Critical Angle — phryzby

Q. Light in water meets air at i. At what angle does it stop getting out?

What the draft establishes

Fills in once a draft is checked.

The web of concepts

given read from the words equation asked

The equations it used

None yet.

Check an answer

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

The playground

Inputs

angle of incidence

i

30.00°

Outputs

critical angle

c

48.75°

angle of refraction

r_{r}

41.68°

no refracted ray?no

The solution

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 Ruby

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

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

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

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?