Chapter 33: Q. 48 (page 957)

For your science fair project you need to design a diffraction grating that will disperse the visible spectrum $(400 - 700 n m)$ over $30 . 0^{\circ}$ in first order.
$(a)$ How many lines per millimeter does your grating need?
$(b)$ What is the first-order diffraction angle of light from a sodium lamp $(λ = 589 n m)$ ?

Short Answer

Expert verified

Part $(a)$

$(a)$ The lines per mm needed for grating is $n = 1231 .$

Part $(b)$

$(b)$ The angle of light from sodium lamp is $θ_{1} = 46.5 .$

Step by step solution

Step: 1 Equating equation: (part a)

In diffraction grating,has

$d \sin (θ_{1}) = λ$

The small angle at each wavelength as

localid="1649135282502" $\begin{array}{l} d \sin (θ) = 400 nm \\ d \sin (θ + 30^{\circ}) = 700 nm \\ \frac{\sin (θ)}{\sin (θ + 30^{\circ})} = \frac{4}{7} \\ \sin (θ) = \frac{4}{7} \sin (θ + 30^{\circ}) \end{array}$

Step: 2 Finding angle: (part a)

By trignomentry,

$\begin{array}{l} \sin (a + b) = \sin a \cos b + \cos a \sin b \\ \frac{7}{4} \sin θ = (\sin θ \cos 30^{\circ}) + (\cos θ \sin 30^{\circ}) \\ \frac{7}{4} \sin θ = \frac{\sqrt{3}}{2} \sin θ + \frac{\cos θ}{2} \\ (\frac{7}{4} - \frac{\sqrt{3}}{2}) \sin θ = \frac{\cos θ}{2} \\ 0.884 \sin θ = \frac{\cos θ}{2} \\ \frac{\sin θ}{\cos θ} = \tan θ = \frac{1}{2 \times 0.884} \\ θ = \tan^{- 1} (\frac{1}{2 \times 0.884}) \\ θ = {29.5}^{\circ} . \end{array}$

Step: 3 Calculating numbers: (part a)

Calculating distance and numbers by

$d = \frac{400 nm}{\sin ({29.5}^{\circ})} d = 812.3 nm d = 812.3 \times 10^{- 6} mm . n = \frac{1}{812.3 \times 10^{- 6} mm} n = 1231 .$

Step: 4 Angle of light in sodium lamp: (part b)

The angle of fringe for any wavelength

$\begin{array}{l} \sin (θ_{1}) = \frac{λ}{d} \\ θ_{1} = \sin^{- 1} (\frac{λ}{d}) \\ θ_{1} = \sin^{- 1} (\frac{589 nm}{812.3 nm}) \\ θ_{1} = {46.5}^{\circ} . \end{array}$

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Short Answer

Step by step solution

Step: 1 Equating equation: (part a)

Step: 2 Finding angle: (part a)

Step: 3 Calculating numbers: (part a)

Step: 4 Angle of light in sodium lamp: (part b)

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For your science fair project you need to design a diffraction grating that will disperse the visible spectrum 400-700nmover30.0∘ in first order.a How many lines per millimeter does your grating need?bWhat is the first-order diffraction angle of light from a sodium lamp λ=589nm?

Short Answer

Step by step solution

Step: 1 Equating equation: (part a)

Step: 2 Finding angle: (part a)

Step: 3 Calculating numbers: (part a)

Step: 4 Angle of light in sodium lamp: (part b)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Waves Physics

Translational Dynamics

Quantum Physics

Measurements

Electrostatics

Magnetism

Study anywhere. Anytime. Across all devices.