PHYS272_Lab9

=Physics for Scientists and Engineers - Lab 9 PreLab =  Answer the questions below. The prelab is due by the beginning of the lab period. These questions will assist you in performing the error analysis for the lab.

1. The distance from a diffraction grating to a screen is D = 30.0 ± 0.5 cm and the distance from the central bright spot to the first order maxima is y 1 = 16.2 ± 0.2 cm. By referring to Figure 1 in the lab writeup below, you can see that tan θ m = y m /D. Calculate the angle θ 1 **in radians** and the uncertainty in θ 1. Then, using d sin θ m = m λ and knowing that in this case m = 1 and d = 1/(750000 lines/m), calculate the wavelength of light λ and the uncertainty in λ. For the uncertainty calculation use the fact that, if y=f(x 1, x 2 ) then

math \Delta y = \sqrt{(\partial f / \partial x_1)^2 \Delta x_1^2 + (\partial f / \partial x_2)^2 \Delta x_2^2}. math

2. When a light shines through a thin slit with width w, it creates a pattern of bright and dark spots. The angle between a line connecting the slit and the center of the pattern and one connecting the slit with a dark spot follows w sin θ m = m λ where m = ± 1, ± 2, ± 3, ... The angle can also be determined (see Figure 1 below) from tan θ m = y m /D. The lab writeup makes the claim that one can make a plot of y m vs m and obtain a slope equal to λ D/w. What are the assumptions behind this claim? Do a derivation to show that the claim is reasonable.

3. Suppose, in determining the size of Lycopodium spores, you measure the distance between a slide covered with the spores and a screen to be L = 1.10 ± 0.01 m. A laser beam with light wavelength of λ = 632.8 nm is sent through the slide and makes a circular diffraction pattern on the screen with a radius R = 2.5 ± 0.2 cm. Given that the diameter d of the spores should follow d = 1.22 λ L/R, what is the average spore diameter and the uncertainty in the diameter? HINT: don't forget to convert all measurements to meters before calculating!

=Physics for Scientists and Engineers – Lab 9 =

Objectives:

 * To measure the wavelength of light emitted by a Helium Neon laser.
 * To observe the character of single slit diffraction.
 * To observe the character of double slit diffraction.
 * To measure the groove spacing of a CD
 * To measure the size of Lycopodium spores
 * To measure the Law of Malus for polarization

Equipment:

 * Helium Neon laser
 * Desk lamp
 * Diffraction gratings: 750 lines per mm
 * CD
 * Slide of single slits, .02, .04, .08, .16 mm width
 * Slide of double slits, .04, .08 mm width; .25, .5 mm separation
 * Foam slit holders
 * Masonite board attached to wood block (for laser diffraction screen) (2)
 * Meter stick, two meter stick
 * Lycopodium powder and glass slide (kept at front of class)
 * Graphical Analysis software

On a single table for the Polarization Experiment:
 * Two polarizer slides mounted on a goniometer
 * Multimeter, medium leads (2) and alligator clip connectors (2)
 * 100 W light bulb
 * screen holders (2)
 * aperture slide
 * table clamp
 * rods (two medium, one short)
 * LED in small rod
 * right angle clamps (6)

Diffraction Grating
 A diffraction grating consists of a series of opaque and transparent strips. Light passing through the grating is broken up into portions which come through each slit. The light from the various slits interfere with one another producing dark and bright fringes. Bright fringes occur when the path length of the light from adjacent slits to the screen is an integral multiple of the wavelength. When the light is incident upon the grating perpendicular to its surface this condition is satisfied at an angle from this direction given by

math d \sin \theta = m \lambda math

This is the condition for interference maximum, where d is the distance between adjacent slits (d = 1/750000 m = 1.333×10 -6 m), θ is the angle from grating source line, and m is the number of wavelengths of path length difference for light paths through adjacent slits (m is called the order of the line).

Double Slit Interference
 The double slit is a special case of the diffraction grating with 2 slits. Therefore the same equation gives the condition for the location of the bright fringes.

Single Slit interference
 When light passes through a single slit the light from different portions of the slit interferes again producing a series of bright and dark fringes. Total destructive interference will occur when

math w \sin \theta = m \lambda math

This is the condition for diffraction minima, where w is the width of the slit, θ is the angle of the dark center from the source slit line, and m is the number of wavelengths of path length difference for light paths from opposite sides of the slit. In the case of a small angle, θ m, the sine of the angle can be approximated the tangent of the angle = y m / D, where y m is the distance of the dark center from the central maximum, and D is the distance from the slit to the screen.

Polarization
 Light consists of electromagnetic waves with electric and magnetic fields that are perpendicular to each other and to the direction of the travel of the wave. Many light sources are unpolarized, where the directions of the fields are randomly oriented in the plane perpendicular to the travel direction. A polarizer is a material that allows waves to travel through when the electric field is along the direction of the polarization axis. Such a polarizer will produce plane polarized light waves. When this light passes through a second polarizer that has its polarization axis rotated at an angle θ from the polarization axis of the first polarizer only the component of the electric field that is along the polarization axis of the second polarizer will pass through it. Since the intensity of the light is proportional to the square of the electric field the final intensity will be given by the relation

math I = I_0 \cos^2 \theta math

Determination of the wavelength of the helium-neon laser
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 * 1) Place the diffraction grating into the foam holder block.
 * 2) Tape a blank sheet of paper to the Masonite board screen.
 * 3) Position the grating a distance D = 0.15 m from the screen and parallel with it.
 * 4) Direct the laser beam through the diffraction grating so that the undeflected beam strikes the screen perpendicularly.
 * 5) Rotate the grating around a vertical axis until it is perpendicular to the beam and the first order maxima are symmetrical about the central maximum.
 * 6) Record the positions y m of the first order maxima and y c the central spot using a ruler.
 * 7) Using the given information that 1/d = 750,000 lines / m and θ m = tan -1 (y m / D), calculate the wavelength of the helium-neon laser red line from equation 1 and the uncertainty in the wavelength. Compare the value of λ to the accepted value of 632.8 nm and compute the percent error from this value.

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">Measurement of groove spacing of a CD
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 * 1) Insert a CD into the foam holder so that the beam reflects off the CD and onto the paper taped to the two wooden stands. (see CD setup). The two boards should be separated by a small slit to let the laser light through. Reflect the reflected laser light directly back into the laser (this is the central maximum), the refracted light should be seen on the wooden stands (these are the diffuse maximum).
 * 2) Mark the centers of the two diffuse maxima on either side of the laser.
 * 3) Measure the distance separating the diffuse maximum, (y), and divide by 2, (y 1 ).
 * 4) Record the distance from the CD to the paper, (L).
 * 5) Calculate θ and then sinθ using: θ = tan -1 (y 1 /L)
 * 6) Using λ = 632.8 nm calculate the spacing, (d) and the uncertainty in the spacing, noting that in this case m = 1.
 * 7) Compare with the industry standard of 1.6 μm for a CD. (The standard for a DVD is 0.74μm and for a Blu-ray disc is 0.32 μm)

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">Single slit diffraction pattern (0.02 mm slit width)
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 * 1) Place the single slit into the foam holder block.
 * 2) Tape a blank sheet of paper to the Masonite board screen
 * 3) Position the slit a distance D about 0.6 m from the screen and parallel with it.
 * 4) Direct the laser beam through the diffraction grating so that the undeflected beam strikes the screen perpendicularly. Rotate the slit around a vertical axis until it is perpendicular to the laser beam and the first order maxima are symmetrical about the central maximum.
 * 5) Draw short lines on the positions ym on the screen of the locations of the first four or five order minima and yc the central spot.
 * 6) Remove the sheet from the wall and mark the center of the central diffraction maximum as half way between the adjacent minima.
 * 7) Measure and record the distance y m of each minimum from the center of the central maximum in the row with the order number m = 1, 2, 3, 4 and 5, and record your data in a table.
 * 8) Measure and record the distance, D, of the slit from the screen.
 * 9) Make a plot of y m on the y axis versus the order m on the x axis. Include error bars in your plot. The slope of the line should be Dλ/w. Record the slope and the error in the slope.
 * 10) From the slope and the accepted value of λ = 632.8 nm calculate the width w of the slit. How well does this compare with the labeled value of the slit width?

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">Single slit diffraction pattern (various slit widths)
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 * 1) Move the slide of slits until the laser beam is passing through the .04 mm wide slit and the pattern is bright and sharp and replace the sheet on the screen, with the central maximum near the center.
 * 2) With a sharp pencil mark the minima bounding the central maximum.
 * 3) Mark the center of the central maximum as halfway between these two minima. Also, mark the fifth minimum.
 * 4) Measure the distance between this fifth minimum and the central maximum and record it as y m.
 * 5) Repeat this for the .08 mm and .16 mm wide slits.
 * 6) Record the particular minimum that you are working with (suggested m = 5).
 * 7) Plot y m on the y axis versus 1/w on the x axis for the four data points (three from this section, one from the prior section). Include error bars. The slope of this line, according to equation (5) should be mDλ. Record the slope and the error in the slope.
 * 8) Find the % error between the slope that you found and this predicted value. State if the uncertainty intervals overlap.

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">Double slit interference (.25 mm slit separation)
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 * 1) Place a double slit slide in the foam holder and position it about 1.5 m from a wall.
 * 2) Direct the laser through the double slit that has a slit separation of .25 mm and a slit width of .04 mm and form an interference pattern on the wall. Tape a sheet of paper on the wall with the long side horizontal and the central maximum on one side of the paper.
 * 3) With a sharp pencil mark the center of the central maximum and of the eighth order maximum.
 * 4) Use the equation of the condition for interference maxima to predict the distance x m = x 8 for the eighth interference maximum. Compare these results.

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">Double slit interference (various slit separations)
<span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> Repeat this with the laser coming through the double slit that has a slit spacing of .5 mm and a slit width of .04 mm. Observe carefully to note that because of the diffraction minimum some of the interference maximum are missing. Test these observations against the following argument. Rewriting equations (1) and (2) for small angles we see that θ = n(λ/d)=m(λ/w) so that n=(d/w)m. In other words, the diffraction minimum occurs when the number n of wave lengths of path difference between the two slits is an integer times the ratio of d/w thus eliminating the n-th interference maximum. For the first slit this means .25/.04 = 6.25 or about every 6th maximum would be missing. Test this for all four sets of double slits.

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">Law of Malus
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 * 1) Place a light source with a 40 W light bulb in front of an aperture stop.
 * 2) Mount the goniometer polarizer-analyzer between the aperture and an LED light detector.
 * 3) Use a multimeter to measure the voltage output from the led
 * 4) Align the second polarizer to obtain the largest voltage reading. The voltage is proportional to the light intensity.
 * 5) Take intensity measurements at five different angles of the second polarizer from 0° to 90°.
 * 6) Make a graph of V vs. cos 2 θ. Include error bars.
 * 7) From the graph does the Law of Malus hold? Why can we use V in the place of I?

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">Width of Lycopodium Spores
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 * 1) Dip the end of your microscope slide into the Lycopodium powder to a depth of .5 inch and knock of any excess so that a thin approximately even distribution is obtained.
 * 2) Set the laser about 1 m from the stand and shine the beam through the powder on the slide. Your slide should be placed close to the laser, giving an L of close to 1 meter.
 * 3) Tape a piece of paper on the stand and mark the central bright spot that appears.
 * 4) Measure the diameter of the circumference of the central bright spot and divide by 2 to get the radius, (R).
 * 5) Measure the distance from the slide to the screen, (L).
 * 6) Calculate the size of the diffracting objects (powder spores) using: d = 1.22(λL/R) where λ is 632.8 nm. Also find the uncertainty in d.
 * 7) Compare your answer to the diameter of lycopodium spores, 25 to 40 μm.