Total Internal Reflection

Q. Light in a medium of index μ₁ meets one of μ₂ at i. How much turns back past the critical angle?

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The web of concepts

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

The playground

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%

The solution

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 Ruby

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

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

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

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?