Question

A physics instructor wants to project a spectrum of visible-light colors from 400 nm to 700 nm as part of a classroom demonstration. She shines a beam of white light through a diffraction grating that has 600 lines per mm, projecting a pattern on a screen 2.9 m behind the grating.How much distance separates the end of the m = 1 spectrum and the start of the m = 2 spectrum?I already determined that the m=1 spectrum is 0.63 meters wide.

Answer 1

Answer:

Spectrum width = 0.784m

Explanation:

We are given that;

λ1 = 400 nm = 400 x 10^(-9)

λ2 = 700 nm = 700 x 10^(-9)

Slit width; d = 10^(-3)/700 = 1.428 x 10^(-6) m

Distance of screen from light source; D = 2.9m

Now, in slit Experiment, we know that;

d•sinθ = nλ

where;

θ =angle between the path and a line from the slits to the screen.

Thus,

1.428 x 10^(-6) x sinθ = 1 x 400 x 10^(-9)

sinθ = [1 x 400 x 10^(-9)]/(1.428 x 10^(-6))

sinθ = 0.28011

θ = sin^(-1)0.28011

θ = 16.27°

Now,tan θ = y/D

where y is the position of First maxima corresponding

Thus,

y1 = D.tanθ

y1 = 2.9 x tan16.27

y1 = 0.846m

Now, let's do the same for λ2;

1.428 x 10^(-6) x sinθ' = 1 x 700 x 10^(-9)

sinθ' = [1 x 700 x 10^(-9)]/(1.428 x 10^(-6))

sinθ' = 0.4901

θ = sin^(-1)0.4901

θ' = 29.35°

Now,tan θ' = y'/D

where y is the position of First maxima corresponding

Thus,

y'1 = D.tanθ

y'1 = 2.9 x tan29.35

y'1 = 1.63 m

Spectrum width = y'1 - y1 = 1.63 - 0.846 = 0.784m

Answer 2

Spectrum width between the end of the m = 1 spectrum and the start of the m = 2 spectrum is 0.784m.

What is diffraction?

When a light ray passes through a narrow slit or strikes to any obstacle in its path, the light gets diffracted into different colors of electromagnetic spectrum that are seen on a screen placed at some apart from the slit.

Give is the wavelength λ₁ = 400 nm = 400 x 10⁻⁹m and λ₂ = 700 nm = 700 x 10⁻⁹ m

Slit width d = 10⁻³/700 = 1.428 x 10⁻⁶m, distance of screen from light source D = 2.9m

Relation between slit width and wavelength is

d•sinθ = nλ

where, θ  is the angle between the path and a line from the slit to the screen.

Substituting the values, we get

1.428 x 10⁻⁶x sinθ = 1 x 400 x 10⁻⁹

sinθ = 0.28011

θ = sin⁻¹ (0.28011)

θ = 16.27°

Now, tan θ = y/D  where y is the position of First maxima

So,  y₁ = D.tanθ

y₁ = 2.9 x tan16.27

y₁ = 0.846m

Similarly for wavelength λ₂, we have

1.428 x 10⁻⁶ x sinθ' = 1 x 700 x 10⁻⁹

sinθ' = 0.4901

θ' = sin⁻¹ (0.4901)

θ' = 29.35°

Now, tan θ' = y'/D

where y is the position of First maxima

So, y₁' =D.tanθ

y₁' = 2.9 x tan29.35

y₁' = 1.63 m

Spectrum width will be

y₁' - y₁ = 1.63 - 0.846 = 0.784m

Thus, the spectrum width will be  0.784m

Learn more about diffraction.

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