Because the Angles Are Small, We Have

Homework 12

Ch24: P 25, 29, 41, 45, 47, 55, 65, 77

Problems (Ch24):

25. (II) If a slit diffracts 650-nm light so that the diffraction maximum is 4.0 cm wide on a screen 1.50 m away, what will be the width of the diffraction maximum for light of wavelength 420 nm?

Solution

Because the angles are small, we have

The condition for the first minimum is

If we form the ratio of the expressions for the two wavelengths, we get

which gives

29.(II) How many lines per centimeter does a grating have if the third-order occurs at an 18.0° angle for 630-nm light?

Solution

We find the slit separation from

which gives

The number of lines/cm is

41. (II) What is the smallest thickness of a soap film that would appear black if illuminated with 480-nm light? Assume there is air on both sides of the soap film.

Solution

We have - destructive interference. For m=1:

45.(II) How thick (minimum) should the air layer be between two flat glass surfaces if the glass is to appear bright when 450-nm light is incident normally? What if the glass is to appear dark?

Solution

- constructive interference. The minimum non-zero thickness is (m=0)

- destructive interference. The minimum non-zero thickness is (m=1)

47.(III) A thin film of alcohol lies on a flat glass plate When monochromatic light, whose wavelength can be changed, is incident normally, the reflected light is a minimum for and a maximum for What is the minimum thickness of the film?

Solution

Constructive interference:

Destructive interference:

The wave that reflects from the top surface of the alcohol has aphase change of

,or in terms of wave length.

The wave that reflects from the glass at the bottom surface of the alcohol has a phase change of

,or in terms of wave length.

Because of that:for constructive interference

for destructive interference

When we combine these two equations, we get:,

or

Thus we see that, and the thickness of the film is

55.(II) What is Brewster’s angle for a diamond submerged in water if the light is hitting the diamond while traveling in the water?

Solution

Because the light is coming from water to diamond, we find the angle from the vertical from

which gives

65.Light of wavelength 590 nm passes through two narrow slits 0.60 mm apart. The screen is 1.70 m away. A second source of unknown wavelength produces its second-order fringe 1.33 mm closer to the central maximum than the 590-nm light. What is the wavelength of the unknown light?

Solution

For constructive interference, the path difference is a multiple of the wavelength:

.

We find the location on the screen from

For small angles, we have

which gives

For the second-order fringes we have

When we subtract the two equations, we get

which gives

77.What is the minimum (non-zero) thickness for the air layer between two flat glass surfaces if the glass is to appear dark when 640-nm light is incident normally? What if the glass is to appear bright?

Solution

The phase change at the lower surface means:

Destructive interference .

We set to find the smallest nonzero value of t:

Constructive interference .

We set to find the smallest value of t: