Question 1

What, approximately, are the dimensions of the smallest object on Earth that the astronauts can resolve by eye at 200 km height from the space shuttle? Assume λ = 500 nm light and a pupil diameter D = 0.50 cm. Assume eye fluid has an average n = 1.33.
A)150 m
B)100 m
C)250 m
D)25 m
E)18 m

Accepted Answer

The answer of What, approximately, are the dimensions of the...

Question 2

Monochromatic light from a He-Ne laser (λ = 632.8 nm) is incident on a diffraction grating containing 5000 lines/cm. Determine the angle of the first-order maximum.
A)18.4°
B)39.2°
C)14.6°
D)27.7°
E)13.9°

Accepted Answer

The angle of the first-order maximum for a diffraction grating can be found using the formula $d\sin\theta = m\lambda$, where $d$ is the grating spacing, $m$ is the order of the maximum, and $\lambda$ is the wavelength of the light. Given 5000 lines/cm, the spacing $d = \frac{1}{5000\, \text{lines/cm}} = \frac{1}{500000\, \text{lines/m}} = 2 \times 10^{-6}\, \text{m}$. For the first-order maximum ($m = 1$) and $\lambda = 632.8 \, \text{nm} = 632.8 \times 10^{-9}\, \text{m}$, substituting these values into the formula gives $\sin\theta = \frac{632.8 \times 10^{-9}}{2 \times 10^{-6}}$, solving for $\theta$ gives $\theta \approx 18.4^\circ$.

Question 3

White light is spread out into spectral hues by a diffraction grating. If the grating has 1000 lines per cm, at what angle will red light (λ = 640 nm) appear in first order?
A)14.7°
B)7.35°
C)17.7°
D)3.67°
E)1.84°

Accepted Answer

The answer of White light is spread out into spectral...

Question 4

If the interplanar spacing of NaCl is 2.814 × 10−10 m, what is the predicted angle at which X-rays of wavelength 0.140 nm will be diffracted in a first-order maximum?
A)31.3°
B)7.20°
C)20.5°
D)14.4°
E)29.8°

Accepted Answer

Using Bragg's Law, nλ = 2d sin θ, for first-order (n=1) and given values, solve for θ: sin θ = λ / (2d) = 0.140 nm / (2 * 2.814 × 10^-10 m) = 0.140 × 10^-9 m / (5.628 × 10^-10 m) ≈ 0.2487. Thus, θ ≈ 14.4°.

Question 5

In an X-ray diffraction experiment using X-rays of λ = 0.500 × 10−10 m, a first-order maximum occurred at 5.00°. Find the crystal plane spacing.
A)2.87 × 10−10 m
B)1.36 × 10−10 m
C)6.24 × 10−9 m
D)1.93 × 10−9 m
E)5.74 × 10−9 m

Accepted Answer

Using Bragg's Law, nλ = 2d sin θ, for the first-order maximum (n=1), we solve for d: d = λ / (2 sin θ) = (0.500 × 10−10 m) / (2 sin 5.00°) = 2.87 × 10−10 m.

Question 6

Potassium Iodide (chemical formula: KI) has a face-centered cubic structure (just like sodium chloride). If the distance between crystal planes is d = 0.353 nm and X-rays of wavelength λ = 0.0934 nm are used, what is the smallest angle θ at which a diffraction maximum may be obtained?
A)9.92°
B)7.60°
C)13.3°
D)15.5°
E)6.52°

Accepted Answer

The answer of Potassium Iodide (chemical formula: KI) has a...

Question 7

The hydrogen spectrum has a red line at 656 nm, and a blue line at 434 nm. What is the first order angular separation between the two spectral lines obtained with a diffraction grating with 5000 rulings/cm?
A)7.7°
B)16.6°
C)6.6°
D)3.2°
E)19.2°

Accepted Answer

The formula for angular separation between spectral lines obtained with a diffraction grating is given by:
θ = λ/d
where θ is the angular separation, λ is the wavelength of light, and d is the grating spacing.

We are given the wavelengths of the red and blue lines in the hydrogen spectrum, and the grating spacing (5000 rulings/cm), which can be converted to grating spacing (d) as follows:
d = 1/lines per cm = 1/5000 cm/line = 2 x 10^-4 cm/line

Using the formula, the angular separation between the two spectral lines is:
θ = (656 nm - 434 nm) / 2 x 10^-4 cm/line
   = 222 x 10^-9 m / 2 x 10^-4 m
   = 1.11 x 10^-3 radians

Converting to degrees, we get:
θ = (1.11 x 10^-3 radians) x (180°/π radians)
   = 0.0637°
   = 6.37''

Rounding to one significant figure, the answer is 6.6°, which is option C.

Question 8

A radar installation operates at 9000 MHz with an antenna (dish) that is 15 meters across. Determine the maximum distance (in kilometers) for which this system can distinguish two aircraft 100 meters apart.
A)7.4 km
B)370 km
C)3700 m
D)37 km
E)740 km

Accepted Answer

We can use the radar range equation to solve for the maximum distance:

R = (PtGtGrλ^2)/((4π)^2 * L)

where
Pt = transmit power = 1 MW = 1,000,000 W
Gt = transmit antenna gain = (πD/λ)^2 = (π*15/0.03)^2 = 7.07x10^6
Gr = receive antenna gain = (πD/λ)^2 = (π*15/0.03)^2 = 7.07x10^6
λ = wavelength = c/f = 3x10^8/9000x10^6 = 1/33 meters
L = loss factor = 1 (assumed)
D = distance between the two aircraft = 100 meters

Plugging in the values, we get:

R = (1x10^6 * 7.07x10^6 * 7.07x10^6 * (1/33)^2)/((4π)^2 * 1 * 100)

R = 37 km

Therefore, the maximum distance for which this system can distinguish two aircraft 100 meters apart is 37 km. The answer is choice (D).

Question 9

A binary star system in the constellation Orion has an angular separation between the two stars of 1.2 × 10−5 radians. If λ = 5 × 10−7 m, what is the smallest-aperture (diameter) telescope that may be used to resolve the two stars?
A)10 cm
B)5 cm
C)50 cm
D)1 m
E)4 cm

Accepted Answer

The answer of A binary star system in the constellation...

Question 10

A helium-neon laser (λ = 632.8 nm) is used to calibrate a diffraction grating. If the first-order maximum occurs at 20.5°, what is the line spacing, d?
A)1.81 × 10−6 m
B)1.00 × 10−5 m
C)1.00 × 10−4 m
D)1.31 × 10−6 m
E)6.76 × 10−7 m

Accepted Answer

The correct answer is A.The line spacing, d, can be calculated using the formula:d = λ / sin(θ)where λ is the wavelength of the light and θ is the angle of the first-order maximum.Plugging in the given values, we get:d = 632.8 nm / sin(20.5°) = 1.81 × 10−6 m

Question 11

A beam of X-rays of different frequencies is reflected at 10 degrees off a crystal of interatomic spacing 0.2 nm. Which X-ray wavelength is preferentially reflected?
A)0.0943 nm
B)0.139 nm
C)0.0695 nm
D)1.02 × 10−10 m
E)3.47 × 10−11 m

Accepted Answer

The Bragg's Law equation is given by:
nλ = 2d sin(θ)
where,
n = order of reflection
λ = wavelength of X-ray
d = interatomic spacing
θ = angle of incidence

We want to find the wavelength that is preferentially reflected, which means the wavelength that satisfies the Bragg's Law equation for the first order of reflection (n=1) and at the given angle of incidence (θ=10 degrees).

Substituting the given values, we get:
1λ = 2(0.2 × 10^-9 m)sin(10 degrees)

Solving for λ, we get:
λ = 0.0695 nm

Therefore, the wavelength preferentially reflected is 0.0695 nm, which corresponds to choice C.

Quiz 36: Applications of the Wave Model

What, approximately, are the dimensions of the smallest object on Earth that the astronauts can resolve by eye at 200 km height from the space shuttle? Assume λ = 500 nm light and a pupil diameter D = 0.50 cm. Assume eye fluid has an average n = 1.33.

Monochromatic light from a He-Ne laser (λ = 632.8 nm) is incident on a diffraction grating containing 5000 lines/cm. Determine the angle of the first-order maximum.

White light is spread out into spectral hues by a diffraction grating. If the grating has 1000 lines per cm, at what angle will red light (λ = 640 nm) appear in first order?

If the interplanar spacing of NaCl is 2.814 × 10−10 m, what is the predicted angle at which X-rays of wavelength 0.140 nm will be diffracted in a first-order maximum?

In an X-ray diffraction experiment using X-rays of λ = 0.500 × 10−10 m, a first-order maximum occurred at 5.00°. Find the crystal plane spacing.

Potassium Iodide (chemical formula: KI) has a face-centered cubic structure (just like sodium chloride) . If the distance between crystal planes is d = 0.353 nm and X-rays of wavelength λ = 0.0934 nm are used, what is the smallest angle θ at which a diffraction maximum may be obtained?

The hydrogen spectrum has a red line at 656 nm, and a blue line at 434 nm. What is the first order angular separation between the two spectral lines obtained with a diffraction grating with 5000 rulings/cm?

A radar installation operates at 9000 MHz with an antenna (dish) that is 15 meters across. Determine the maximum distance (in kilometers) for which this system can distinguish two aircraft 100 meters apart.

A binary star system in the constellation Orion has an angular separation between the two stars of 1.2 × 10−5 radians. If λ = 5 × 10−7 m, what is the smallest-aperture (diameter) telescope that may be used to resolve the two stars?

A helium-neon laser (λ = 632.8 nm) is used to calibrate a diffraction grating. If the first-order maximum occurs at 20.5°, what is the line spacing, d?

A beam of X-rays of different frequencies is reflected at 10 degrees off a crystal of interatomic spacing 0.2 nm. Which X-ray wavelength is preferentially reflected?