Chapter 36: Q19P (page 1110)

How far from grains of red sand must you be to position yourself just at the limit of resolving the grains if your pupil diameter is $1.5 mm$ , the grains are spherical with radius $50 μ m$ , and the light from the grains has wavelength $650 n m$ ? (a) If the grains were blue and the light from them had wavelength $400 n m$ , would the answer to (b) be larger or smaller?

Short Answer

Expert verified

The distance between the grains and the eye is $19.0 c m$ .
Length L will be larger

Step by step solution

Introduction

Wavelength is the distance between identical points (adjacent crests) in the adjacent cycles of a waveform signal propagated in space or along a wire.

Calculation

The pupil’s diameter $' d' = 1.55 mm$

The radius of sphere $R = 50 μ m$ and wavelength $' λ' = 650 mm$

Since $θ_{R} = 1.22 \frac{λ}{d} = θ_{i}$

$d = 1.22 \times \frac{650 \times 10^{- 9} m}{1.5 \times 10^{- 3} m} θ_{R} = 5.286 \times 10^{- 4} r a d$

Diameter of the sphere is

$D = 2 (R) = 2 (50 μ m) = 100 μ m$

(a) Determine the distance between grains and the eye

The distance between the grains and the eye is

$\ begingathered L = \frac{D d}{1.22 λ} = \frac{D}{θ_{R}} = \frac{100 \times 10^{- 4} m}{5.2867 \times 10^{- 4} r a d} = 18.9 c m L = 19.0 c m$

Hence, the distance is $19.0 c m$ .

(b) Determine the wavelength is larger & smaller

If wavelength $λ = 400 m m$

Since $L = \frac{D d}{1.22 λ}$

$L α \frac{1}{λ}$

Since $λ = 400 n m$ is smaller than $650 n m$ ,

Hence, length L will be larger larger.

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Short Answer

Step by step solution

Introduction

Calculation

(a) Determine the distance between grains and the eye

(b) Determine the wavelength is larger & smaller

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