Refractive Index

solved Reflection · Refraction

Q. Light passes from a medium of index $μ_{1}$ into one of $μ_{2}$ at 30° $i = 30 °$ . Where does the bending come from?

Inputs

refractive index of first medium

μ_{1}

1.33water

refractive index of second medium

μ_{2}

1air

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

$i = 30 °$
$μ_{1}$ , $μ_{2}$ 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

From one number each. Every medium has its own, the law asks for both, and everything the last five chapters held fixed was a pair of them.

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 :refractive_index do4  describes "Light passes from a medium of index %{first_index} into one of %{second_index} at %{angle}. Where does %{refracted} come from?"5  given :first_index6  given :second_index7  given angle: "30 deg"8  asks refracted: "the bending"9end

solution.rb

1require_relative "../questions/refractive_index"2require_relative "../science/ray_optics_and_optical_instruments/reflection"3require_relative "../science/ray_optics_and_optical_instruments/refraction"45solve :refractive_index, using: [ Reflection, Refraction ] do6  take first_index: :mu_1, second_index: :mu_2, angle: :i7  find_value :c8  take refracted: :r_r9  find_value :refracted10  find_whether :no_refracted_ray?1112  conclusion "From one number each. Every medium has its own, the law asks for both, and everything the last five chapters held fixed was a pair of them."13end

playground.rb

1require_relative "../solutions/refractive_index"2require_relative "../science/ray_optics_and_optical_instruments/refractive_media"3require_relative "../pictures/light"45INDEX = 1.0..4.067play_with :refractive_index do8  title "Refractive Index"910  input :mu_1, INDEX, default: 1.33, marks: REFRACTIVE_MEDIA11  input :mu_2, INDEX, default: 1.0, marks: REFRACTIVE_MEDIA1213  output :c14  output :r_r15  output :no_refracted_ray?, alarm: true1617  draw Light::Picture, rising: true do18    media :mu_1, :mu_219    ray "incident",  arriving_at: :i, angle: true20    ray "reflected", leaving_at:  :r_l21    ray "refracted", crossing_at: :r_r, angle: true, unless: :no_refracted_ray?22    mark "critical", arriving_at: :c23    note "all of it turns back", when: :no_refracted_ray?24  end25end

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?