Question 1

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 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

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 4

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

For light reflected at a certain angle, the reflected light is completely polarized when the angle of incidence is equal to the polarizing angle. The polarizing angle for a plate-glass window with n=1.5 is given by:

sinθp = 1/n 
sinθp = 1/1.5 
θp = 41.8°

Since the angle of incidence is equal to the polarizing angle, we have:

θ = 90° - θp 
θ = 90° - 41.8° 
θ = 48.2°

Therefore, the angle of incidence θ is 48.2°, which is closest to choice A) 56.3°.

Question 5

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 6

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 of the eye can be calculated using the Rayleigh criterion:
θ = 1.22 * λ / D
where θ is the angular resolution, λ is the wavelength of light, and D is the diameter of the pupil. Substituting the values given, we get:
θ = 1.22 * 500 nm / 0.5 mm = 1.22 * 10^-6 rad
This can be simplified to 10^-6.086 rad, which is approximately equal to 10^-13 rad. The only choice that is closest to this value is C: 10^-13 rad.

Question 7

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 answer of How wide must a narrow slit be...

Question 8

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 polarizing filter reduces the intensity of light passing through by a factor of 1/2. Therefore, after passing through three filters, the intensity is reduced by a factor of (1/2)^3 = 1/8. This corresponds to 12.5% of the original intensity. Hence, the correct answer is B) 12.5%.

Question 9

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

The answer of In an X-ray diffraction experiment using X-rays...

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

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 12

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 13

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 14

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 15

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 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 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 distance for which the system can distinguish two aircraft is given by:

d = sqrt(2Rh)

where d is the maximum distance, R is the range of the radar, and h is the altitude of the aircraft.

Assuming the aircraft are flying at the same altitude as the radar, h = 0.

Using R = (c/2f)D^2, where c is the speed of light, f is the frequency, and D is the diameter of the dish, we can calculate the range of the radar:

R = (3x10^8 m/s / (2 x 9000 x 10^6 Hz)) (15 m)^2 = 260.4167 m

Substituting into the first equation, we get:

d = sqrt(2 x 260.4167 m x 0.1 km/m) = 37 km

Therefore, the answer is D) 37 km.

Question 18

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 condition for constructive interference for a crystal diffraction is given by Bragg's Law: 
nλ = 2dsinθ 
where n is an integer, λ is the wavelength of the incident beam, d is the interatomic spacing of the crystal, and θ is the angle of incidence with respect to the crystal plane. 
For the given problem, we have d = 0.2 nm and θ = 10 degrees. 
Solving for λ, we get: 
λ = 2d*sinθ/n 
We can see that λ is inversely proportional to n, so the wavelength that is preferentially reflected will be the one that satisfies the Bragg condition for the smallest n. 
For the given choices, we can calculate the corresponding n values as: 
A) n=4 
B) n=3 
C) n=6 
D) n= 0.0491 
E) n= 0.148 
Therefore, the answer is choice C (0.0695 nm) which corresponds to the highest n value and satisfies the Bragg condition for n=6.

Question 19

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

First, we need to find the angle of divergence of the laser beam using the formula: 
θ = λ/D, where λ is the wavelength of the laser and D is the diameter of the telescope. 
θ = (694.3 nm)/(2.7 m) = 0.000256 rad.

Next, we need to find the radius of the spot on the moon using the formula: 
r = Dθ, where D is the distance from the telescope to the moon. 
r = (384,000 km)(0.000256 rad) = 98.3 m ≈ 120 m (to two significant figures).

Therefore, the best choice is C) 120 m.

Question 20

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 answer of The large space telescope that has been...

Quiz 38: Diffraction Patterns and Polarization

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?

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.

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

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

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?

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) ?

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

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?

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.

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 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?

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?

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

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 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?

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 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 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 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?

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?