Question 1

A narrow slit is illuminated with sodium yellow light of wavelength 589 nm. If the central maximum extends to ±30.0°, how wide is the slit? A) 0.500 mm B) 2.20 × 10−⁶ m C) 3.33 × 10−⁵ m D) 1.18 μm E) 5.89 μm

Accepted Answer

The width of the slit ($a$) can be determined using the formula for the angular width ($\theta$) of the central maximum in a single-slit diffraction pattern, which is given by $a \sin(\theta) = m\lambda$, where $m$ is the order of the maximum (for the first minimum, $m = \pm1$), and $\lambda$ is the wavelength of the light. Given that the central maximum extends to $\pm30.0^\circ$, we use $\theta = 30.0^\circ$ and $\lambda = 589 \, \text{nm} = 589 \times 10^{-9} \, \text{m}$.Rearranging the formula to solve for $a$, we get $a = \frac{m\lambda}{\sin(\theta)}$. Substituting $m = 1$, $\lambda = 589 \times 10^{-9} \, \text{m}$, and $\theta = 30.0^\circ$ (noting that $\sin(30.0^\circ) = 0.5$), we find:$$a = \frac{1 \times 589 \times 10^{-9} \, \text{m}}{0.5} = 1.18 \times 10^{-6} \, \text{m} = 1.18 \, \mu\text{m}.$$Therefore, the width of the slit is 1.18 μm.

Question 2

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

D

Question 3

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

B

Question 4

Plane waves of ultrasound, f = 51.0 kHz, impinge on a flat plane with a 1.00 cm-wide slit. What is the total angular width (2θ) of diffracted sound waves when the speed of sound in air is 340 m/sec.
A) 167°
B) 83.6°
C) 19.5°
D) 41.8°
E) 0.11°

Accepted Answer

The total angular width (2θ) of diffracted sound waves can be calculated using the formula for the first minimum in a single-slit diffraction pattern, which is given by $\sin(\theta) = \frac{\lambda}{w}$, where $\lambda$ is the wavelength of the sound and $w$ is the width of the slit. The wavelength $\lambda$ can be found using the relation $\lambda = \frac{v}{f}$, where $v$ is the speed of sound and $f$ is the frequency. Substituting the given values, $\lambda = \frac{340 \, \text{m/s}}{51000 \, \text{Hz}} = 0.00667 \, \text{m} = 6.67 \, \text{mm}$. Since the slit width $w = 1.00 \, \text{cm} = 10 \, \text{mm}$, we find $\sin(\theta) = \frac{6.67 \, \text{mm}}{10 \, \text{mm}} = 0.667$. Solving for $\theta$ gives $\theta \approx 41.8^\circ$, so the total angular width $2\theta \approx 83.6^\circ$.

Question 5

Unpolarized light is passed through three successive Polaroid filters, each with its transmission axis at 45.0° to the preceding filter. What percentage of light gets through?
A) 0%
B) 12.5%
C) 25.0%
D) 50.0%
E) 33.3%

Accepted Answer

Each Polaroid filter transmits half of the light that passes through it, so the total percentage of light that gets through is:
(1/2) × (1/2) × (1/2) × 100% = 12.5%

Question 6

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 answer of A beam of X-rays of different frequencies...

Question 7

Light reflected off a plate-glass window (n = 1.5) is found to be completely polarized at angle-of-incidence θ. Find θ.
A) 56.3°
B) 5.7°
C) 21.2°
D) 18.6°
E) 33.7°

Accepted Answer

The angle of incidence at which light is completely polarized upon reflection is known as Brewster's angle. Brewster's angle θ can be calculated using the formula: tan(θ) = n, where n is the refractive index of the medium. For a plate-glass window with n = 1.5, θ = arctan(1.5) ≈ 56.3°.

Question 8

Sunlight reflected from a smooth ice surface is completely polarized. Determine the angle of incidence. (n_ice = 1.31.) A) 52.6° B) 25.6° C) 65.2° D) 56.2° E) 49.8°

Accepted Answer

The angle of incidence at which light is completely polarized upon reflection is given by Brewster's angle, calculated using the formula tan(θ) = n, where n is the refractive index of the material. For ice, with n = 1.31, solving for θ gives θ = tan^(-1)(1.31) ≈ 52.6°.

Question 9

The large space telescope that has been placed into an Earth orbit has an aperture diameter of 2.40 meters. What angular resolution will this telescope achieve for visible light of wavelength λ = 4.80 × 10−⁷ m? A) 2.44 × 10−⁷ rads B) 2.44 × 10−⁴ rads C) 2.44 × 10−³ rads D) 1.22 × 10−⁶ rads E) 2.00 × 10−⁶ rads

Accepted Answer

The angular resolution of a telescope is given by the Rayleigh criterion:$$	heta = 1.22 \frac{\lambda}{D}$$where θ is the angular resolution, λ is the wavelength of light, and D is the diameter of the telescope's aperture.Plugging in the given values, we get:$$	heta = 1.22 \frac{4.80 	imes 10^{-7} 	ext{ m}}{2.40 	ext{ m}} = 2.44 	imes 10^{-7} 	ext{ rad}$$Therefore, the correct answer is A.

Question 10

At what distance could one theoretically distinguish two automobile headlights separated by 1.5 meters? Assume a pupil diameter of 0.50 cm and yellow headlights seen at the wavelength 5.0 × 10−⁷ m. Assume eye fluid has an average n = 1.33. A) 6.0 km B) 12 km C) 9.0 km D) 3.0 km E) 16 km

Accepted Answer

The answer of At what distance could one theoretically distinguish...

Question 11

Helium-neon laser light (λ = 6.33 × 10−⁷ m) is sent through a 0.30 mm-wide single slit. What is the width of the central maximum on a screen 1.0 m from the slit? A) 2.0 cm B) 4.2 mm C) 1.1 cm D) 2.0 mm E) 0.70 mm

Accepted Answer

The width of the central maximum ($W$) in a single-slit diffraction pattern is given by the formula $W = 2L\lambda / a$, where $L$ is the distance to the screen, $\lambda$ is the wavelength of the light, and $a$ is the width of the slit. Substituting the given values: $W = 2 \times 1.0 \, \text{m} \times (6.33 \times 10^{-7} \, \text{m}) / (0.30 \times 10^{-3} \, \text{m}) = 4.22 \times 10^{-3} \, \text{m} = 4.2 \, \text{mm}$.

Question 12

The pupil of a cat's eye narrows to a slit width of 0.5 mm in daylight. What is the angular resolution of the cat's eye in daylight (λ = 500 nm)? A) 0.01 rads B) 10−⁵ rads C) 10−³ rads D) 10−⁴ rads E) 0.1 rads

Accepted Answer

The angular resolution can be calculated using the formula:

Angular resolution = 1.22 * λ / D

where λ is the wavelength of light and D is the diameter of the pupil.

Substituting the given values, we get:

Angular resolution = 1.22 * (500 nm) / (0.5 mm) = 1.22 * 10^-6 rad

This can be expressed in scientific notation as 10^-1.087, which is closest to 10^-1.13. Therefore, the best choice is C) 10^-13 rad.

Question 13

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 14

How wide must a narrow slit be if the first diffraction minimum occurs at ±12° with laser light of 633 nm? A) 3.0 × 10−⁶ m B) 3.0 × 10−⁵ m C) 6.1 × 10−⁶ m D) 6 .1× 10−⁵ m E) 1.5 × 10−⁶ m

Accepted Answer

The correct answer is A.The first diffraction minimum occurs at an angle given by:sin θ = λ/wwhere λ is the wavelength of light, w is the width of the slit, and θ is the angle of the minimum.Solving for w, we get:w = λ/sin θSubstituting the given values, we get:w = (633 nm)/(sin 12°) = 3.0 × 10−⁶ m

Question 15

How far above the horizon is the moon when its image, reflected in a calm lake, is completely polarized? (n_water = 1.33.) A) 22.2° B) 7.7° C) 16.6° D) 36.9° E) 53.1°

Accepted Answer

The answer of How far above the horizon is the...

Question 16

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

The Bragg's law equation, $n\lambda = 2d\sin\theta$, where $n$ is the order of maximum (1 in this case), $\lambda$ is the wavelength of X-rays (0.140 nm), and $d$ is the interplanar spacing (2.814 × 10^-10 m), can be rearranged to solve for $\theta$. Plugging in the given values and solving for $\theta$ gives an angle close to 14.4°.

Question 17

A ruby laser beam (λ = 694.3 nm) is sent outwards from a 2.7-m diameter telescope to the moon, 384,000 km away. What is the radius of the big red spot on the moon?
A) 500 m
B) 250 m
C) 120 m
D) 1.0 km
E) 2.7 km

Accepted Answer

The radius of the spot on the moon can be determined using the formula:
$r=\frac{1.22\lambda D}{2L}$
where r is the radius of the spot on the moon, λ is the wavelength of the laser beam, D is the diameter of the telescope, and L is the distance to the moon.
Substituting the given values, we get:
$r=\frac{1.22	imes 694.3	imes 10^{-9}	imes 2.7}{2	imes 384,000	imes 10^3}=120$ m
Therefore, the radius of the spot on the moon is 120 m. Thus, the answer is C.

Question 18

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

The maximum range of a radar system can be determined using the radar equation:

$R=\sqrt[4]{\frac{P_tG^2\lambda^2\sigma}{P_r(4\pi)^3L}}$

where:
$R$ = maximum range in kilometers
$P_t$ = transmitter power in watts (not given but assumed to be standard for this type of radar)
$G$ = gain of the antenna (not given but assumed to be standard for this type of radar)
$\lambda$ = wavelength, which can be calculated by dividing the speed of light by the frequency (c/f) 
$\sigma$ = radar cross section, which is a measure of how reflective the target is 
$P_r$ = received power, which depends on the target range and is given by the radar equation 
$L$ = system loss, which includes losses due to atmospheric absorption, antenna pointing, etc. (not given but assumed to be negligible)

Substituting the given values: 
$\lambda = \frac{c}{f} = \frac{3	imes10^8}{9000	imes10^6} = 0.033	ext{ meters}$

$\sigma$ is not given but assumed to be the same for both targets, so it cancels out of the equation.

$R=\sqrt[4]{\frac{P_tG^2\lambda^2}{P_r(4\pi)^3}}$

To distinguish two aircraft 100 meters apart, the radar must be able to resolve a target at least 50 meters away from another target. This means that the range resolution $\Delta R$ must be less than or equal to 50 meters.

$\Delta R = \frac{c}{2B}$

where $B$ is the bandwidth of the radar system (not given but assumed to be standard for this type of radar).

Solving for $B$:

$B = \frac{c}{2\Delta R} = \frac{3	imes10^8}{2	imes50} = 3	imes10^6	ext{ Hz}$

Now we can substitute $B$ into the radar equation:

$R=\sqrt[4]{\frac{P_tG^2\lambda^2}{P_r(4\pi)^3(\frac{c}{2B})}}$

$R=\sqrt[4]{\frac{P_tG^2\lambda^2B}{P_r(4\pi)^3c}}$

Substituting the given values:

$R=\sqrt[4]{\frac{P_tG^2(0.033)^2	imes3	imes10^6}{P_r(4\pi)^3	imes3	imes10^8}}$

$R=\sqrt[4]{\frac{P_tG^2}{P_r(4\pi)^3}	imes0.0033}$

Assuming standard values for $P_t$ and $G$ for this type of radar, we can simplify the equation to:

$R=\sqrt[4]{\frac{0.1}{P_r}}$

where $P_r$ is the received power in watts.

To distinguish two targets at a range of 50 meters, we need to assume a target radar cross section in order to calculate the received power. Assuming a typical value of 1 square meter, we get:

$P_r = \frac{P_tG^2\lambda^2\sigma}{(4\pi)^3R^4} = \frac{(3000)^2	imes0.0033	imes1}{(4\pi)^3	imes0.05^4} = 2.56	imes10^{-16}	ext{ watts}$

Substituting this value back into the equation for range:

$R=\sqrt[4]{\frac{0.1}{2.56	imes10^{-16}}} = 36.9	ext{ kilometers}$

Therefore, the maximum range for this radar system to distinguish two aircraft 100 meters apart is 37 kilometers.

The correct choice is D) 37 km.

Question 19

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 line spacing, $d$, of a diffraction grating can be calculated using the formula $d \sin(\theta) = m\lambda$, where $\theta$ is the angle of the diffraction maximum, $m$ is the order of the maximum, and $\lambda$ is the wavelength of the light. For the first-order maximum ($m = 1$) at $20.5^\circ$ with $\lambda = 632.8 \, \text{nm} = 632.8 \times 10^{-9} \, \text{m}$, solving for $d$ gives:$$d = \frac{\lambda}{\sin(\theta)} = \frac{632.8 \times 10^{-9}}{\sin(20.5^\circ)} \approx 1.81 \times 10^{-6} \, \text{m}$$

Question 20

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.

Quiz 38: Diffraction Patterns and Polarization

A narrow slit is illuminated with sodium yellow light of wavelength 589 nm. If the central maximum extends to ±30.0°, how wide is the slit?

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 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?

Plane waves of ultrasound, f = 51.0 kHz, impinge on a flat plane with a 1.00 cm-wide slit. What is the total angular width (2θ) of diffracted sound waves when the speed of sound in air is 340 m/sec.

Unpolarized light is passed through three successive Polaroid filters, each with its transmission axis at 45.0° to the preceding filter. What percentage of light gets through?

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?

Light reflected off a plate-glass window (n = 1.5) is found to be completely polarized at angle-of-incidence θ. Find θ.

Sunlight reflected from a smooth ice surface is completely polarized. Determine the angle of incidence. (n_ice = 1.31.)

The large space telescope that has been placed into an Earth orbit has an aperture diameter of 2.40 meters. What angular resolution will this telescope achieve for visible light of wavelength λ = 4.80 × 10−⁷ m?

At what distance could one theoretically distinguish two automobile headlights separated by 1.5 meters? Assume a pupil diameter of 0.50 cm and yellow headlights seen at the wavelength 5.0 × 10−⁷ m. Assume eye fluid has an average n = 1.33.

Helium-neon laser light (λ = 6.33 × 10−⁷ m) is sent through a 0.30 mm-wide single slit. What is the width of the central maximum on a screen 1.0 m from the slit?

The pupil of a cat's eye narrows to a slit width of 0.5 mm in daylight. What is the angular resolution of the cat's eye in daylight (λ = 500 nm) ?

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.

How wide must a narrow slit be if the first diffraction minimum occurs at ±12° with laser light of 633 nm?

How far above the horizon is the moon when its image, reflected in a calm lake, is completely polarized? (n_water = 1.33.)

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 ruby laser beam (λ = 694.3 nm) is sent outwards from a 2.7-m diameter telescope to the moon, 384,000 km away. What is the radius of the big red spot on the moon?

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 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?

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.