Internal Reflection

solved Reflection · Refraction

Q. Light in water $μ_{1} = 1.33$ meets air $μ_{2} = 1$ at $i$ . What happens to the light that is not refracted?

Inputs

angle of incidence

i

60.00°

Outputs

angle of reflection

r_{l}

60.00°

angle of refraction

r_{r}

—

no refracted ray?yes

How it's solved

Solved with Reflection, Refraction

Given

$μ_{1} = 1.33$ from “water”
$μ_{2} = 1$ from “air”
$i$ yours to vary

Find $r_{l}$ , $r_{r}$ , whether there is no refracted ray.

Working

law of reflection $i = r_{l}$
$60 ° = r_{l}$
$r_{l} = 60.00°$
snells law when grazing exit $μ_{1} \cdot \sin c = μ_{2} \cdot \sin 90 °$
$1.33 \cdot \sin c = 1 \cdot \sin 90 °$
$c = 48.75°$
$r_{r} = —$
no refracted ray? $i > c$ yes

Conclusion

It turns back into the water, and past this angle that is all of the light: the surface is a mirror. Slide down and the other ray returns, because some of it turns back at every angle.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :internal_reflection do4  describes "Light in water meets air at %{angle}. %{reflected}?"5  given :angle6  asks reflected: "What happens to the light that is not refracted"7end

solution.rb

1require_relative "../questions/internal_reflection"2require_relative "../science/ray_optics_and_optical_instruments/reflection"3require_relative "../science/ray_optics_and_optical_instruments/refraction"4require_relative "../science/ray_optics_and_optical_instruments/refractive_media"56solve :internal_reflection, using: [ Reflection, Refraction ] do7  take angle: :i8  deduce_from "water", mu_1: REFRACTIVE_MEDIA.fetch("water")9  deduce_from "air", mu_2: REFRACTIVE_MEDIA.fetch("air")10  take reflected: :r_l11  find_value :reflected12  find_value :r_r13  find_whether :no_refracted_ray?1415  conclusion "It turns back into the water, and past this angle that is all of the light: the surface is a mirror. Slide down and the other ray returns, because some of it turns back at every angle."16end

playground.rb

1require_relative "../solutions/internal_reflection"2require_relative "../science/ray_optics_and_optical_instruments/refractive_media"3require_relative "../pictures/light"45play_with :internal_reflection do6  title "Internal Reflection"78  input :i, default: 60.deg910  output :r_l11  output :r_r12  output :no_refracted_ray?, alarm: true1314  draw Light::Picture, rising: true do15    media "water", "air"16    ray "incident",  arriving_at: :i,  angle: true17    ray "reflected", leaving_at:  :r_l, angle: true18    ray "refracted", crossing_at: :r_r, angle: true, unless: :no_refracted_ray?19    note "all of it turns back", when: :no_refracted_ray?20  end21end

The laws it drew on

quantity	written	units	within	from
angle to the normal	$θ$	—	0 … 1.571	Incidence
angle of incidence	$i$	—	0 … 1.571	Incidence
angle of reflection	$r_{l}$	—	0 … 1.571	Reflection
refractive index	$μ$	—	—	Refraction
angle of refraction	$r_{r}$	—	0 … 1.571	Refraction
refractive index of first medium	$μ_{1}$	—	—	Refraction
refractive index of second medium	$μ_{2}$	—	—	Refraction
critical angle	$c$	—	0 … 1.571	Refraction

i \equiv θ_{i} (before)

angle of incidence

r_{l} \equiv θ_{i} (after)

angle of reflection

r_{r} \equiv θ_{i} (after)

angle of refraction

μ_{1} \equiv μ_{b}

refractive index of first medium

μ_{2} \equiv μ_{a}

refractive index of second medium

i = r_{l}

law of reflection

μ_{1} \cdot \sin i = μ_{2} \cdot \sin r_{r}

snells law

μ_{1} \cdot \sin c = μ_{2} \cdot \sin 90 °

snells law when grazing exit

r_{r} = 90 °

grazing exit?

i > c

no refracted ray?

r_{r} < i

bends toward the normal?

r_{r} > i

bends away from the normal?