Question

A narrow beam of light containing red (660 nm) and blue (470 nm) wavelengths travels from air through a 1.00-cm-thick flat piece of crown glass and back to air again. The beam strikes at a 30.0° incident angle.
(a) At what angles do the two colors emerge?
(b) By what distance are the red and blue separated when they emerge?

Answer 1

Answer

given,

wavelength of red light = 660 nm

wavelength of blue light = 470 nm

thickness = 1 cm = 0.01 m

angle of incident = 30°

using Snell's law

n₁ sin θ₁ = n₂ sin θ₂

refractive index for red and blue color for crown glass

n_r = 1.512     n_b = 1.524

now,

incident ray is red

$sin (\theta_{ir})=\dfrac{1\times sin(30^0)}{1.512}$

when incident ray is blue

$sin (\theta_{ib})=\dfrac{1\times sin(30^0)}{1.524}$

so,

$(\theta_e)_r=sin^{-1}(\dfrac{1.512 sin (\theta_{ir})}{1})$

$(\theta_e)_r=sin^{-1}(\dfrac{1.512\times \dfrac{1\times sin(30^0)}{1.512}}{1})$

on solving

$(\theta_e)_r = 30^0$

similarly for blue ray the angle of emerge is 30°

b)

now, refracting angle of blue and red ray

$sin (\theta_{ir})=\dfrac{1\times sin(30^0)}{1.512}$

$\theta_{ir}=19.316^0$

for blue ray

$sin (\theta_{ib})=\dfrac{1\times sin(30^0)}{1.524}$

$\theta_{ib}=19.158^0$

now,

 d₁ = 1 x tan(19.316°) = 0.3505 m

 d₂ = 1 x tan (19.158°) = 0.3474 m

now, the distance is separated by

  Δ d = d₁ - d₂

  Δ d = 0.3505 - 0.3474

 Δ d =0.0031 cm

Answer 2

(a) The angle of emergence of the two colors is 30°

(b) The separation between the red and blue light is 0.0031 cm

Calculating the separation and angle of emergence:

Given that the narrow beam of light contains red and blue light with wavelengths λ = 660nm and λ' = 470nm respectively.

The thickness of the crown glass is, t = 1cm = 0.01m

(a) The angle of emergence of the beam will be the same as the angle of incidence which is 30°, only the red and blue light will emerge parallel to each other but at the same angle of 30°

(b) The refractive index of crown glass for red light is, n = 1.512

and for blue it is, n' = 1.524

so according to Snell's law:

sin(i) / sin(r) = 1.512  ...... for red light

where i is the angle of incidence and r is the angle of refraction for red light

r = sin⁻¹(sin30°/1.512)

r = 19.316°

similarly for blue light:

sin(i) / sin(b) = 1.524

where b is the angle of refraction for blue light

b = sin⁻¹(sin30°/1.524)

b = 19.158°

now, the lateral displacement of red light is:

d = t × tan(19.316°) = 0.3505 cm

d' = t × tan (19.158°) = 0.3474 cm

so the separation between the red and blue light is:

Δd = d - d' = 0.3505 - 0.3474

Δd = 0.0031 cm

Learn more about Snell's Law:

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