Physics 1102 In-Class Problems: Interference and Diffraction

1. To measure the distance between two large metal walls, a weak radio transmitter set at 105 MHz is placed between them. A small radio receiver is then set to the same frequency and is moved along the shortest straight line between the walls which also passes through the transmitter. It is found that there are twenty dead areas between the walls (no including the walls) where nothing is heard on the radio. Sketch the wave and determine the distance between the walls.

2. For the same setup as problem #1, the frequency of the transmitter is changed. There are now twenty-two points of radio silence. What is the new frequency?

3. What is the thinnest film of a 1.40 refractive index coating on glass (n = 1.50) for which destructive interference of the green component (500 nm) of an incident white light beam in air can take place by reflection?
   

4. A layer of oil of thickness 200 nm floats on top of a layer of water of thickness 400 nm resting on a flat metallic mirror. The index of refraction of the oil is 1.24 and that of the water is 1.33. A beam of light is normally incident on these layers. What must be the wavelength of the beam if the light reflected by the top surface of the oil is to interfere destructively with the light reflected by the mirror?
   

5. Two flat 10 cm long glass plates are shown in the diagram below. At one end of the plates are in contact, while at the other they are separated by a distance of 0.020 mm. Incident light (λ = 500 nm) shines normally onto the glass and is reflected of the bottom layer of the top plate and the top layer of the of the bottom plate. What is the spacing of the interference fringes? Is the fringe adjacent to the line of contact bright or dark?
   

6. If in the above problem, the space between the plates is filled with transparent silicone grease (n = 1.5), the top plate has refractive index, n = 1.4, and the lower n = 1.6, what is the spacing of the dark fringes? Is the fringe at the line of contact bright or dark?

7. A thin film of thickness 0.450 mm and index of refraction n = 1.50 is placed in one arm of a Michelson Interferometer. Light of wavelength λ = 550 nm is used. How many fringes does the pattern shift? In the entire setup was immersed in a fluid of index n = 1.20, how many fringes would the pattern shift?

8. An airtight chamber 5.0 cm long with glass windows is placed in one arm of a Michelson Interferometer. Light of wavelength 500 nm is used. Air is slowly evacuated from the chamber using a vacuum pump. This causes a shift of 60 fringes. Find the index of refraction of air at atmospheric pressure and room temperature. A fringe shift is when a point goes from dark to light to dark or vice versa.

9. Young's experiment is performed with sodium light (λ = 589 nm). Fringes are measured carefully on a screen 1.50 m away from the double slit, and the centre of the twentieth bright fringe (not counting the central bright fringe) is found to be 11.9 mm from the centre of the central bright fringe. What is the separation of the two slits?

10. In Young's double slit experiment, when a thin film of transparent material is placed over one of the slits, the central bright fringe of the white-light fringe system is displace by 3.6 fringes. The refractive index of the material is 1.40, and the wavelength of the light is 550 nm. (a) By how much does the film increase the optical path? (b) What is the exact thickness of the film?

11. On a flat screen 2.0 m from a single slit, the distance between the first minima is 13.3 cm for light of wavelength λ = 550 nm. What is the width of the slit? For light of wavelength λ = 600 nm, how far apart are the first minima?

12. The intensity of the fringes which make up a single slit diffraction pattern is given by
    
    I = I₀sin²(β)/(β²r₀²),

    Where β = aπsinθ/λ. Show that the intensity maxima are given by the formula

    (βcosβ - sinβ)/β² = 0.

    Note that the next three maxima after the central maxima are located at β = 1.43π, β = 2.46π, and β = 3.47π. The maxima after the central maxima is only 4.7% as bright as the central peak.

13. An interference pattern consists of two slits of width a = 8λ separated by a centre-to-centre distance of 26λ. Determine the intensity of the m=1 and m=2 interference maxima relative to the central maxima.

14. It is claimed that some professional baseball players can see which way the ball is spinning as it travels toward home plate. One way to judge this is to estimate the distance at which a batter can first hope to resolve two points on opposite sides of a baseball, which has a diameter of 0.0738 m. (a) Estimate this distance, assuming the pupil of the eye has a diameter of 2.0 mm, the material within the eye has a refractive index of 1.36, and the wavelength of light is 550 nm in a vacuum. (b) Considering that the distance between the pitcher's mound and home plate is 18.4 m, can you rule out such a claim?

15. Ignoring atmospheric effects, how far apart must two objects be on the moon before the eye can resolve them? The distance to the moon's surface is 3.77 × 10⁸ m. Assume λ = 550 nm and n = 1.34. Assume that the diameter of the eye's pupil is 2.00 mm.

16. Sodium light with wavelengths 588.99 nm and 589.59 nm is incident on a grating with 5500 lines per centimetre. A screen is placed 3.0 m beyond the grating. What is the distance between the two spectral lines in the first-order spectrum on the screen? In the second order spectrum?

Questions? mike.coombes@kwantlen.ca