Total Internal Reflection

solved Reflection · TotalInternalReflection

Q. Light in a medium of index $μ_{1}$ meets one of $μ_{2}$ at $i$ . How much turns back past the critical angle?

Inputs

angle of incidence

i

50.00°

refractive index of first medium

μ_{1}

1.5glass

refractive index of second medium

μ_{2}

1air

Outputs

no refracted ray?yes

reflectance

r

100.0%

How it's solved

Solved with Reflection, TotalInternalReflection

Given

$i$ , $μ_{1}$ , $μ_{2}$ yours to vary

Find whether there is no refracted ray, $r$ .

Working

snells law when grazing exit $μ_{1} \cdot \sin c = μ_{2} \cdot \sin 90 °$
$1.5 \cdot \sin c = 1 \cdot \sin 90 °$
$c = 41.81°$
everything reflects $r = 1$
$r = 100.0%$
no refracted ray? $i > c$ yes

Conclusion

All of it turns back. Not most of it: the share is exactly one, and it is one because there is nowhere else for the light to go.

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 :total_internal_reflection do4  describes "Light in a medium of index %{first_index} meets one of %{second_index} at %{angle}. %{reflected}?"5  given :angle6  given :first_index7  given :second_index8  asks reflected: "How much turns back past the critical angle"9end

solution.rb

1require_relative "../questions/total_internal_reflection"2require_relative "../science/ray_optics_and_optical_instruments/reflection"3require_relative "../science/ray_optics_and_optical_instruments/total_internal_reflection"45solve :total_internal_reflection, using: [ Reflection, TotalInternalReflection ] do6  take angle: :i, first_index: :mu_1, second_index: :mu_27  find_whether :no_refracted_ray?8  take reflected: :r9  find_value :reflected1011  conclusion "All of it turns back. Not most of it: the share is exactly one, and it is one because there is nowhere else for the light to go."12end

playground.rb

1require_relative "../solutions/total_internal_reflection"2require_relative "../science/ray_optics_and_optical_instruments/refractive_media"3require_relative "../pictures/light"45INDEX = 1.0..4.067play_with :total_internal_reflection do8  title "Total Internal Reflection"910  input :i, default: 50.deg11  input :mu_1, INDEX, default: 1.5, marks: REFRACTIVE_MEDIA12  input :mu_2, INDEX, default: 1.0, marks: REFRACTIVE_MEDIA1314  output :no_refracted_ray?, alarm: true15  output :r, in: "%"1617  draw Light::Picture, rising: true do18    media :mu_1, :mu_219    ray "incident",  arriving_at: :i, angle: true20    ray "reflected", leaving_at: :i, weight: :r21    ray "refracted", crossing_at: :r_r, angle: true,22        weight: :tau, unless: :no_refracted_ray?23    mark "critical", arriving_at: :c24    note "trapped — every ray turns back", when: :no_refracted_ray?25  end26end

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
reflectance s polarised	$r_{s}$	—	—	Reflectance
reflectance p polarised	$r_{p}$	—	—	Reflectance
reflectance	$r$	—	—	Reflectance
transmittance	$τ$	—	—	Reflectance

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_{s} = {(\frac{μ_{1} \cdot \cos i - μ_{2} \cdot \cos r_{r}}{μ_{1} \cdot \cos i + μ_{2} \cdot \cos r_{r}})}^{2}

s polarised

r_{p} = {(\frac{μ_{1} \cdot \cos r_{r} - μ_{2} \cdot \cos i}{μ_{1} \cdot \cos r_{r} + μ_{2} \cdot \cos i})}^{2}

p polarised

r = \frac{r_{s} + r_{p}}{2}

unpolarised

r + τ = 1

reflected and transmitted share the light

r = 1

everything reflectswhen no refracted ray

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?