Question 1

X-ray diffraction: Certain planes of a crystal of halite have a spacing of 0.399 nm. The crystal is irradiated by a beam of X-rays. First order constructive interference occurs when the beam makes an angle of 20° with the planes. What is the wavelength of the X-rays?
A) 0.14 nm
B) 0.17 nm
C) 0.21 nm
D) 0.24 nm
E) 0.27 nm

Accepted Answer

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Question 2

Diffraction grating: What is the angular separation of the two second-order spectral lines having wavelengths 417 nm and 388 nm using a diffraction grating having 456 lines/mm?
A) 1.63°
B) 1.52°
C) 0.815°
D) 3.26°

Accepted Answer

The angular separation can be calculated using the grating equation $d\sin\theta = m\lambda$, where $d$ is the grating spacing, $m$ is the order of the spectrum, and $\lambda$ is the wavelength. The grating spacing $d$ is the reciprocal of the number of lines per meter, so for 456 lines/mm, $d = \frac{1}{456 \times 10^3}$ meters. For the second order ($m = 2$), the difference in angles for the two wavelengths can be found and converted to degrees, resulting in an angular separation of approximately 1.63°.

Question 3

X-ray diffraction: If an X-ray beam of wavelength 1.4 × 10^-10 m makes an angle of 20° with a set of planes in a crystal causing first order constructive interference, at what angle will the second order line appear? A) 40° B) 20° C) 43° D) 4.0° E) 11°

Accepted Answer

The Bragg's law states that nλ = 2d sinθ, where n is the order of diffraction, λ is the wavelength of the X-ray beam, d is the distance between the planes, and θ is the angle of incidence.

For the first order diffraction, n = 1 and θ = 20°. Therefore,

1.4 × 10^-10 m = 2d sin20°

Solving for d gives d ≈ 2.55 × 10^-10 m.

For the second order diffraction, n = 2 and we need to find θ. Using the same equation, we have

2 × 1.4 × 10^-10 m = 2(2.55 × 10^-10 m) sinθ

Solving for sinθ gives sinθ ≈ 0.866, which corresponds to an angle of θ ≈ 43°. Therefore, the second order line will appear at an angle of 43°.

The correct answer choice is C.

Question 4

Diffraction grating: An 18-mm wide diffraction grating has rulings of 880 lines per mm. Monochromatic light of 590 nm wavelength is incident normally on the grating. What is the largest angle, measured from the central bright spot, at which an intensity maximum is formed?
A) 31°
B) 29°
C) 27°
D) 25°
E) 23°

Accepted Answer

The largest angle for an intensity maximum can be found using the grating equation $d\sin\theta = m\lambda$, where $d$ is the grating spacing, $\theta$ is the diffraction angle, $m$ is the order of the maximum, and $\lambda$ is the wavelength of the light. The grating spacing $d$ is the inverse of the number of lines per mm, so $d = \frac{1}{880 \text{ lines/mm}} = \frac{1}{880 \times 10^3 \text{ lines/m}}$. Given $\lambda = 590 \text{ nm} = 590 \times 10^{-9} \text{ m}$, we find the maximum order $m$ that satisfies the equation before $\sin\theta$ exceeds 1. For the highest order maximum that can be observed, $\sin\theta$ approaches 1, which corresponds to $\theta = 90^\circ$, but practically, we look for the highest $\theta < 90^\circ$. Rearranging the equation for $\theta$, and incrementing $m$ from 1 upwards, we find the maximum $m$ that yields a valid $\theta$. For the largest angle less than 90°, we calculate $\theta$ for the highest possible $m$. Given the constraints, the calculation will show that the angle for the highest observable order is closest to 31°, making option A correct.

Question 5

Diffraction grating: A diffraction grating has rulings of 890 lines/mm. When white light is incident normally on the grating, what is the longest wavelength that forms an intensity maximum in the fifth order?
A) 225 nm
B) 200 nm
C) 250 nm
D) 275 nm
E) 300 nm

Accepted Answer

The longest wavelength that forms an intensity maximum in the fifth order is given by:$$\lambda = \frac{d}{m}$$where:* $\lambda$ is the wavelength* $d$ is the grating spacing* $m$ is the order of the maximumIn this case, $d = 1/890 	ext{ mm} = 1.124 	imes 10^{-3} 	ext{ mm}$ and $m = 5$. Therefore,$$\lambda = \frac{1.124 	imes 10^{-3} 	ext{ mm}}{5} = 224.8 	ext{ nm}$$which is closest to choice A, 225 nm.

Question 6

Circular aperture: Light of wavelength 500 nm illuminates a round 0.50-mm diameter hole. A screen is placed 6.3 m behind the slit. What is the diameter of the central bright area on the screen?
A) 15 mm
B) 270 μm
C) 7.7 mm
D) 3800 μm

Accepted Answer

The diameter of the central bright area is given by:$$d = 2.44 \lambda \frac{D}{a}$$where:$$d$$ is the diameter of the central bright area$$\lambda$$ is the wavelength of light$$D$$ is the distance from the aperture to the screen$$a$$ is the diameter of the aperturePlugging in the given values, we get:$$d = 2.44 (500 	imes 10^{-9} 	ext{ m}) \frac{6.3 	ext{ m}}{5.0 	imes 10^{-4} 	ext{ m}} = 15 	ext{ mm}$$

Question 7

Resolving power: If the headlights on a car are separated by 1.3 m, how far down the road can they be resolved if the angular resolution of the eye is 5.0 × 10^-4 rad and the person has excellent vision? A) 1.3 km B) 5.0 km C) 4.8 km D) 0.65 km E) 2.6 km

Accepted Answer

The resolving power is given by the formula θ = s / d, where θ is the angular resolution, s is the separation, and d is the distance. Solving for d gives d = s / θ = 1.3 m / (5.0 × 10^-4 rad) = 2.6 km.

Question 8

Resolving power: What is the limiting angle of resolution for the eye if the pupil diameter of the eye is 4.0 mm, the wavelength of the light is 600 nm, and index of refraction of the liquid in the eye is 1.34?
A) 0.42 mrad
B) 0.21 mrad
C) 0.14 mrad
D) 0.11 mrad
E) 0.26 mrad

Accepted Answer

The limiting angle of resolution can be calculated using the formula θ = 1.22 * (λ / (n * D)), where λ is the wavelength of light, n is the index of refraction, and D is the diameter of the pupil. Substituting the given values: θ = 1.22 * (600 x 10^-9 m / (1.34 * 4.0 x 10^-3 m)) = 0.14 mrad.

Question 9

Diffraction grating: A metallic sheet has a large number of slits, 5.0 mm wide and 20 cm apart, and is used as a diffraction grating for microwaves. A wide parallel beam of microwaves is incident normally on the sheet. If the microwave wavelength is 6.0 cm, what is the largest angle away from the central maximum at which an intensity maximum occurs.
A) 64°
B) 69°
C) 74°
D) 79°
E) 84°

Accepted Answer

The largest angle away from the central maximum at which an intensity maximum occurs is given by the equation:sin θ = mλ/dwhere m is the order of the maximum, λ is the wavelength of the microwaves, and d is the distance between the slits.For the first-order maximum (m = 1), we have:sin θ = λ/d = 6.0 cm / 20 cm = 0.3θ = sin^-1(0.3) = 17.5°For the second-order maximum (m = 2), we have:sin θ = 2λ/d = 12.0 cm / 20 cm = 0.6θ = sin^-1(0.6) = 36.9°The largest angle away from the central maximum at which an intensity maximum occurs is therefore 36.9°, which is closest to choice A) 64°.

Question 10

Diffraction grating: A metallic sheet has a large number of slits, 5.0 mm wide and 20 cm apart, and is used as a diffraction grating for microwaves. A wide parallel beam of microwaves is incident normally on the sheet. What is the smallest microwave frequency for which only the central maximum occurs? (c = 3.00 × 10⁸ m/s) A) 0.50 GHz B) 0.70 GHz C) 1.0 GHz D) 1.5 GHz E) 2.0 GHz

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Question 11

Diffraction grating: A diffraction grating has 450 lines per mm. What is the highest order m that contains the entire visible spectrum from 400 nm to 700 nm?
A) m = 2
B) m = 3
C) m = 4
D) m = 5
E) m = 6

Accepted Answer

The highest order m that contains the entire visible spectrum can be found using the grating equation for diffraction maxima, $d\sin\theta = m\lambda$, where $d$ is the grating spacing, $\theta$ is the diffraction angle, $m$ is the order of the spectrum, and $\lambda$ is the wavelength of light. The grating spacing $d$ can be found from the given 450 lines per mm, which is equivalent to $d = \frac{1}{450} mm = \frac{1}{450,000} m$. To ensure the entire visible spectrum (400 nm to 700 nm) is within the diffraction limit, we use the condition for the maximum possible order, which occurs at $\sin\theta = 1$, leading to $d = m\lambda$. For the longest wavelength (700 nm or 700x10^-9 m) to fit within an order, we solve for $m$ using the maximum $d$ value: $m = \frac{d}{\lambda} = \frac{1}{450,000} / 700x10^{-9} \approx 3.2$. Since $m$ must be an integer, the highest possible order that can contain the entire visible spectrum is $m = 3$.

Question 12

X-ray diffraction: If an X-ray beam of wavelength 1.4 × 10^-10 m makes an angle of 30° with a set of planes in a crystal causing first order constructive interference, what is the plane spacing? A) 0.28 nm B) 0.070 nm C) 0.32 nm D) 0.16 nm E) 0.14 nm

Accepted Answer

Using Bragg's law, $n\lambda = 2d\sin\theta$, where $n=1$ (first order), $\lambda = 1.4 \times 10^{-10} m$ (wavelength), and $\theta = 30°$. Solving for $d$ (plane spacing), $d = \frac{\lambda}{2\sin\theta} = \frac{1.4 \times 10^{-10}}{2\sin(30°)} = \frac{1.4 \times 10^{-10}}{2 \times 0.5} = 1.4 \times 10^{-10} m = 0.14 nm$.

Question 13

Resolving power: A certain astronomical telescope has a diameter of 5.60 m. Considering only the limitation due to diffraction, what is the minimum angle of resolution for this telescope at a wavelength of 620 nm?
A) 0.111 µrad
B) 0.311 µrad
C) 0.270 µrad
D) 0.135 µrad
E) 0.405 µrad

Accepted Answer

The minimum angle of resolution for a telescope, considering only the limitation due to diffraction, can be calculated using the formula $\theta = 1.22 \frac{\lambda}{D}$, where $\theta$ is the angle in radians, $\lambda$ is the wavelength of light, and $D$ is the diameter of the telescope's aperture. Substituting $\lambda = 620 \, \text{nm} = 620 \times 10^{-9} \, \text{m}$ and $D = 5.60 \, \text{m}$ into the formula gives $\theta = 1.22 \times \frac{620 \times 10^{-9}}{5.60} \approx 1.35 \times 10^{-7}$ radians, which is equivalent to 0.135 µrad.

Question 14

Diffraction grating: A thin beam of laser light of wavelength 514 nm passes through a diffraction grating having 3952 lines/cm. The resulting pattern is viewed on a distant curved screen that can show all bright fringes up to and including ±90.0° from the central spot. If the experiment were performed with all of the apparatus under water (which has an index of refraction of 1.33), what would be the TOTAL number of bright spots that would show up on the screen?
A) 6
B) 7
C) 12
D) 13
E) 14

Accepted Answer

The equation for the number of bright spots in a diffraction grating pattern is:

N = (d sinθ)/λ

Where N is the number of bright spots, d is the distance between grating lines, θ is the angle between the central maximum and the nth bright spot, and λ is the wavelength of the light.

In this case, we can use the given values to solve for N:

d = 1/3952 cm (since there are 3952 lines per cm)
λ = 514 nm (since the laser light has a wavelength of 514 nm)
n = ? (we need to solve for the maximum value of n that produces a bright spot within the viewing range of ±90.0°)

To find the maximum value of n, we can use the equation for the angle θ:

sinθ = nλ/d

Rearranging, we get:

n = dsinθ/λ

Substituting the values given, we have:

n = (1/3952 cm)(sin 90°)(1.33)/514 nm = 13

Therefore, the total number of bright spots that would show up on the screen is 13, which is answer choice D.

Question 15

Circular aperture: Light of wavelength 633 nm from a He-Ne laser passes through a circular aperture and is observed on a screen 4.0 m behind the aperture. The diameter of the central bright area is 5.4 cm. What is the diameter of the aperture?
A) 110 μm
B) 2.0 μm
C) 6600 μm
D) 960 μm

Accepted Answer

The diameter of the central bright area (first dark ring) in the diffraction pattern from a circular aperture is given by the formula $D = 2.44 \lambda \frac{L}{d}$, where $D$ is the diameter of the central bright area, $\lambda$ is the wavelength of the light, $L$ is the distance to the screen, and $d$ is the diameter of the aperture. Rearranging this formula to solve for $d$, we get $d = 2.44 \lambda \frac{L}{D}$. Plugging in the given values ($\lambda = 633 \times 10^{-9}$ m, $L = 4.0$ m, and $D = 5.4 \times 10^{-2}$ m), we find $d \approx 110 \times 10^{-6}$ m, or 110 μm.

Question 16

Resolving power: A radio telescope 200 m in diameter is used to investigate sources emitting a 21-cm wavelength wave. What is the minimum angular separation of the sources that can be resolved by this system?
A) 0.073°
B) 0.030°
C) 0.0013°
D) 0.154°
E) 0.0026°

Accepted Answer

The Rayleigh criterion for the minimum angular separation of two sources that can be resolved by a telescope is given by:θ = 1.22 * (λ/D)where θ is the minimum angular separation, λ is the wavelength of the wave, and D is the diameter of the telescope.In this case, λ = 21 cm = 0.21 m and D = 200 m. Substituting these values into the equation, we get:θ = 1.22 * (0.21 m / 200 m) = 0.0013°Therefore, the minimum angular separation of the sources that can be resolved by this system is 0.0013°.

Question 17

Diffraction grating: A thin beam of laser light of wavelength 514 nm passes through a diffraction grating having 3952 lines/cm. The resulting pattern is viewed on a distant curved screen that can show all bright fringes up to and including ±90.0° from the central spot. What is the TOTAL number of bright fringes that will show up on the screen?
A) 4
B) 5
C) 8
D) 9
E) 10

Accepted Answer

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Question 18

Diffraction grating: A diffraction grating is to be used to find the wavelength of the light in the emission spectrum of a gas. The grating spacing is not known, but a light having a known wavelength of 632.8 nm is deflected by 43.2° away from the central maximum in the second order by this grating. Light of the wavelength to be measured is deflected by 48.2° away from the central maximum in the second order. What is the wavelength of this light?

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Question 19

X-ray diffraction: A researcher is investigating a cubic crystal with X-rays. He is looking at Bragg reflection from the planes parallel to the cube faces. He finds that when using X-rays of 0.165 nm a strong first maximum occurs when the beam makes an angle of 23.5° with the planes. What is the spacing of adjacent atoms in the crystal?

Accepted Answer

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Question 20

Resolving power: A 10-inch telescope (25.4 cm in diameter) is used to determine if what appears to be one star is actually two stars. Stars are so far away that they are essentially point sources. How close (in angle) can the two stars be and still be resolved by this telescope if it is focusing light of wavelength of 550 nm? (Consider only the limitation due to diffraction.) A) 4.2 × 10^-8 degree B) 2.6 × 10^-6 degree C) 3.0 × 10^-4 degree D) 1.5 × 10^-4 degree E) 6.6 × 10^-8 degree

Accepted Answer

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Quiz 28: Sources of Magnetic Field

X-ray diffraction: Certain planes of a crystal of halite have a spacing of 0.399 nm. The crystal is irradiated by a beam of X-rays. First order constructive interference occurs when the beam makes an angle of 20° with the planes. What is the wavelength of the X-rays?

Diffraction grating: What is the angular separation of the two second-order spectral lines having wavelengths 417 nm and 388 nm using a diffraction grating having 456 lines/mm?

X-ray diffraction: If an X-ray beam of wavelength 1.4 × 10-10 m makes an angle of 20° with a set of planes in a crystal causing first order constructive interference, at what angle will the second order line appear?

Diffraction grating: An 18-mm wide diffraction grating has rulings of 880 lines per mm. Monochromatic light of 590 nm wavelength is incident normally on the grating. What is the largest angle, measured from the central bright spot, at which an intensity maximum is formed?

Diffraction grating: A diffraction grating has rulings of 890 lines/mm. When white light is incident normally on the grating, what is the longest wavelength that forms an intensity maximum in the fifth order?

Circular aperture: Light of wavelength 500 nm illuminates a round 0.50-mm diameter hole. A screen is placed 6.3 m behind the slit. What is the diameter of the central bright area on the screen?

Resolving power: If the headlights on a car are separated by 1.3 m, how far down the road can they be resolved if the angular resolution of the eye is 5.0 × 10-4 rad and the person has excellent vision?

Resolving power: What is the limiting angle of resolution for the eye if the pupil diameter of the eye is 4.0 mm, the wavelength of the light is 600 nm, and index of refraction of the liquid in the eye is 1.34?

Diffraction grating: A diffraction grating has 450 lines per mm. What is the highest order m that contains the entire visible spectrum from 400 nm to 700 nm?

X-ray diffraction: If an X-ray beam of wavelength 1.4 × 10-10 m makes an angle of 30° with a set of planes in a crystal causing first order constructive interference, what is the plane spacing?

Resolving power: A certain astronomical telescope has a diameter of 5.60 m. Considering only the limitation due to diffraction, what is the minimum angle of resolution for this telescope at a wavelength of 620 nm?

Circular aperture: Light of wavelength 633 nm from a He-Ne laser passes through a circular aperture and is observed on a screen 4.0 m behind the aperture. The diameter of the central bright area is 5.4 cm. What is the diameter of the aperture?

Resolving power: A radio telescope 200 m in diameter is used to investigate sources emitting a 21-cm wavelength wave. What is the minimum angular separation of the sources that can be resolved by this system?

X-ray diffraction: If an X-ray beam of wavelength 1.4 × 10^-10 m makes an angle of 20° with a set of planes in a crystal causing first order constructive interference, at what angle will the second order line appear?

Resolving power: If the headlights on a car are separated by 1.3 m, how far down the road can they be resolved if the angular resolution of the eye is 5.0 × 10^-4 rad and the person has excellent vision?

X-ray diffraction: If an X-ray beam of wavelength 1.4 × 10^-10 m makes an angle of 30° with a set of planes in a crystal causing first order constructive interference, what is the plane spacing?