Chapter 33: Q. 20 (page 955)

For what slit-width-to-wavelength ratio does the first minimum of a single-slit diffraction pattern appear at (a) $30 °$ , (b) $60 °$ , and (c) $90 °$ $?$

Short Answer

Expert verified

a. Slit width to wavelength for $30 °$ is $2$ .

b. Slit width to wavelength for $60^{°}$ is $1.15$ .

c. Slit width to wavelength for $90^{°}$ is $1$ .

Step by step solution

Formula for first order grating

The angles at which dark fringes is,

asin (θ_{p}) = pλ

Slit width to wavelength ratio is,

\frac{a}{λ} = \frac{1}{\sin (θ_{1})}

Calculation foraλ (part a)

(a).

For $θ = 30^{°}$ :

The first order grating is,

$\frac{a}{λ} = \frac{1}{\sin (θ_{1})}$

Substitutelocalid="1649242325679" $θ$ value,

we get,

$\sin (30^{°}) = \frac{1}{2}$

$\frac{a}{λ} = \frac{1}{(1 / 2)}$

$\frac{a}{λ} = 2$

Calculation foraλ(part b)

(b).

For $θ = 60^{°}$ :

The first order grating is,

$\frac{a}{λ} = \frac{1}{\sin (θ_{1})}$

Substitutelocalid="1649242354752" $θ$ value,

We get,

$\sin (60^{°}) = \frac{\sqrt{3}}{2}$

$\frac{a}{λ} = \frac{1}{(\sqrt{3} / 2)}$

$\frac{a}{λ} = \frac{2}{\sqrt{3}}$

$\approx 1.15$

Calculation foraλ(part c)

(c).

For $θ = 90^{°}$ :

The first order grating is,

$\frac{a}{λ} = \frac{1}{\sin (θ_{1})}$

$\sin (90^{°}) = 1$

$\frac{a}{λ} = 1$

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For what slit-width-to-wavelength ratio does the first minimum of a single-slit diffraction pattern appear at (a) $30 °$ , (b) $60 °$ , and (c) $90 °$ $?$

Short Answer

Step by step solution

Formula for first order grating

Calculation foraλ (part a)

Calculation foraλ(part b)

Calculation foraλ(part c)

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For what slit-width-to-wavelength ratio does the first minimum of a single-slit diffraction pattern appear at (a) 30°, (b) 60°, and (c) 90°?

Short Answer

Step by step solution

Formula for first order grating

Calculation foraλ (part a)

Calculation foraλ(part b)

Calculation foraλ(part c)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Circular Motion and Gravitation

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Study anywhere. Anytime. Across all devices.

For what slit-width-to-wavelength ratio does the first minimum of a single-slit diffraction pattern appear at (a) $30 °$ , (b) $60 °$ , and (c) $90 °$ $?$