Into Air — phryzby

solved Reflection · Refraction

Q. Light leaves a medium of index $μ_{1}$ for air $μ_{2} = 1$ at 30° $i = 30 °$ . Does every medium have its own angle?

How it's solved

Solved with Reflection, Refraction

Given

$i = 30 °$
$μ_{2} = 1$ from “air”
$μ_{1}$ yours to vary

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

Working

snells law when grazing exit $μ_{1} \cdot \sin c = μ_{2} \cdot \sin 90 °$
$1.33 \cdot \sin c = 1 \cdot \sin 90 °$
$c = 48.75°$
snells law $μ_{1} \cdot \sin i = μ_{2} \cdot \sin r_{r}$
$1.33 \cdot \sin 30 ° = 1 \cdot \sin r_{r}$
$r_{r} = 41.68°$
no refracted ray? $i > c$ no

Conclusion

Every medium has one of its own, and the denser the medium the sooner it stops letting light out.

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 :into_air do4  describes "Light leaves a medium of index %{index} for air at %{angle}. Does every medium have %{critical_angle}?"5  given :index6  given angle: "30 deg"7  asks critical_angle: "its own angle"8end

solution.rb

1require_relative "../questions/into_air"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 :into_air, using: [ Reflection, Refraction ] do7  take index: :mu_1, angle: :i8  deduce_from "air", mu_2: REFRACTIVE_MEDIA.fetch("air")9  take critical_angle: :c10  find_value :critical_angle11  find_value :r_r12  find_whether :no_refracted_ray?1314  conclusion "Every medium has one of its own, and the denser the medium the sooner it stops letting light out."15end

playground.rb

1require_relative "../solutions/into_air"2require_relative "../science/ray_optics_and_optical_instruments/refractive_media"3require_relative "../pictures/light"45play_with :into_air do6  title "Into Air"78  input :from, REFRACTIVE_MEDIA, default: "water"910  input(:mu_1) { from }1112  output :c13  output :r_r14  output :no_refracted_ray?, alarm: true1516  draw Light::Picture, rising: true do17    media :from, "air"18    ray "incident",  arriving_at: :i, angle: true19    ray "reflected", leaving_at:  :r_l20    ray "refracted", crossing_at: :r_r, angle: true, unless: :no_refracted_ray?21    mark "critical", arriving_at: :c22    note "all of it turns back", when: :no_refracted_ray?23  end24end

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?