PCS 213 LAB 2 (3)

Introduction: The objective of this lab is to observe the interference and diffraction of microwaves. These observations will then be used to determine the wavelength of the microwaves. To determine the wavelength of the microwaves the equations and must ????θ = ?λ ???θ = ? λ 𝑎 first be understood, a derivation of these equations is shown in the theory section. There are three parts to this lab. In the first part the microwaves will be shot through a double slit setup, the slit separation (d) for this experiment will be 7.5cm. Data will then be recorded every 5° until an angle of 60° is reached on both sides of the center. This data will then be graphed with the intensity being a function of the angle theta. Using the graph and equation (3) the wavelength will be calculated. In the second part the microwaves will be shot through a single slit setup. This will be done twice, the size of the opening between the metal plates is 1.5cm for the first trial and and the size of the opening between the metal plates will be 6cm for the second trial. Data will then be recorded every 5° until an angle of 60° is reached on both sides of the center. This data will then be graphed with the relative power being a function of the angle theta. Using the graph and equation (4) the wavelength will be calculated. The objective of the third part of the lab is to observe water’s ability to absorb microwaves. This will be done by way of three trials, in the first trial the intensity will be recorded when there is no material between the transmitter and the receiver. In the second trial the data will be recorded when a piece of paper towel is placed in the receiver horn. In the third trial a piece of wet paper towel will be placed in the receiver horn this data will also be recorded.

Theory: The equation for the double slit experiment and the equation for the single slit are needed to be understood to be able to perform the lab. A derivation and explanation of both are shown below Equation for the double slit experiment: Figure 1: Diagram of the double slit experiment.[3] (1) ????θ = ? For constructive interference to occur must be an integer number of wavelengths greater than ? 2 knowing this equation (2) is created. ? 1 ? 2 = ? 1 + ?λ (2) ? 2 − ? 1 = ?λ = ? Subbing equation (1) into equation (2) we get. (3) ????θ = ?λ Therefore the equation for the double slit experiment is [1] ????θ = ?λ

Procedure: Part 1 Double Slit Experiment: Figure 3: Double slit Experiment 1. Set up the goniometer with both the transmitter and receiver mount at either end. The transmitter should be mounted at least 30cm away from the center. The receiver should be mounted as far away from the center as possible. 2. Set the double slit using one small metal piece and two larger ones set in the middle of the magnetic mount. The spacing of each slit should be around 1.5cm wide for both slits. 3. Plug in the transmitter and turn on the receiver to the 30x range or whichever range supplies sufficient readings. Adjusting the variable sensitivity knob as required if readings are unstable. 4. Survey the range which you will be taking data from (120 to 240 degrees). You should be seeing patterns of maxima and minimas as the degrees change. 5. Record the receiver’s readings as a function of the angle theta at each 5 degree interval up to 60 degrees on both sides of the center. 6. Using a ruler measure the center to center separation (d) of the two slits.

Part 2 Single Experiment: Figure 4: Single slit Experiment 1. Using only the two large metal screens create a 1.5cm opening for the slit. Then mount the slit onto the center of the magnetic component holder. The transmitter should be mounted at least 30cm away from the center. The receiver should be mounted as far away from the center as possible. 2. Record the receivers readings as a function of the angle at each 5 degree interval up to 60 degrees on both sides of the center. 3. Repeat all of the above steps with a single slit of width 6cm. Part 3 Absorption of Microwaves: 1. Remove the magnetic mount in the center. 2. Record the reading on the receiver with no paper towel between the transmitter and receiver. 3. Crumple up the paper towel and place it inside the receiver then record the reading. 4. Remove the paper towel from the horn and wet it. Place the damp paper towel inside the receiver then record the reading. Results and calculations: To see if the lab was successful the experimental wavelength will be compared with the theoretical wavelength. The experimental wavelength was gathered by graphing the intensity vs the angle theta. Using that graph and equation (3) and (4) the wavelength was calculated.

Graph 1: Double Slit (Intensity vs Angle Theta) Calculation of Wavelength: Using equation (3) the experimental wavelength will be calculated. (3) ????θ = ?λ First we must get , to do so the average angular separation from the center must be θ 𝑎𝑣? calculated. To find the average angular separation, the positions of all the maximums on the graph must first be identified. The smallest measurement on the Goniometer was 1 dividing this number by two will give yield ° the uncertainty. Therefore the uncertainty is ± 0. 5° The four maximums are located at θ 1 = 65° ± 0. 5° θ 2 = 80° ± 0. 5° θ 3 = 100° ± 0. 5° θ 4 = 110°± 0. 5°

(90±0.5°−θ 1 )+(90±0.5°−θ 2 )+(90±0.5°−θ 3 )+(90±0.5°−θ 4 ) ? = θ 𝑎𝑣? = (90±0.5°−65±0.5°)+(90±0.5°−80±0.5°)+ 90±0.5°−100±0.5° | |+ 90±0.5°−110±0.5° | | 4 θ 𝑎𝑣? When Adding or subtracting two numbers with uncertainties, the uncertainties will add together. (25±1°)+(10±1°)+(10±1°)+(20±1°) 4 = θ 𝑎𝑣? 65±4° 4 =θ 𝑎𝑣? θ 𝑎𝑣? = 16. 25°± 1° Uncertainty for the sine functions [4] ∆???θ 𝑎𝑣? = ???θ 𝑎𝑣? · ∆? ∆? = (1°· π 180 ) ∆???16. 25° = ???16. 25° · π 180 =0.0168 ∆???16. 25° ????θ 𝑎𝑣? = ?λ 7. 5???(16. 25° ± 1°) = ?λ Uncertainty for the distance between the slits (d) The smallest measurement on the ruler used was 1mm, therefore the uncertainty for (d) is 0.5mm or 0.05cm λ = (7.5??±0.05??)(0.2798±0.0168) 4 λ = (7.5± 0.05 7.5 ·100%)(0.2798± 0.0168 0.2798 ·100%) 4 λ = (7.5±0.667%)(0.2798±6%) 4 Adding the uncertainty and multiplying we get (0. 667% + 6%) (7. 5 · 0. 2798) λ = 2.0985±6.67% 4 6. 67% ?? 2. 0985 ?? 0. 14

60° 0.3 Table 2: Angle Theta and Intensity (a=1.5cm) Graph 2: Single Slit First Trial (Intensity vs Angle Theta) Calculation of Wavelength: Using equation (4) the experimental wavelength for the first trial will be calculated. The four minimas are located at θ 1 = 70° ± 0. 5° θ 2 = 80° ± 0. 5° θ 3 = 100° ± 0. 5° θ 4 = 110°± 0. 5°

(90±0.5°−θ 1 )+(90±0.5°−θ 2 )+(90±0.5°−θ 3 )+(90±0.5°−θ 4 ) ? = θ 𝑎𝑣? = (90±0.5°−70±0.5°)+(90±0.5°−80±0.5°)+ 90±0.5°−100±0.5° | |+ 90±0.5°−110±0.5° | | 4 θ 𝑎𝑣? (20±1°)+(10±1°)+(10±1°)+(20±1°) 4 = θ 𝑎𝑣? 60±4° 4 =θ 𝑎𝑣? θ 𝑎𝑣? = 15°± 1° Uncertainty for the sine functions [4] ∆???θ 𝑎𝑣? = ???θ 𝑎𝑣? · ∆? ∆? = (1°· π 180 ) ∆??? 15° = ??? 15° · π 180 =0.0169 ∆??? 15° ???θ 𝑎𝑣? = ? λ 𝑎 𝑎 = 1. 5?? ? = 1 1. 5???(15° ± 1°) = ?λ Uncertainty for the slit width (a) The smallest measurement on the ruler used was 1mm, therefore the uncertainty for (a) is 0.5mm or 0.05cm λ = (1. 5 ± 0. 05)(0. 2588 ± 0. 0169) λ = (1. 5 ± 0.05 1.5 · 100%)(0. 2588 ± 0.0169 0.2588 · 100%) 3.33%)(0.2588 6.53%) λ = (1. 5± ± Adding the uncertainty and multiplying we get (3. 33% + 6. 53%) (1. 5 · 0. 2588) 0.388 9.86% λ = ±

80° 0.73 75° 0.8 70° 0.72 65° 0.6 60° 0.62 Table 3: Angle Theta and Intensity (a=6cm) Graph 3: Single Slit Second Trial (Intensity vs Angle Theta) Calculation of Wavelength: Using equation (4) the experimental wavelength for the second trial will be calculated.

θ 1 = 65° ± 0. 5° θ 2 = 80° ± 0. 5° θ 3 = 95° ± 0. 5° θ 4 = 110°± 0. 5° (90±0.5°−θ 1 )+(90±0.5°−θ 2 )+(90±0.5°−θ 3 )+(90±0.5°−θ 4 ) ? = θ 𝑎𝑣? = (90±0.5°−65±0.5°)+(90±0.5°−80±0.5°)+ 90±0.5°−95±0.5° | |+ 90±0.5°−110±0.5° | | 4 θ 𝑎𝑣? (25±1°)+(10±1°)+(5±1°)+(20±1°) 4 = θ 𝑎𝑣? 60±4° 4 =θ 𝑎𝑣? θ 𝑎𝑣? = 15°± 1° Uncertainty for the sine functions [4] ∆???θ 𝑎𝑣? = ???θ 𝑎𝑣? · ∆? ∆? = (1°· π 180 ) ∆??? 15° = ??? 15° · π 180 =0.0169 ∆??? 15° ???θ 𝑎𝑣? = ? λ 𝑎 𝑎 = 6?? ? = 1 6???(15° ± 1°) = ?λ Uncertainty for the slit width (a) The smallest measurement on the ruler used was 1mm, therefore the uncertainty for (a) is 0.5mm or 0.05cm

From this graph it can be seen that the intensity becomes greater the closer the angle gets to 90 . ° It can also be seen that the center maximum has the greatest intensity for all three experiments and that center maximum is located at 90 for all three experiments . ° Part 3 Absorption of Microwaves: When no material was between the transmitter and the receiver the microwaves had an intensity of 4mA. When the dry paper towel was placed in the receiver horn, the intensity of the microwaves were 4mA. When a wet paper towel was placed in the receiver horn, the microwaves had an intensity of zero. The reason behind this is that microwaves resonate with water leading the microwaves to be absorbed by the water before making contact with the receiver resulting in an intensity of 0mA . Since there was no water in the dry paper towel trial there was no absorption resulting in an intensity of 4mA. Conclusion: In conclusion the data gathering for the three initial parts of this experiment, using distanced slits, went well. When graphing the results of the slit parts, clear maximas and minimas were present. For the absorption of microwaves, the hypothesis that the water would absorb the microwaves was found to be true as the final reading with water was zero. The wavelengths for each section of the experiment are as follows. The double slit experiment wavelength is 0.525 cm 0.035cm. The 1.5cm single slit experiment wavelength is 0.388 cm 0.0383cm, The 6cm ± ± single slit experiment is 1.55 cm 0.114cm. When looking at the experimental wavelength of ± both single slit experiments it can be seen that these results make sense. This is because the single slit equation ( ) shows that when the slit width ( a ) is increased the wavelength ???θ = ? λ 𝑎 ( 𝝺 ) will also have to increase since ( k) is constant. So theoretically for the second single slit trial

when a=6cm it should have a bigger wavelength than the first trial where a=1.5cm. This is the case the second trial has a bigger experimental wavelength than the first trial. When comparing the experimental wavelength for the double slit experiment, the first trial of the single slit experiment and the second trial of the single slit experiment ),( (λ = 0. 525??± 0. 035?? ,( ) respectively to the theoretical λ = 0. 388?? ± 0. 0383??) λ = 1. 55?? ± 0. 114?? wavelength it can be seen that the range of uncertainty for the experimental λ = 2. 85?? wavelength does not include the manufacturer’s value for the wavelength. This could be due to errors made when performing the experiment. There were sources of error in this experiment. There were two main sources of error. The first was that the battery of the gun transmitter was low, which affected the accuracy of the data. The second main source of error was that the receiver was impacted by noise and when the experiment was conducted the noise around the receiver was not constant therefore impacting the accuracy of the data. All in all the theory for this experiment was found to hold true with the wavelength calculations and absorption of microwaves.

The smallest measurement on the ruler used was 1mm, therefore the uncertainty for is 0.5mm or 0.005cm Angle calculation: ?𝑎?θ = ??? 𝑎?? = (−0.085±0.0005)? (0.71±0.0005)? (7) (?±δ?)(?±δ?) = ( ? | | ? | | )( δ? ? | | + δ? ? | | ) When dividing two numbers with uncertainty we can use equation (7) ?𝑎?θ =− 0. 1197 ± 0. 00078 θ = ?𝑎? −1 (− 0. 1197 ± 0. 00078) Uncertainty for inverse tan: ?𝑎? −1 (− 0. 1197 + 0. 00078) · π 180 = 0. 1119 ?𝑎? −1 (− 0. 1197 − 0. 00078) · π 180 = 0. 1204 ( )= 0. 1204 − 0. 1119