Question 1

For a given light source and collimator slit width, the spectral lines obtained using a prism are brighter than those using a diffraction grating because
A) the grating absorbs more light than the prism.
B) light is dispersed more by a prism than by a grating.
C) the prism forms a single spectrum and the grating forms multiple spectra.
D) light passes through the prism at minimum deviation.
E) light is dispersed more by a grating than by a prism.

Accepted Answer

A prism disperses light by refracting it through a prism of a certain material that bends the light at different angles based on its wavelength. On the other hand, a diffraction grating disperses light by diffracting it through a series of closely spaced, parallel lines that create interference when the light passes through it. The interference causes the light to split into its various wavelengths. The prism disperses light more than a grating, which causes the spectral lines to appear brighter.

Question 2

When a first-order spectrum is examined by means of a one-inch grating having 10,000 lines/inch, the resolving power is R. When a second-order spectrum is examined by means of a two-inch grating having 20,000 lines/inch, the resolving power is
A) R/4
B) R/2
C) 2R
D) 4R
E) 8R

Accepted Answer

The resolving power (R) of a grating is given by the formula R = nN, where n is the order of the spectrum and N is the total number of lines on the grating. For the first case, n = 1 and N = 10,000, so R = 1*10,000. For the second case, n = 2 and N = 2*20,000 = 40,000, so the resolving power is 2*40,000 = 80,000, which is 8 times the original R.

Question 3

The spacing between the adjacent lines of a diffraction grating is the same as the distance between the two slits in the Young's double slit experiment. Both are illuminated by light of the same wavelength. In which way do the interference fringes seen on the screen differ for the two methods?
A) There is no difference in the interference fringes.
B) The fringes from the grating are farther apart.
C) The fringes from the double slit are farther apart.
D) Each of the fringes from the grating is narrower than the double slit.
E) Each of the fringes from the double slit is narrower than the grating.

Accepted Answer

The diffraction grating produces narrower fringes due to the constructive and destructive interference from multiple slits, leading to sharper and more defined peaks compared to the broader fringes produced by the double slit experiment.

Question 4

The angle at which a telescope must be set to locate the second-order image of light of wavelength 589.3 nm is approximately
A) 18º
B) 45º
C) 75º
D) 85º
E) 37º

Accepted Answer

The answer of  The angle at which a telescope...

Question 5

A diffraction grating has 2000 lines per centimeter and is 3.0 centimeters wide. When visible light is incident on the grating, a maximum is observed at an angle of 10º from the normal. The wavelength of the incident radiation is
A) 435 nm
B) 525 nm
C) 600 nm
D) 725 nm
E) 870 nm

Accepted Answer

The formula for finding the wavelength of the incident radiation for a diffraction grating is:

λ = d sinθ / N

where λ is the wavelength of light, d is the distance between adjacent lines on the grating, θ is the angle from the normal at which the maximum is observed, and N is the number of lines per unit length.

Plugging in the values, we get:

λ = (3.0 cm / 2000 lines/cm) sin(10º) 
   = 5.25 x 10^-5 cm 
   = 525 nm

Therefore, the wavelength of the incident radiation is 525 nm, which corresponds to choice B.

Question 6

If the prism spectrum of a source is a line spectrum, the grating spectrum would
A) have lines identical in intensity and position.
B) have lines identical in position but not in intensity.
C) be an absorption spectrum.
D) also be a line spectrum.
E) be a continuous spectrum.

Accepted Answer

A line spectrum is characterized by specific wavelengths or frequencies of light being emitted, resulting in discrete lines in the spectrum. A diffraction grating works in a similar way to a prism, but it separates the light into its component wavelengths at different angles. Therefore, the grating spectrum would also show these discrete lines, resulting in another line spectrum.

Question 7

A discharge tube is known to produce two spectral lines, one at 640 nm and the other at 640.4 nm. The total number of rulings that a grating must have to just resolve these lines in the second order is
A) 400
B) 800
C) 1600
D) 3200
E) 6400

Accepted Answer

The resolution of spectral lines using a diffraction grating is determined by the grating equation and the Rayleigh criterion for resolution. According to the Rayleigh criterion, two wavelengths, λ and λ + Δλ, are just resolved in the nth order of diffraction when their diffraction patterns are separated by the angular distance corresponding to the first minimum of one pattern coinciding with the maximum of the other. The formula for the resolving power (R) of a grating is given by R = λ/Δλ = nN, where n is the order of diffraction and N is the total number of rulings on the grating. Given: λ = 640 nm, Δλ = 0.4 nm, and n = 2 (second order).R = λ/Δλ = 640/0.4 = 1600Therefore, nN = 1600. Since n = 2, we find N by rearranging the equation to N = 1600/n = 1600/2 = 800. Thus, the grating must have at least 800 rulings to just resolve these two spectral lines in the second order.

Question 8

White light is in one case dispersed by refraction on passing through a glass prism and in a second case diffracted by means of a grating. When the red component and the blue component are considered, it is found that
A) red is both refracted and diffracted at greater angles than blue.
B) blue is both refracted and diffracted at greater angles than red.
C) red is refracted at a greater angle than blue, but blue is diffracted at a greater angle than red.
D) blue is refracted at a greater angle than red, but red is diffracted at a greater angle than blue.
E) both red and blue are refracted at the same angle and diffracted at the same angle.

Accepted Answer

In refraction, the angle of deviation depends on the refractive index of the material and the wavelength of the light. Red light has a longer wavelength than blue light, so it bends less when passing through the prism. Thus, red light is refracted at a smaller angle than blue light. This means that blue is refracted at a greater angle than red. Therefore, option A and C are incorrect.

In diffraction, the angle of deviation depends on the spacing of the grating and the wavelength of the light. Again, red light has a longer wavelength than blue light, so it diffracts at a greater angle than blue light. This means that red is diffracted at a greater angle than blue. Therefore, option B is also incorrect.

Option E is not possible as we have just shown that red and blue have different angles of refraction and diffraction.

This leaves option D, which is correct.

Question 9

A parallel beam of sodium light of wavelength 589 nm is normally incident on a diffraction grating. If the second order diffraction maximum is observed at 50.25 degrees to the normal, then calculate the number of lines per cm of the grating. A) 5.69 * 10⁵ cm^-1 B) 1.53 * 10⁶ cm^-1 C) 6.53 * 10³ cm^-1 D) 1.31 * 10⁶ cm^-1 E) 1.31 * 10⁴ cm^-1

Accepted Answer

The formula for calculating the line spacing (d) of a diffraction grating is:
d = λ/sinθ
where λ is the wavelength of light, θ is the angle of diffraction, and d is the spacing between adjacent lines on the grating.

In this problem, λ = 589 nm and θ = 50.25 degrees.

Using the formula above, we can solve for d:
d = λ/sinθ
d = (589 x 10^-9 m)/(sin(50.25))
d = 6.53 x 10^-6 m = 6.53 μm

The number of lines per cm (N) can be calculated using the formula:
N = 1/d

Substituting the value of d that we calculated above, we get:
N = 1/d
N = 1/(6.53 x 10^-6 m)
N = 153,074 lines/cm

Rounding this to the nearest whole number, we get 153,074 lines/cm, which matches choice C.

Question 10

Which of two diffraction gratings, one with N₁ and the other with N₂ slits per centimeter, has the greater resolving power if N₁ is greater than N₂? A) the grating with N₁ slits per centimeter B) the grating with N₂ slits per centimeter C) They both have the same resolving power, but the one with N₁ slits per centimeter has sharper spectral lines. D) They both have the same resolving power, but the one with N₂ slits per centimeter has sharper spectral lines. E) The width of the individual slits must be known to answer this question.

Accepted Answer

The resolving power of a diffraction grating increases with the number of slits per centimeter. Therefore, the grating with more slits per centimeter (N₁) has greater resolving power.

Question 11

Red light ($\lambda$ = 700 nm) is incident normally on a diffraction grating having 1000 lines/cm. A maximum is observed at a deflection angle of 12º. The order in which this occurs is
A) 1
B) 2
C) 3
D) 4
E) 5

Accepted Answer

The answer of Red light ($\lambda$ = 700 nm) is...

Question 12

$\lambda$(nm)
N
350
1)725
400
1)600
500
1)525
600
1)500
700
1)495
You obtain the data in the table for a certain prism in a spectrometer experiment and determine that the light of an unknown wavelength has an index of refraction of 1.562. From a dispersion curve you find the wavelength to be
A) 375 nm
B) 430 nm
C) 450 nm
D) 480 nm
E) 550 nm

Accepted Answer

The index of refraction of 1.562 corresponds to a wavelength of 430 nm based on the dispersion curve data provided.

Question 13

Hydrogen emits violet light with a wavelength of 410 nm and red light with a wavelength of 656 nm. A parallel beam of hydrogen light is normally incident on a diffraction grating that has 5500 lines per cm. What is the angle between the second order red line and the third order violet line that appear close together?
A) 3.62 degrees
B) 2.58 degrees
C) 4.23 degrees
D) 5.79 degrees
E) None of these is correct.

Accepted Answer

Using the formula for the angle of diffraction,
$$\sin(	heta)=\frac{n\lambda}{d}$$
where $	heta$ is the angle of diffraction, $n$ is the order of diffraction, $\lambda$ is the wavelength of light, and $d$ is the distance between the lines on the grating.

For the red line, $n=2$, $\lambda=656$ nm and $d=1/5500$ cm. For the violet line, $n=3$, $\lambda=410$ nm, and $d=1/5500$ cm.

\begin{align*}
\sin(	heta_	ext{red}) &= \frac{2\cdot656\,	ext{nm}}{1/5500\,	ext{cm}} \
\sin(	heta_	ext{violet}) &= \frac{3\cdot410\,	ext{nm}}{1/5500\,	ext{cm}}
\end{align*}

Taking the difference of the two angles of diffraction,
$$	heta=	heta_	ext{violet}-	heta_	ext{red}=\sin^{-1}\bigg(\frac{3\cdot410}{1/5500}-\frac{2\cdot656}{1/5500}\bigg)$$

Using a calculator to evaluate the expression inside the $\sin^{-1}$ function, we get $	heta\approx\boxed{	extbf{(A) }3.62	ext{ degrees}}$.

Question 14

A student looks through a transmission grating at the light from a helium light source. He sees the red, yellow, and green light from the source superimposed on a meterstick. If the yellow lines are the ones indicated in the figure, then
A) 1 and 2 are green; 3 and 4 are red.
B) 1 and 4 are red; 2 and 3 are green.
C) 1 and 4 are green; 2 and 3 are red.
D) 1 and 3 are red; 2 and 4 are green.
E) 1 and 3 are green; 2 and 4 are red.

Accepted Answer

The answer of  A student looks through a transmission...

Question 15

The yellow line of a Sodium lamp is two lines very close together at $\lambda$₁ = 589.592 nm and $\lambda$₂ = 588.995 nm. A diffraction grating with 10,000 lines/cm is used to resolve the lines. What is the angular separation of the two lines for the first order diffraction? A) 0.212$\degree$ B) 0.0423$\degree$ C) 0.0613$\degree$ D) 0.0814$\degree$ E) There is no angular separation.

Accepted Answer

The angular separation of two adjacent spectral lines of a diffraction grating can be calculated using the grating equation:
d sinθ = mλ
where d is the grating spacing (1/10000 cm in this case), θ is the angle of diffraction, λ is the wavelength of light, and m is the order of diffraction.
For the first order diffraction, m = 1.
The difference in wavelength between the two spectral lines is Δλ = 589.592 nm - 588.995 nm = 0.597 nm.
Substituting these values in the grating equation and solving for θ, we get:
sinθ = (mλ)/d = (1) (0.597 x 10^-7 cm)/(1/10000 cm) = 0.0597
θ = sin^-1(0.0597) = 3.42°
The angular separation between the two lines can be calculated using the formula:
Δθ = (Δλ/λ) (180°/π) / θ
Substituting the values, we get:
Δθ = (0.597/589.794) (180/π) / 3.42 = 0.0423°
Therefore, the angular separation between the two lines for the first order diffraction is 0.0423°, which corresponds to choice B.

Question 16

The yellow line of a Sodium lamp is two lines very close together at $\lambda$₁ = 589.592 nm and $\lambda$₂ = 588.995 nm. If a diffraction grating has to resolve the first order diffraction lines by an angular separation of 0.1$\degree$, what is the least number of lines per cm for the diffraction grating? A) 12,400 cm^-1 B) 13,600 cm^-1 C) 14,700 cm^-1 D) 15,300 cm^-1 E) 16,700 cm^-1

Accepted Answer

We can use the formula for angular dispersion of the grating:

θ = λ/d

where θ is the angular separation between two adjacent lines, λ is the wavelength of the light, and d is the spacing between the lines on the grating.

For the first order diffraction lines, we have:

θ = 0.1° = 0.00175 radians (given in the question)
λ = 589.592 nm (from the question)

Using the first wavelength (589.592 nm), we can solve for d:

d = λ/θ = 589.592 nm / 0.00175 radians = 336714 nm

We convert this to units of cm:

d = 3367.14 nm = 0.0336714 cm

Now we need to find the number of lines per cm:

n = 1/d = 1/0.0336714 cm = 29.70 cm^-1

To get the least number of lines per cm that satisfies the condition, we round up to the nearest integer: 30 cm^-1. However, this is not one of the choices listed.

Since the question asks for the "least" number of lines per cm, we want to find the choice with the smallest value that is greater than or equal to our calculated value of 29.70 cm^-1. The only choice that meets this criterion is C, which has a value of 14700 cm^-1 (or 14,700 lines per cm).

Quiz 13: Interference and Diffraction

For a given light source and collimator slit width, the spectral lines obtained using a prism are brighter than those using a diffraction grating because

When a first-order spectrum is examined by means of a one-inch grating having 10,000 lines/inch, the resolving power is R. When a second-order spectrum is examined by means of a two-inch grating having 20,000 lines/inch, the resolving power is

The spacing between the adjacent lines of a diffraction grating is the same as the distance between the two slits in the Young's double slit experiment. Both are illuminated by light of the same wavelength. In which way do the interference fringes seen on the screen differ for the two methods?

The angle at which a telescope must be set to locate the second-order image of light of wavelength 589.3 nm is approximately

A diffraction grating has 2000 lines per centimeter and is 3.0 centimeters wide. When visible light is incident on the grating, a maximum is observed at an angle of 10º from the normal. The wavelength of the incident radiation is

If the prism spectrum of a source is a line spectrum, the grating spectrum would

A discharge tube is known to produce two spectral lines, one at 640 nm and the other at 640.4 nm. The total number of rulings that a grating must have to just resolve these lines in the second order is

White light is in one case dispersed by refraction on passing through a glass prism and in a second case diffracted by means of a grating. When the red component and the blue component are considered, it is found that

A parallel beam of sodium light of wavelength 589 nm is normally incident on a diffraction grating. If the second order diffraction maximum is observed at 50.25 degrees to the normal, then calculate the number of lines per cm of the grating.

Which of two diffraction gratings, one with N₁ and the other with N₂ slits per centimeter, has the greater resolving power if N₁ is greater than N₂?

Red light ( $\lambda$ = 700 nm) is incident normally on a diffraction grating having 1000 lines/cm. A maximum is observed at a deflection angle of 12º. The order in which this occurs is

$\lambda$ (nm) N 350 1) 725 400 1) 600 500 1) 525 600 1) 500 700 1) 495 You obtain the data in the table for a certain prism in a spectrometer experiment and determine that the light of an unknown wavelength has an index of refraction of 1.562. From a dispersion curve you find the wavelength to be

A student looks through a transmission grating at the light from a helium light source. He sees the red, yellow, and green light from the source superimposed on a meterstick. If the yellow lines are the ones indicated in the figure, then

The yellow line of a Sodium lamp is two lines very close together at $\lambda$ ₁ = 589.592 nm and $\lambda$ ₂ = 588.995 nm. A diffraction grating with 10,000 lines/cm is used to resolve the lines. What is the angular separation of the two lines for the first order diffraction?