Relativity of Time and Length

Introduction:
          The purpose of this experiment is analyze and understand the concept for the time interval between two events with different frame of reference and with the concept of simultaneity. Furthermore, this experiment will also analyze the diffreence between two points when each of them depends on the different frame of reference.
          The concept of time interval was summed up by the followed equation for time dilation:

∆t = γ∆t₀,

where ∆t₀ represents the time of the observer at rest, which is also the known as the rest frame of this observer. ∆t , on the other hand, is known as the frame which is moving with a constant speed relative to the rest frame mentioned earlier.

  

The above equation shows us that ∆t is always larger than ∆t₀ since speed u can never be larger than the speed of light.

          The concept of length constration is explained by the following equation:

l= γ/l₀

where, the γ is the same symbol used for the time dilation. Therefore, with the same concept that the speend of light is alwasy faster than the  moving speed, the lenght of the stationary frame l₀ , which is also known as the rest frame is alwasy shorter than the lenght of the moving frame l, and that is why the equation is used for determining the length constration.

          In order to fully undertand the concept behind its equation, the online source , Active Physics would be used to see and prove how the time and the lenght changes with the constant speed applied in different frame of reference.

Pictures for Time Interval:

The left picture shows the stationary frame of reference.

The right picture shows the moving frame of reference.

Image represents the time interval when γ = 1.40

Image represents the time interval when γ = 1.30

Image represents the time interval when γ = 1.20

Analysis&Conclusion:

          The above three images has different γ values, since γ depends on the ratio of

u²/c², therefore, when the values γ is larger, this means u, the speed would be faster. Therefore, with equation ∆t = γ∆t₀, it shows that when γ is larger, the time taken for the light to travel back is longer. The three above images proves the equation, the time tacken for value γ 1.40, 1.30,1.10 are 9.34 (μs), 8.67(μs), and 7.33(μs). On the other hand, the time taken by the stationary clock is same for all three cases, which is 6.67 (μs) since the position where the light leaves is the same as the position when the light reaches back. Therefore,the round-trip distance for the stationary clock will be 2d, and the round-trip ditance for the moving clock, on the other hand,  is denoted as the radical of (d²+ ((u * ∆t )/2)^{2 .}We notice that the round-trip distance for the moving clock will get longer with the increasing of velocity. Hence it also shows that the round-trip distance would be propotional to the time interval.

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Pictures for Time Interval:

The top picture shows the stationary frame of reference.

The bottom picture shows the moving frame of reference.

Image represents the interval when γ = 1.40

Image represents the interval when γ = 1.30

Image represents the interval when γ = 1.40

Analysis&Conclusion:

The above three images has different γ values, since γ depends on the ratio of

u²/c², and since the equation l= γ/l₀ , it shows that the length contraction would be the reciprocal of the proper lentgh.  Therefore, when the values γ is larger, which also means u, the speed would be faster. Therefore, with the constanct length of the stationany clock, it shows that the constraction length will get shorter as the velocity of the light beam decreases.  This result is also proved by the above three images, where the value of γ is 1.40, 1.30, and 1.10, the constraction lenght follows as 714(m), 769(m), and 909(m). These values show that the value γ is recipricol to the length at the moving frame of inference. On the other hand,  the time taken at the frame of inference for the stationary clock is all 6.67 (μs) for each case, and the proper lenght for each case for the stationary clock is all 1000m. Furthermore, by applying the equation of ∆t = γ∆t₀, it shows that it shares the same concept of time interval when it was applied for the time interval tacken at different frames of inference mentioned earlier in this experiment.

Light and Matter Waves

Introduction:

In this experiment, the language program, Vpython, would be introduced and be used to get the visualization of the electric field through the space. Besides that, the countour plot of the electric field could also be obtained by using Vpython. With all the visualizations, it allows us to analyze how the electric changes when different wavelengths, or different seperation between the two sources was adjusted. The equation used in this experiment was

E₀=A*cos(2*pi*r/wavelength)/r

From here, it shows that the electric filed would be changed based  on the wavelength, and also the distance between the two sources, which is also denoted as r in this equation.

 
Here shows the sample Vpython code:

 

from visual import *

import pylab as p
import mpl_toolkits.mplot3d.axes3d as p3

wavelength = 2.0e-3

scrnDist = 5.0e-2
scrnWdth = 2.4e-2
xs=0
ys=0
A=1
N=100
dX=scrnDist/N
Xcoords=arange(dX,scrnDist+2*dX,dX)
dY=scrnDist/N
Ycoords=arange(-scrnDist/2,scrnDist/2+2*dY,dY)
[xd,yd]=meshgrid(Xcoords,Ycoords)
r=sqrt((xd-xs)**2+(yd-ys)**2)
E0=A*cos(2*pi*r/wavelength)/r
#print E0

For this code, each Eo will be the ouput, and when we change some of the code, such as adding

fig=p.figure()

Efield=p3.Axes3D(fig)

Efield.plot_wireframe(xd,yd,E0)

Efield.set_xlabel('Xd')

Efield.set_ylabel('Yd')

Efield.set_zlabel('E0')

fig2=p.figure()

p.contour(xd,yd,E0)

p.show()

 

The program then will output the 3-D visualization and also the contour plot.

 For wavelengh 2mm, the electirc field:

For wavelengh 4mm, the electirc field:

For wavelengh 8mm, the electirc field:

From the three above cases, it shows that when the wavelength is increased, the lines on the coutour plot became less intense, and the same situation was also applied to the electric wave on the 3D visualization. 

The 3-D diagram and countour plot for wavelength as 4mm was obtained:

Now, the wavelengh would be adjusted and if it agrees to the prediction, when the wavelength is increased, the the intensity between lines would become larger, and the intensity between lines would become smaller when the wavelengh is decreased.

The 3-D diagram and countour plot for wavelength as 8mm was obtained:

As predicted, the lines became less intense when the wavelengh is increased.

The 3-D diagram and countour plot for wavelength as 8mm was obtained:

As predicted, the lines became more intense when the wavelengh is decreased.

Now the visulizations would be analyzed when the distance between two sources was adjusted:

 As we can see here, the distance between two most intence points became wider as desired.

Now the parameters were changed, the wavelengh now was 600 nm, screen distance was 50nm, screen width was 30 nm, and the souce seperation wsa 0.016nm
This time the intsensity would only be plot at points on the screen, the code was replaced and changed to :

p,plot(yd[:,N], Itot[:,N]

p.show()

Now the new visulization of intensity for the new parameters was:

The left diagram showed the intensity when the wavelengh is small, and the right diagram showed the intensity when the wavelengh was increased. Therefore, the space between lines got wider as the wavelengh was increased.

With these point only diagrams, it still showed that the wavelengh was one of the main factors which caused the changing of intensity.

  

Experiment 12:CD Diffraction

Introduction:

In this experiment, the distance between the grooves on the DCDs had to be measured by using the laser and the concept of diffraction.

The equipments used for this experiment were the laser, Compact Disk, large screen (wooden blocks in our experiment), meter stick.

The really conducting of this experiment was using the laser and pointing it to the shinning side of the CD or DVD, the laser would then be reflecated to a certain distance on the screen. In our experiment, DVD was used and the standard distance between the grooves is 760 (nm).When the experiment was doing, the reflected center point had to be reflected back to where the laser pointer sent out the laser. Then both of the reflected dots, the first order maxima would be found on the screen. Conceptually, the distance between each dot and the central point had to be the same. Therefore, the distance x, which is the distance between one of the first order maxima dot and the central dot, would then be measured. Without the screen being moved, measured the distance between the DCD and the screen. Since the laser pointer being used is the Helium-neon laser, the wavelength of 632.8 (nm) then would be applied.

The distance for grooves needed to be obtaind

 the setup of this experiment

 the laser is reflected to the screen(the wood)

both dots on the wood represented the first order mixima

Given Data:

λ=632.8x10^-9

standard value for the grooves' distance: 760 (nm)

Measured Data:

L=11.7 +/- 1 (cm)

x=19.65 +/- 1(cm)

Calculation:

Calculate θ first: tan(θ)=10.65/22.7

                             θ=tan^-(19.65/11.7)=59.23

                             sin(θ)=λ/a 

                             a= λ/sin(θ)=632.8x10^-9/sin(59.63)=736(nm) 

Percent Error:

|(736-760)/760|x100%= 3.158%

 

Conclusion:

When the experiment was conducting, the relationshoip between the distance between the CD and the screen, and the distance between the first order maxima and the central point is proportional to each other. In other words,. no matter how far the screen was moved, the angel for the first order maxima would not be changed. When the screen was moved further, the distance between the central dot and the first order dot would also increase, hence the angle between stayed constant. In this experiment, the percent error calculated was  3.158% , which was within 10% error, the error could be contributed by the efficiency of the laser pointer, and also the accuracy of the distance's measurements. Based on the diffraction concept, the area between these two first ordr maxima dots was the dark area, as well as where the wave destructed, if one much larger screen was used, then the second order maxima, or even the third order maxima could also be obtained, too.

Experiment 11: Measuring a human hair

Introduction:


For this experiment, the concept of interference would be applied to measure a human hair with the helium-neon laser.
The concept of inference of light is that  when the interference is constructive, the intensity is maximum, hence showing the bright region. On the other hand, when the interference is destructive, the intensity is minimum, hence showing the dark region. In this experiment, the equation, y_m=Rmλ/d,would be applied to calculate the thickness of the hair. For this equation, the only known variable was the wavelength of  the helium-neon laser, which was known as 632.8 (nm). Therefore, in this experiment, the distance between the light source, as well as where the hair was put had to be measured. Besides it, the position of the fringes had also be measured. This distance would be measured from one bright region to the adjacent bright region. IT could also be measured from the center of the interference pattern to the m^th bright region. After obtaining this distance, it had to be divided by the numbers of bright regions so that the distance y_m could then be obtained. In order to get the interference fringes, the laser had to point through the hair directly, and perpendicularly to the white board. After obtaining those measurements mentioned above, the thickness of the hair would then be measured by using a micrometer to see if the result would agree to each other, and the better method had to be chosen if the result did not quite agree to each other.

Laser point to the hair

the laser would point through the hole where the hair was put

This picture showed clearly that laser point directly and perpendicularly through the hole

Use the marker to mark the distance between the fringes, then measure it afterwards

Use Micrometer to measure the thickness of the hair

Data:

Wavelength λ	632.8 (nm)
L	109±(cm)
y	0.8±0.05(cm)
d	90±3 (µm)

Calculation:

λ=d*y/L

-->d=λ*L/y

-->d=(6.328×10^-7(cm)×109(cm))/0.8(cm) = 8.62×10^-3(cm) = 86.2(µm)

Uncertainty:

√((∂ d/∂ L*∂ L)²+(∂ d/∂ y*∂ y )²)= √((λ/y*∂L)²+(-λL/(y^2)* ∂y)²)

√((632.8×10^-7/ 0.8*0.5)²+(-632.8×10^-7×109/0.8²*0.4 )²)=4.31×10^-4(cm)=4.3(µm)

Result:

d(laser)	±4.3 (µm)	90.5(µm)
86.2(µm)		81.9(µm)
d(micrometer)	89.7(µm)~90.3(µm)

Conclusion: 

From the thickness of hair calculated by using the helium-neon laser, the result was 86.2 (µm). However, the result measured by the micrometer was  90±3 (µm) , With the uncertainty plus or minus 4.3 (µm), the result obtained by using the helium-neon laser become the range from 81.9 (µm) ~ 90.5 (µm). Therefore, the range  was within the number obtained by using the micrometer. It showed us that by using the  helium-neon laser and by applying the concept of interference of light, the thickness of the hair could be measured. Since the thickness of the hair was really small, and since the distance between the whiteboard and the slit was large compared to the distance between two bright fringes. Therefore, the equation y_m=Rmλ/d could be applied, and the experiment also proved that this equation worked for measuring the thickness of the hair. Since this equation could be used in this experiment, and based on what the concept of the two-source interference of light, it showed us that light waves constructed at where the brightest points were, and destructed at where the darkest points were. Therefore, with the applying of interference of light concept, the result of the hair's thickness could be more accurate compared to the hair's thickness measured by the micrometer. However, if the experiment had to be done with applying the interference of light theory, the high intensity source, such as the laser had to be fetched easily. It is the disadvantage of the interference of light. However, on the other hand, if the micrometer was not provided, the concept of interference could be applied to measure the target object. 

Experiment 10 : Lenses

Introduction:

In this experiment, the image appeared on the white board would be analyzed based on the distance, as well as the length of the distance taken from different numbers of the focal point.

Before the experiment was started, the focal point of the lens had to be obtained from an infinite source, which the chosen source was the sun. In order to find the focal point, one point would be found very intense where the light from the sun converged. After the focal point being obtained, the object distance would be varied so that the image appeared on the white board could be observed and analyzed. The object distance would be changed into the distance of one, two, three, four, and five times of the focal point. Since the object height would always be constant, which was also the height of the Socket lamp with V-shaped filament. Therefore, the magnification, M, would depend on the height of the image since its equation is the image height divided by the object height. With the different object distance and with the focal point, the image distance and the image height should be only varied based on the object distance. Therefore, with the experimental observations, the theory of lens should be agreed experimentally.

Materials:

Socket lamp with V-shaped filament

Large converging lens

Large split lens or masking tape

Lens holder for large lens

cardboard

Meter stick

the image would appear on the white board, and the clearest image had to be obtained 

the light source would go through the lens

The object (also the light source)

The object distance started from the light source

The image appeared on the white paper, the image height could be obtained

The thick of the lens produced reflections in itself

Clearly the size and the height of the image was different

Data:

Converging lens		Focal Point:     9.22(cm)
# of focal point distance	Object distance do, cm	Image distance di, cm	Object height ho, cm	Image height hi, cm	M	Type of image
5f	46.1± 0.5	8.5± 0.5	9.0± 0.5	2.2± 0.5	~7/30	Real /Inverted
4f	36.8± 0.5	9.7± 0.5	9.0± 0.5	3.1± 0.5	~1/3	Real /Inverted
Reverse the lens	36.8± 0.5	9.7± 0.5	9.0± 0.5	3.1± 0.5	~1/3	Same as above
3f	27.7± 0.5	10.0± 0.5	9.0± 0.5	3.4± 0.5	~1.1/3	Real /Inverted
2f	18.4± 0.5	11.8± 0.5	9.0± 0.5	6.0± 0.5	~2/3	Real /Inverted
1.5f	13.8± 0.5	17.2± 0.5	9.0± 0.5	12.0±0.5	~4/3	Real /Inverted

Equation for Magnification = h'/ h or -s'/s

                        in this case, the magnification would be obtained by calculating h_i/h_o

---The image was observed to be the same when the object distance was kept constant, and the lens was reversed.

---When the top of the lens was masked with the tape without moving the lens, the image would still from on the white board, however, the image would become dimmer compared to the original appeared image.

---When the object was put a distance to 0.5 f, there was no image appeared on the white board. Then when the image was observed through the lens, the image turned to be erect. Therefore, in this case, the image was virtual since it could not appear on the white board and this virtual image should appear behind the object since the object was inside the focal point which could not form a real image the other side of the lens.

The picture below shows how the object distance is less than the focal point length:

 

Conclusion:

From the observations based on the experiment, it showed us that when the object distance was further than the focal point, the image formed on the white board, and from the magnifications calculated above, the sequence of the magnification was 4/3, 2/3, 1.1/3, 1/3, 7/30 with the increase of the object distance. Hence, we notice that the image size decreased as the object distance increased. However, even if we did not test the distance the same as the focal point, the image should be real and erect but might probably be too small to see. Based on the experiment, when the object image decreased and became smaller than the focal point, the image would be erect but not form on the whiteboard. The image would then be virtual and on the same side as the object.
In addition, after the experiment, when parts of the lens was masked, the image could still be formed on the whiteboard, however, the image was dimmer. That's because the light was still be able to go through the lens, however, the amount of the light went through the lens was less, so that the image appeared was dimmer.

From all the tests in this experiment, it showed us that the light was converging light with the application of convex lens. If the light could not converge because of the object distance, then the virtual image would appear, in other word, the light would virtually converged the opposite direction of the incident light, since it was not technically converging, the real image could not be formed.