Refraction: Water to Air

solved Refraction

Q. Light in water $μ_{1} = 1.33$ meets air $μ_{2} = 1$ at $i$ . Which way does it bend coming out?

Inputs

angle of incidence

i

45.00°

Outputs

angle of refraction

r_{r}

70.13°

no refracted ray?no

How it's solved

Solved with Refraction

Given

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

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

Working

snells law $μ_{1} \cdot \sin i = μ_{2} \cdot \sin r_{r}$
$1.33 \cdot \sin 45 ° = 1 \cdot \sin r_{r}$
$r_{r} = 70.13°$
snells law when grazing exit $μ_{1} \cdot \sin c = μ_{2} \cdot \sin 90 °$
$1.33 \cdot \sin c = 1 \cdot \sin 90 °$
$c = 48.75°$
no refracted ray? $i > c$ no

Conclusion

It bends away from the normal instead, and far enough round there is no way across at all.

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 :water_to_air do4  describes "Light in water meets air at %{angle}. %{refracted}?"5  given :angle6  asks refracted: "Which way does it bend coming out"7end

solution.rb

1require_relative "../questions/water_to_air"2require_relative "../science/ray_optics_and_optical_instruments/refraction"3require_relative "../science/ray_optics_and_optical_instruments/refractive_media"45solve :water_to_air, using: Refraction do6  take angle: :i7  deduce_from "water", mu_1: REFRACTIVE_MEDIA.fetch("water")8  deduce_from "air", mu_2: REFRACTIVE_MEDIA.fetch("air")9  take refracted: :r_r10  find_value :refracted11  find_whether :no_refracted_ray?1213  conclusion "It bends away from the normal instead, and far enough round there is no way across at all."14end

playground.rb

1require_relative "../solutions/water_to_air"2require_relative "../science/ray_optics_and_optical_instruments/refractive_media"3require_relative "../pictures/light"45play_with :water_to_air do6  title "Refraction: Water to Air"78  input :i, default: 45.deg910  output :r_r11  output :no_refracted_ray?, alarm: true1213  draw Light::Picture, rising: true do14    media "water", "air"15    ray "incident",  arriving_at: :i,  angle: true, extended: true16    ray "refracted", crossing_at: :r_r, angle: true, unless: :no_refracted_ray?17    note "no refracted ray", when: :no_refracted_ray?18  end19end

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
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_{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

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?