Answer Happy

1. diffraction gratings are commonly used in instruments called spectrometers. Spectrometers
take light and separate it into its various color components (similar to a prism). If the goal is
to separate the different colors/wavelengths as much as possible, which grating (of the two
you used in this experiment) would do a better job, and why? Give an equation which proves
you are correct.
please answer question 1. diffraction gratings are commonly used in instruments called spectrometers. Spectrometerstake light and separate it into its various color components (similar to a prism). If the goal isto separate the different colors/wavelengths as much as possible, which grating (of the twoyou used in this experiment) would do a better job, and why? Give an equation which provesyou are correct.
take light and separate it into its various color components (similar to a prism). If the goal is
to separate the different colors/wavelengths as much as possible, which grating (of the two
you used in this experiment) would do a better job, and why? Give an equation which proves
you are correct.
Part 1: Double Slits Since the beam is coherent, the two waves passing through the respective slits start in phase. The two waves will then travel different paths (with different path lengths) to different points on the screen. In most locations on the screen the two waves will be out of phase based on having traveled different path lengths. When the waves are out of phase they cancel each other out and no light is observed. However, for certain locations, the path lengths will differ by an integer multiple of the wavelength of the light, and the waves will be in phase. When the waves are in phase, they will add together and produce a bright spot. The condition for this is d sin(a) = nx, (11.1) where n is an integer and d is the distance between the slits. In the present experiment, the distance d is MUCH smaller than the distance L between the screen and the slits. Consequently, the angle a is very small and we can approximate tan(a) sin(a). (11.2) By substituting this approximation and, based on Figure 11.3, tan(a)= y/L, into Equation 11.1, we obtain yd nλ, (11.3) L where y is the position of a bright spot on the screen. Thus, there is a bright spot for each value of n, or for y = yn, where nLX Yn = (11.4) d For each successive integer n, Equation 11.4 specifies the location of a bright spot on the screen and, therefore, the distance between adjacent spots is given by
Ay=X ^(1). (11.5) Rearranging, the wavelength is thus λ = dAy L (11.6) Part 2: Diffraction Grating Now, a slight inconvenience arises in that the distance between, say, yn and Yn+1, or yn and Yn-1 is extremely small because the factor A/d in Equation 11.4 is very small. For example, with a slit of width d= 0.25mm, and using visible light of wavelength λ = 550nm, the ratio X/d≈ 0.0022. Fortunately, this ratio can be increased by decreasing the denominator d, the spacing between the slits. However, the spacing between the slits cannot be less than the slit width. And, the slit width must be big enough to let sufficient light though; otherwise the slit images are too dim to be seen. A diffraction grating can be used to overcome this problem. A grating has thousands of parallel pairs of narrow slits. Even though only a little light passes through each narrow slit, the sum of the light from many pairs of slits is bright. The gratings provided have 300 grooves/mm and 600 grooves/mm, respectively. For the for- mer, the distance d between the slits is d= .001m/500 = 2 x 10-6m. (11.7) For each pair of slits, Equation 11.1 is still valid. However, with d small, the angle a is large, so Equation 11.2 is NOT valid. Instead, we must calculate the actual value of a tan (11.8) L before plugging in to Equation 11.1. Using this value for a and rearranging Equation 11.1 to solve for the wavelength yields d sin a X (11.9) n
Experimental Procedures Part 1: Double slits Double Slits Y d sin(a) Figure 11.3: Diffraction by two slits when L>>y. 132 Ay Screen Yut
Figure 11.4: The central pattern of dots is used. Ignore the faint dots off to the sides. 1. Rotate the wheel of double slits so that the set of slits with a = 0.04mm and d = 0.25mm is in the 9 o'clock position. 2. Place the laser on the left end of the optical bench. Place the double slits just in front of the laser. Place the screen at the right end of the optical bench. 3. Turn on the laser and adjust its aim using the two thumbscrews on the back of the laser such that the beam strikes the double slits and a clear diffraction pattern is produced on the screen. The pattern should look like Figure 11.4. 4. Note how there are groupings of dots on the screen. This variation in brightness is a result of the fact that the slits are not infinitely narrow. In this part of the experiment, we will ignore this effect by including only the central group of bright dots. For the first set of slits, you should see 13 bright spots in the central pattern. However, if you see 11 or 15 instead, that's completely fine. For the second set of slits, you should see 23 bright spots. But again, if you see more or less, that is fine too. 5. Record the actual number of dots you observe/use in the central group. Since, for example, if there are 13 dots then there are 12 spaces between them, subtract one from your actual number of dots to get the number of spaces. 6. Determine the spacing Ay by measuring the total distance tot between the middles of the two end dots and dividing by the number of spaces. 7. Measure the distance L between the double slits and the screen. 8. Use Equation 11.5 to determine the wavelength of the laser light. Compare this value to the nominal value of 650nm. 9. Repeat this process for the other three sets of double slits provided on the wheel.
Part 2: Diffraction Grating Incoming plane wave of light. First-order maximum (n=+1) Central or zeroth-order maximum (n=0) First-order maximum (n = -1) Diffraction grating 8= dsin Figure 11.5: Illustration showing diffraction from a grating. 1. Replace the double slits with the grating holder and 600 grooves/mm diffraction grating. 2. Since y will be large, replace the screen with the meter stick holder and meter stick. 3. The meter stick should be approximately 30 cm away from the grating. This distance is L. The distance between the laser and the grating does not matter. 4. Mount the meter stick so that the center light beam (n = 0) and the first (n = ±1) and second (n = ±2) diffracted light beams on each side hit the meter stick (5 spots total). 5. Measure y for each spot individually relative to the n = 0 spot, and then determine a from Equation 11.8. 6. Using each calculated value for a, determine lambda using Equation 11.1. 7. Calculate the average of these four values of X. 8. Repeat this process using the 300 grooves/mm diffraction grating. Note that you should see more diffraction spots on the meter stick than with the 600 grooves/mm grating.