 8.004 µs predicted

actual Δt= 8.00µs very close to the predicted, 8.004s.

     6. If the time interval between departure and return of the light pulse is measured to be 7.45 µs by an earth-bound observer, what is the Lorentz factor of the light clock as it moves relative to the earth?
 

Δt = γΔt_proper  

Δt = 7.45 µs; Δt_proper  = 6.67 µs

γ =1.12

proved!

 

Length contraction
     1. The measurement of this round-trip time interval depend on whether the light clock is moving or stationary relative to the earth.

     2. The round-trip time interval for the light pulse as measured on the earth be longer than the time interval measured on the light clock.

 

     3. In order for the time intervals to obey this law, the length of the moving light clock had to be made smaller.
     4. Lp = 1000m, γ =1.3. L=?
         L= Lp/γ = 769m

proved!

 

Conclusion

When one event occurs, people in different frames observed it at different times. People who are in a relative stationary fram will observe a longer time than people who are in a moving frame, and the proper time means the time interval measure on a clock located at the events. However, people who are in the relative stationary fram will observe a shorter length, and the proper length is the length of an object measured in the reference fram in which the object is at rest.



4/10 CD Diffraction

Purpose

The purpose of this experiment was to determine the distance between grooves on a CD by shining a laser on the grooves, observing the interference pattern resulted on a piece of paper, and making measurements to calculate the d.

Experiment

 Set up to calculate the wavelength of the laser used

 

The wavelengh of the laser was calculated to be 668 nm

 

CD used in experiment

 

Set up to measure the distance between grooves on the blue CD.

 

 

Conclusion

    

The theoretical groove distance was 1600 nm while the experimental grooves distance was determined to be 2992 nm ± 212 nm, with a percent error of 87%. Therefore, the experimental data did not lie within the uncertainty of the equipment, and it is off by a factor of 2. Possible errors could be there was angular separation: the CD was not parallel to paper screen; the three diffracted light dots on the screen were not completely horizontal. Also, the scratches on the CD could cause a significant deviation of the groove distance from its true value because the scartches can destroy the uniform distance between grooves, thus the distance we measured here could be the interference pattern from a couple scratches on the disk instead of the grooves.



4/3 Measuring a human hair

Purpose

     The purpose of this experiment was to measure the thickness of a human hair with a laser and micrometer and compare and values from this two methods.

Experiment

Set up

 

Interference pattern

 

Measuring the thickness of a human hair with micrometer.

 

Data

 

Wavelength,λ (nm)	633
Distance,L (cm)	150 ± 1
Distance between 2 fringes,y (mm)	11.906 ± 0.01
Experimental diameter of the hair (μm)	79.7 ± 0.65
Actual Diameter of the hair (μm)	65 ± 10
% error (%)	22.6

 

Conclusion

 

The experimental diameter of the hair was measured to be 79.7 μm while the actual diameter was 65 μm. Also, these values were within the uncertainties of each other since the smallest experimental diameter was 79 μm and the largest actual diameter was 75 μm. The method using micrometert to measure the thickness of a human hair was more accurate and faster than the laser method because this method involved less measurements, thus less uncertainty and errors.       



3/20 Elecrtromagnetic Radiation

Purpose
The purpose of this experiment was to investigate the behavior of EM radiation due to a simple antenna.

Experiment

 

Set up

 

 

Three ways to verify the signal on the oscilloscope is generated by the transmitting antenna:

1. Move the wire back and forth to see if there's a change in wave shown in the oscilloscope
2. Unplug the detector
3. Turn off the function generator

Data:We are measuring the peak to peak amplitude of the received EM wave as a function of the distance from transmitter.

Distance (m)	# of divisions peak to peak	Vertical scale(mV)	Peak to peak amplitude (mV)
0 ± 0.02	3.5 ± 0.1	50	175 ± 5
0.05 ± 0.02	1.2 ± 0.1	50	60 ± 5
0.10 ± 0.02	3.2 ± 0.1	20	64 ± 5
0.15 ± 0.02	2.9 ± 0.1	20	58 ± 5
0.20 ± 0.02	2.8 ± 0.1	20	56 ± 5
0.25 ± 0.02	2.4 ± 0.1	20	48 ± 5
0.30 ± 0.02	2.2 ± 0.1	20	44 ± 5
0.35 ± 0.02	1.9 ± 0.1	20	38 ± 5
0.40 ± 0.02	2.0 ± 0.1	20	40 ± 5
0.45 ± 0.02	1.5 ± 0.1	20	30 ± 5
0.50 ± 0.02	3.8 ± 0.1	10	38 ± 5
0.55 ± 0.02	3.6 ± 0.1	10	36 ± 5

A/r + b plot:



A/r^2 +b plot:



 A/r^3 + b plot:



A/r +b plot best fits our data, but our data differed from a 1/r function by a b component because of the approximation used when calculating field near the transmitter. The near field is a function of A/r^3 and far field is a function of A/r.

Theoretical Ananlysis

From the calc variant,

where:

L = 0.1 m

k = 9 * 10^9 N^2 * m^2 / C^2
Q = 1.24 * 10^(-10) C
V_0 = 40.29 mV

Therefore,

Distance (m)	Theoretical Amplitude (mV)
0.05	59.99751
0.1	51.0499
0.15	47.61179
0.2	45.82335
0.25	44.73271
0.3	43.99963
0.35	43.47352
0.4	43.07777
0.45	42.76936
0.5	42.52229
0.55	42.31994

 

The experimental and theoretical data have the same trend, but the experimental fluctuate within the theoretical data plot.

Uncertainty Analysis:

Voltage uncertainty: ΔV = ± 10 mV (for a scale of 50mV)

                                        = ± 4 mV (for a scale of 20mV)
                                        = ± 2 mV (for a scale of 10mV)

Distance uncertainty: Δz = ± 0.001 m (uncertainty of the ruler)

                                        = + 0.02 m (detector is not a point and has length involved in measurement )

Total distance uncertainty: Δz_total = + 0.021 m / - 0.001 m

Using the uncertainty in distance, we can calculate the theoretical Vmax and Vmin

Distance (m)	Voltage max (mV)	Voltage min (mV)
0.05 (+0.021/-0.001)	60.32	54.96
0.1 (+0.021/-0.001)	51.15	49.29
0.15 (+0.021/-0.001)	47.66	46.74
0.2 (+0.021/-0.001)	45.85	45.31
0.25 (+0.021/-0.001)	44.75	44.39
0.3 (+0.021/-0.001)	44.01	43.76
0.35 (+0.021/-0.001)	43.48	43.29
0.4 (+0.021/-0.001)	43.08	42.94
0.45 (+0.021/-0.001)	42.77	42.66
0.5 (+0.021/-0.001)	42.53	42.43
0.55 (+0.021/-0.001)	42.32	42.25

The experimental graphs does not lie within theoretical with uncertainty.

Discussion:

1. The simplification we made in our model is treating the antenna as a point source but it actually not and it has some length with it. Also, we assume our detector is a point detector but it actually has some length involved in our measurement, too. For example, when we measure the distance from the antenna to the detector is 5 cm, it actually less than 5 cm because of the length of the detector.

2. Maybe improving the model by increasing the distance from the antenna to the detector, so both the antenna ans detector can be treated as a point source.