Homework 8


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        Submitted Answers will be graded W 11/5, Th 11/6, F 11/7, and M 11/10 at 12:00 noon.
        Problem 8.1
          What you should learn: Light is normally (at 0°, that is) incident on slits and the plane of the slits are parallel to a screen a distance L away.  Light is viewed at an angle θ with respect to the undeflected direction of the beam.
        Problem: Consider scattering from double slits. If the screen is a distance L from the slits, what is the viewing angle (theta) at a point P located a distance y from the center of the interference pattern. Express your answer in degrees.  
        Constants and fixed variables:
          Distance from the slits to the screen: L = 7.00 m.
        Variables that must be changed with each submission:
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        Answer range: theta = 1.00 to 1.60 degrees
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        Problem 8.2
          What you should learn: The path difference between two points on the slits is Δr = d sinθ where d is the slit separation.
        Problem: In a double slit experiment, we view a bright spot located at a point P located a distance d from the central bright spot in the pattern. What is the difference in path lengths (Deltar) from each slit to the point P? Express you answer in units of nm.  
        Constants and fixed variables:
          Distance from the slits to the screen: L = 7.00 m
          Distance from the center of the pattern to the bright spot at P : y = 21.4 cm
        Variables that must be changed with each submission: 
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        Answer range: Deltar = 1900 to 2900 nm
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        Problem 8.3
          What you should learn: If the path difference from two slits is Δr, the phase difference between the two rays is Δφ = kΔr where k = 2π/λ.
        Problem: For the same slits as Prob. 8.2, find the phase difference (Deltaphi) in radians.  
        Constants and fixed variables:
          Distance from the slits to the screen: L = 7.00 m
          Distance from the center of the pattern to the bright spot at P : y = 21.4 cm
          Wavelength of the light: lambda = 630 nm
        Variables that must be changed with each submission:
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        Hints: be careful of units in k and Deltar.
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        Answer range: Deltaphi = 19.1 to 28.8 radians
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        Problem 8.4
          What you should learn: You should apply m λ = d sin(θ)  and (m+1/2) λ = d sin(θ)  to find the maxima and minima. This is probably the most important equation of the chapter, so know it well! Fortunately, it's easy to use.
        Problem: For the same slits as Prob. 8.2, find y, the distance from the central (0°) spot to the third bright spot on either side.  
        Constants and fixed variables:
          Distance from the slits to the screen: L = 7.00 m
          Wavelength of the light: lambda = 630 nm
        Variables that must be changed with each submission:
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        Answer range: y  = 14.0 to 21.0 cm
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        Problem 8.5
          What you should learn: The intensity of the two slit pattern is given in terms of the viewing angle as I(θ)=I12(1+cosΔφ).
        Problem: What is the ratio of the intensity at point P to the intensity of a single slit? Give a value for I(θ)/I1 for the value of θ listed below. (You may call this I_I1.)  
        Constants and fixed variables:
        Wavelength: lambda = 600 nm
          Slit separation: d = 1800 nm
        Variables that must be changed with each submission:
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        Hints: Be careful about degrees and radians. When you calculate Δφ = kd sin(θ), Δφ is in radians, not degrees.
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        Answer Range: I_I1 = 0.700 to 4.10
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        Problem 8.6
          What you should learn: You can qualitatively determine a number of things about multiple slits by using phasors.
        Problem: Four identical slits are separated by a distance d. At what viewing angle (theta) is the first minimum to the side of the central maximum at 0°?  Give theta in degrees.  
        Constants and fixed variables:
          Wavelength: lambda = 500 nm
        Variables that must be changed with each submission:
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        Answer Range: theta = 3.90 to 6.00 degrees
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        Problem 8.7
          What you should learn: This is a typical thin film problem. You need to count the number of phase shifts and decide which equation to use.
        Problem: White light is incident on an oil film on water. What is the shortest wavelength of visible light (390 - 700 nm for this problem) than satisfies the condition for maximum reflectance?  
        Constants and fixed variables:
          Index of refraction of the oil: no = 1.17
          Index of refraction of water: nw = 1.33
        Variables that must be changed with each submission:
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        Answer Range: lambda = 390 to 510 nm
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        Problem 8.8
          What you should learn: The formulas give us conditions for reflection from a thin film, but we can also deduce information about transmission through a thin film.
        Problem: For the same layer of oil given in Problem 8.7, calculate the longest wavelength of visible light for which transmission is maximum.  
        Constants and fixed variables:
          Index of refraction of the oil: no = 1.17
          Index of refraction of water: nw = 1.33
        Variables that must be changed with each submission:
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        Answer Range: lambda = 460 to 610 nm
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        Problem 8.9
          What you should learn: This is one more thin film problem.
        Problem: When white light hits a region of a bubble, it intensely reflects light of wavelength λ. What is t, the thickness of the bubble wall? Assume that the wall is the smallest possible thickness.  
        Constants and fixed variables:
           Index of refraction of soap solution: n = 1.29
        Variables that must be changed with each submission: lambda nm
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        Answer Range: t = 75.0 to 140 nm
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        Problem 8.10
          What you should learn: The maxima for multiple slits or diffraction gratings are at the same angles as for double slits. (The minima are different angles, however.)
        Problem: A collimated beam of white light is incident on a diffraction grating. What color of light bends the most?
        • A. Violet
        • B. Blue
        • C. Green
        • D. Yellow
        • E. Orange
        • F. Red
        		  
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        Problem 8.11
          What you should learn: The maxima for multiple slits or diffraction gratings are given by the equation mλ = d sin(θ). Usually we are given the number of lines per cm on the grating rather than d, however.
        Problem: The light from a laser is normally incident on a diffraction grating. The diffraction pattern is seen on a screen located a distance L from the grating. The first bright spots to each side of the central maximum are located a distance y away. What is the spacing between the lines (sp) in lines per cm?  
        Constants and fixed variables:
          Distance from the grating to the screen: L = 7.00 m
          Wavelength of the light: lambda = 629 nm
        Variables that must be changed with each submission:
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        Answer Range: sp = 680 to 1140 lines/cm
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        Problem 8.12
          What you should learn: When white light is incident on a diffraction grating, the grating produces multiple spectra that are seen over different ranges of angles. The integer m in the equation mλ = d sin(θ).  is called the "order" of the spectrum.
        Problem: When white light is incident on a diffraction grating, different order spectra can be seen to overlap. Light of wavelengths lambda1 and lambda2 from adjacent spectra are seen at the same viewing angle. What is order (m) of the lambda2 spectrum?  
        Variables that must be changed with each submission:
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        Answer Range: m = 2 to 5
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        Problem 8.13
          What you should learn: When unpolarized light goes through an ideal polarizer, half the intensity of the unpolarized is lost. When polarized light goes though a polaroid filter with its axis of polarization at an angle θ with respect to the polarization direction, the transmitted intensity is cos2θ times the incident intensity. This is called Malus' Law.
        Problem: Unpolarized light is incident on a polaroid filter. A second polaroid filter has its axis of polarization oriented at an angle theta with respect to the polarization direction. What is the ration of the the intensity of light passing through the second polarizer to the intensity of the unpolarized light incident on the first polarizer? (Call this I_I0.)  
        Variables that must be changed with each submission: theta
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        Answer Range: I_I0 = 0.200 to 0.450
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        Problem 8.14
          What you should learn: Malus' Law can be applied to a sequence of more than two polarizers. Keep in mind that light when light passes through a polarizer, the component of the light's electric field perpendicular to the axis of polarization is absorbed.
        Problem: Two polaroid filters are arranged so that their axes of polarization are perpendicular and no light passes through them. A transparent sample (it absorbs no light except for the light absorbed in the polarization process) is placed between them. The ratio of the intensity of the light passing through the sample and polarizers to the intensity of the light passing through the first polarizer alone is Io_I1. This is given below. Find the angle of the polarization axis of the sample with respect to the initial polarization direction.

        Note that two angles satisfy the conditions of this problem. The sum of the two angles is 90°. You may enter either angle as your answer. This problem is a bit difficult.

         

        		  
        Constants and fixed variables:
          Index of refraction of the oil: no = 1.17
          Index of refraction of water: nw = 1.33
        Variables that must be changed with each submission:
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        Answer range for the smaller angle: theta =  19.6 to 45.0 degrees
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