Critical Angle — phryzby

solved Reflection · Refraction

Q. Light in water $μ_{1} = 1.33$ meets air $μ_{2} = 1$ at $i$ . At what angle does it stop getting out?

Inputs

angle of incidence

i

30.00°

Outputs

critical angle

c

48.75°

angle of refraction

r_{r}

41.68°

no refracted ray?no

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 $c$ , $r_{r}$ , 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

It is Snell's law with the refracted ray lying flat along the surface. Past it, no angle of refraction satisfies the law, so none leaves.

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 :critical_angle do4  describes "Light in water meets air at %{angle}. %{critical_angle}?"5  given :angle6  asks critical_angle: "At what angle does it stop getting out"7end

solution.rb

1require_relative "../questions/critical_angle"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 :critical_angle, 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 critical_angle: :c11  find_value :critical_angle12  find_value :r_r13  find_whether :no_refracted_ray?1415  conclusion "It is Snell's law with the refracted ray lying flat along the surface. Past it, no " \16             "angle of refraction satisfies the law, so none leaves."17end

playground.rb

1require_relative "../solutions/critical_angle"2require_relative "../science/ray_optics_and_optical_instruments/refractive_media"3require_relative "../pictures/light"45play_with :critical_angle do6  title "Critical Angle"78  input :i, default: 30.deg910  output :c11  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_l18    ray "refracted", crossing_at: :r_r, angle: true, unless: :no_refracted_ray?19    mark "critical", arriving_at: :c20    note "all of it turns back", when: :no_refracted_ray?21  end22end

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?