Diffraction - The TA\'s name was Manoj Kumar. This is a full lab report. PDF

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The TA's name was Manoj Kumar. This is a full lab report....

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Woods

Diffraction Lab Report

Dana Woods

Lab Partner: Katherine Andersh Course: PHYS182-015 TA: Manoj Kumar Due Date: 11:00 AM on 4/12/17

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Abstract The lab required us to look at the diffraction of a laser through small holes. It was first observed that as a razor blade gets closer to the laser, more of the light seems to spread out on the screen due to the diffraction of the laser light. We then looked at single slits. As the screen distance increases, the minimum to minimum distance also increases. The opposite trend was seen with increasing slit width. When we changed the laser to one with a larger wavelength, the corresponding slit width values were slightly larger than with the green laser. When using a piece of hair as an obstruction, we measured the diameter using the single slit theory (the hair acted like a slit) and determined that it was 0.00975 cm. Next, we looked at the laser going through a double slit. As the slit width and slit separation increased (independently), the maxima to maxima distance decreased. When the wavelength changed this time, there was not a clear trend in the data. Finally, we experimentally determined the spacing of the slits in a grating slide mount. This was determined to be 0.0028 cm.

Goals of Experiment The goal of this experiment was to investigate how a laser beam diffracted through a small slit or multiple small slits We explored the effects of wavelength, distance, slit width, and slit separation on the distances between dark spots (for single slits) or light spots (for multiple slits).

Introduction When particles from a point source encounter a hole, some of the particles will go through the hole and land on a screen, where we will see the light or the image. For these particles, as the hole gets smaller, the image present on the screen gets smaller. This is how we have treated waves thus far. However, most light acts like a wave in water. When one drop of water falls into a larger body of water, the ripples spread out in circles of waves that seem endless. They continue to expand from the initial source. This is what happens when a light source passes through a small hole. The waves of light pass around the hole and continue to spread out on the other side instead of just going straight through the hole. Therefore, light waves have the opposite effect of particles; as the hole gets smaller, the image on the screen gets bigger because the waves spread out more behind the hole. These light waves also behave differently depending on the number of slits they go through. When light passes through a single slit, the points at which the waves cancel (termed dark spots or minima) are measured because these places have the same distances above and below the midline. When light passes through a double slit, the points with the maximum intensity (termed bright spots or maxima) is measured and the light spreading out through both slits interferes with the other slit. When viewed, the double slit light pattern will also have a single slit light pattern overlaid on it because the two single slits have the same width. It is important to understand diffraction because it has many practical applications, such as x-ray diffraction and holograms.

Theory and Derivations When light travels through the air, it travels as a wave would through water, as discussed in the introduction. When it encounters a slit, the light goes through the slit and spreads out behind the slit, a process known as diffraction. The light behaves differently depending on how many slits there are. When there is one slit, it follows the single slit theory. At distances above or

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Woods below the midline, the waves cancel and create dark spots on the screen. This is because the waves throughout the slit distance going to that spot do not travel the same distance and therefore arrive at the spot at different times. They cancel out if h=

Dλ a

(1)

where h is the distance above or below the midline, D is the distance from the slit to the screen, λ is the wavelength of light, and a is the width of the slit. Because these lights act as waves, it makes sense to express the minima positions as trigonometry functions. For a single slit, this expression is aSinθ =nλ

(2)

where θ is the angle between the midline and the highest wave of light that hits that point, and n indicates which dark spot you are looking at. This equation is used to analyze single slit minima and determine the diameter of a hair in the lab. When there are two slits that the light goes through, the light follows the double slit theory. This theory is very similar to the single slit theory, and is given by dSinθ =mλ

(3)

where d is the slit separation and m indicates which bright spot you are looking at. The main difference is that the double slit theory is that it measures from maxima to maxima instead of minima to minima. The angle can be verified by using the equation tanθ=

h D

(4)

for both a single slit and a double slit. This is because the height of the minima or maxima and the distance to the screen are simply two sides to a right triangle, so the angle found will be the same angle found in the previous two equations. Finally, a percent error calculation can be used to determine what percent of error was in our measurements. This is given by the equation Value | ActualValue−Theoretical |∙ 100 Theoretical Value

% Error=

(5)

and was used for the last part of the lab.

Procedure The first section of the lab required us to look at single slits. The first thing we observed was what happened to the light on the screen when a razor blade was put in front of the laser light and was brought closer to the laser beam (a “half slit”). The screen was placed 100 cm away from the green laser (and was kept at this distance for the entire experiment unless this distance was the variable). The razor blade was put in the binder clip stand 10 cm away from the laser and 3

Woods brought closer to the laser. We then replaced the binder clip stand with the single slit wheel. We observed what the light pattern on the screen looked like when using the green laser passing through the 0.004 cm slit. Several minima to minima measurements were taken in order to determine why this method for measuring was preferable to measuring from the middle. Then, we began altering several variables to determine how the minima to minima distances changed. First, the screen distance varied by sliding the screen closer to the laser. Then, the slit width was altered by rotating the slit wheel to various widths. Finally, we changed the wavelength by removing the green laser and using the red laser instead. The last part of the single slit experiments required us to place a hair on the binder clip stand and place it in the middle of the green laser and determine the diameter of the hair using the single slit theory, as the hair acts like a single slit in front of the laser light. The second part of the lab looked at double slits. We initially observed what the green laser looked like on the screen when the slit width was 0.004 cm and the slit separation was 0.025 cm. This pattern was compared to the pattern from the single slit wheel from the first part of the experiment. We then changed varied the slit width and the slit separation (independently of each other) by rotating the multiple slit wheel to the correct double slit pair. We then switched over to the red laser and did the same width and separation pairs to compare the different wavelengths. Finally, we looked at a grating slide mount and had to experimentally determine the slit separation. The 1000 slits/cm grading mount was used and was placed 10 cm from the red laser. The double slit theory was then used to determine the distance between slits.

Figure 1. General setup for the entire lab. Shows both laser colors, the slit wheels (single and multiple) and the screen.

Sample Calculations and Results To introduce us to how slits affect a laser light, a “half slit” was created by putting a razor blade in front of the laser. We observed that as more of the laser light is covered, the more the

4 Figure 2. Illustration of single slit image on screen.

Woods light spreads out on the screen. Then, we began using full single slits from the single slit wheel. Figure 2 shows the pattern made with the single slit wheel set to the 0.004 cm slit width. The pattern ran horizontal, while the slit in the wheel was vertical. We then began testing variables. Table 1 compiles the data for the varying screen distances. It was found that as the distance decreases, the minima to minima distance also decreases. This trend is verified in Graph 1. According to Equation 4, because both of these values are decreasing, if they are decreasing while still being in the same ratio (as they should), then the angle should remain the same. This can be verified using the Equation. tanθ=

(

)

h 1.5 cm =¿ θ=tan−1 =0.86 ˚ D 100 cm

(6)

The other angles were 0.93˚, 0.95˚, 0.86˚, and 0.86˚. There is some variation in the answers, but this could be due to errors described later. The next thing we did was vary slit width for the different wavelengths of light, as seen in Table 2. For both wavelengths, the trend found was as the slit width increased, the minima to minima distance decreased. The values for the distance were slightly larger for the red wavelength. We can again look at the angle that this creates with the changing slit width. This time, we will use Equation 2. −7

(1)( 532 x 10 cm) =1.52 ˚ aSinθ =nλ =¿ θ=sin 0.002 cm −1

(7)

The other angles for the green laser with increasing slit width are 0.76˚, 0.38˚, and 0.19˚. For the red slit, these values were 1.86˚, 0.93˚, 0.46˚, and 0.23˚. At the end of this first part, we determined the diameter of a hair placed in front of the laser. This value is equivalent to the width of the slit, so we can use Equation 1 to calculate it. −7

h=

(110 cm )(532 x 10 cm) Dλ =0.00975 cm =¿ a= 0.6 cm a

(8)

This value makes sense, as a hair is very small, and this is a very small number. The second part of the experiment had us look at multiple slits. We first looked at double slits. Figure 3 shows the pattern we observed when looking through the Figure 3. Illustration of double slit image on screen. 0.004 cm slit width and 0.025 cm slit separation combination. The pattern looked very similar to the single slit pattern, with the main difference being that the larger bright spots were split up into smaller bright spots. This confirms the fact that the double slit pattern is superimposed by a single slit pattern. We changed similar variables for this section. First, we altered the slit width and slit separation independent of each other. This data is shown in Table 3. As the slit width increases, the maxima to maxima distance decreases. As the slit separation distance increases, the maxima to maxima distance decreases. We ca use these values to calculate the angle like we did for the single slit. This time, we will use Equation 3 because this is the equation for the double slit theory. (9)

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Woods dSinθ=mλ =¿ θ=sin−1

(1)(532 x 10−7 cm ) =0.12˚ 0.025 cm

The other values for the green laser were 0.06˚, 0.12˚, and 0.06˚. For the red laser, the angles were found to be 0.15˚, 0.07˚, 0.15˚, and 0.07˚. The final part of the multiple slit part was looking at a grating mount and determining the separation distance between the slits on the mount. We first found the angle that the image created using the middle to maxima and the distance to the screen. tanθ=

(

)

7.1 cm h =¿ θ=tan−1 =4.06 ˚ D 100 cm

(10)

This angle was then used to determine the slit width using Equation 3. dSinθ=mλ =¿ d=

(1)(650 x 10−7 cm ) =0.000918 cm sin (4.06 ˚ )

(11)

We determined the slit separation by multiplying this number by three, because three bright spots were visible on the screen. This gave us a slit separation of 0.0028 cm. The final part of the experiment was determining the percent error in our slit width. The slit width was given on the mount as 0.001 cm, and we experimentally determined that it was 0.000918 cm, so we can use Equation 5 to determine the percent error. (12) −Theoretical Value 0.000918 cm−0.001 cm ∙ 100=| | ActualValue | |∙ 100= 8.22% Theoretical Value 0.001 cm

% Error=

This is a good percent error, indicating that we were not far off from the actual value.

Discussion and Conclusions The purpose of this lab was to explore diffraction and how it changes with changing variables. The first portion of the lab involved a single slit. When we observed the light on the screen as more of the razor blade covered the laser, we saw that more of the light was spread out over the width of the screen. This was our first introduction to the idea of diffraction; as the distance that the light can go through gets smaller, the light will bend out further and the image will appear larger on the screen. We then used the slit wheel and observed the same thing. This time, we were able to observe the bright and dark spots made by the addition or cancellation of the light waves, respectively. The first variable we tested was varying screen distance. It was found that as screen distance decreases, the minima to minima distance on the screen also decreases. This makes sense because, according to Equation 4, these two variables are directly proportional to each other. When the angles were calculated, they were all almost the same (some error, explained shortly). This also makes sense because if the other two variables change while keeping the same ratio, the angle should remain the same. Next, we changed the slit width during both the green and the red laser. For each, the trend was as the slit width increased, the minima to

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Woods minima distance decreased. This makes sense because, as explained in the introduction, light waves will diffract more if the slit is smaller. This effect also effects the angle that is made. Since Equation 2 only depends on the slit width, and the width and angle are inversely proportional, as the slit width increases, the angle should decrease. This is what was observed. When the wavelength was altered, the angles increased for each corresponding slit width. This makes sense with Equation 2 as well, because the wavelength and the angle are directly proportional. This makes sense according to the theory of diffraction. The final part of the single slit experiments was to obstruct the laser with a single hair and determine its diameter. In this case, the diameter of the hair is equivalent to the width of a slit. Therefore, we were able to use Equation 1 to find the slit width, which corresponded with the hair diameter. It is interesting to note that such a small obstruction, or slit, can cause light to have a large diffraction across the screen. The second set of experiments looked at how multiple slits affect the diffraction of the laser light. We first observed the pattern made on the screen using the multiple slit wheel. It was observed that the double slit pattern was superimposed with a single slit pattern, which matches what was described in the introduction. We varied the slit width and slit separation (independently) with the different laser colors. As the slit width increased, the maxima to maxima distance decreased. The same trend was seen with increasing slit separation. These observations were the same for both laser wavelengths. The angles for these were again measured. Because the angle for the double slit theory only depends on slit separation, the slit width did not matter for this calculation. This is why there are only two different angles for each wavelength. According to Equation 3, slit separation is inversely proportional to the angle, which is why the angle decreases with increasing slit separation. The angles for the larger wavelength (red laser) were slightly larger, due to the fact that the angle is directly proportional to wavelength. This follows the theory of diffraction laid out previously. The final part of this section and the lab was determining the slit separation distance in the grating mount. We used the 1000 slits/cm grating for this experiment. We initially found the slit width using the middle to maxima height, the distance from the slit to the screen, and the wavelength (red laser was used). Even though the slit width was given, we decided to experimentally determine it and use this value for our experimentally determined slit separation. This way, we could also determine a percent error. In order to determine the slit separation, we decided to use the width of the laser by using the number of slits the laser intersected (number of bright spots on the screen). This gave us our answer, which made sense when looking at some of the other slit separation distances from the slit wheel. The main error seen in this lab was using a ruler to measure the distances on the screen. It was difficult to get the ruler lined up exactly where it needed to be and stay there, especially because of the fact that we were not looking directly at the screen. This could give us multiple errors, such as the numbers in the Tables, the angle calculations, or the slit width found in the final portion of the lab (percent error). Another error could be that the lab equipment was not placed at the exact distances that we said they were. They could have been off by a couple centimeters, giving rise to slight errors in the values we found. These were the two main errors that could have cause discrepancies in our data. Other than these slight inaccuracies, the goals of the lab were met with reliable data that matched what the theory states.

Varying Screen Distance (Single Slit)

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Woods Screen Distance (cm) 20 40 60 80 100

Minima to Minima Distance (cm) 0.6 1.2 2 2.6 3

Table 1. The distance from minima to minima as the distance from the laser to the screen changes. As the screen distance increases, the minima to minima distance increases.

Varying Slit Width (Single Slit) Green Laser (532 nm) Red Laser (650 nm) Slit Width Minima to Minima Distance Slit Width Minima to Minima Distance (cm) (cm) (cm) (cm) 0.002 7.2 0.002 8 0.004 3 0.004 2.6 0.008 1.6 0.008 1.8 0.016 0.8 0.016 1 Table 2. The distance from minima to minima as the size of the slit changes. As the slit width increases, the minima to minima distance decreases. The red laser values are slightly higher than the green laser for the corresponding slit widths.

Slit Width (cm) 0.004 0.004 0.008 0.008

Varying Slit Parameters (Double Slit) Green Laser (532 nm) Red Laser (650 nm) Slit Maxima to Maxima Slit Slit Maxima to Maxima Separation Distance (cm) Width Separation Distance (cm) (cm) (cm) (cm) 0.025 0.6 0.004 0.025 0.6 0.05 0.23 0.004 0.05 0.1 0.025 0.33 0.008 0.025 0.6 0.05 0.2 0.008 0.05 0.23

Table 3. The distance from maxima to maxima as slit width and slit separation change. As the slit width increases, the maxima to maxima distance decreases. As the slit separation increases, the maxima to maxima distance decreases. The red laser and green laser values do not seem to have a specific correlation.

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Varying Screen Distance (Single Slit)

Minima to Minima Distance (cm)

3.5 3

f(x) = 0.03 x + 0.02 R² = 0.99

2.5 2 1.5 1 0.5 0 10

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Screen Distance (cm) Graph 1. Indicates how minima to minima distance changes with increasing screen distan...