Relative Refractive Index

solved Reflection · RelativeIndex

Q. Light meets a medium of index $μ_{2}$ at 30° $i = 30 °$ , and the two media's relative index is $μ_{21}$ . Do both numbers matter, or only what they come to?

Inputs

relative refractive index

μ_{21}

0.75

Outputs

critical angle

c

48.59°

angle of refraction

r_{r}

41.81°

no refracted ray?no

How it's solved

Solved with Reflection, RelativeIndex

Given

$i = 30 °$
$μ_{21}$ , $μ_{2}$ yours to vary

Assumed

$μ_{1} = 1$

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

Working

snells law when grazing exit $μ_{1} \cdot \sin c = μ_{2} \cdot \sin 90 °$
$1 \cdot \sin c = 0.75 \cdot \sin 90 °$
$c = 48.59°$
snells law $μ_{1} \cdot \sin i = μ_{2} \cdot \sin r_{r}$
$1 \cdot \sin 30 ° = 0.75 \cdot \sin r_{r}$
$r_{r} = 41.81°$
no refracted ray? $i > c$ no

Conclusion

Only what they come to. One file says so — mu_21 is mu_2 over mu_1 — and here the first medium is called 1, which leaves the second one standing for the pair.

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 :relative_refractive_index do4  describes "Light meets a medium of index %{index} at %{angle}, and the two media's relative index is " \5          "%{relative_index}. Do both numbers matter, or only %{refracted}?"6  given :relative_index7  given :index8  given angle: "30 deg"9  asks refracted: "what they come to"10end

solution.rb

1require_relative "../questions/relative_refractive_index"2require_relative "../science/ray_optics_and_optical_instruments/reflection"3require_relative "../science/ray_optics_and_optical_instruments/relative_index"45solve :relative_refractive_index, using: [ Reflection, RelativeIndex ] do6  take relative_index: :mu_21, index: :mu_2, angle: :i7  assume mu_1: 1.08  find_value :c9  take refracted: :r_r10  find_value :refracted11  find_whether :no_refracted_ray?1213  conclusion "Only what they come to. One file says so — mu_21 is mu_2 over mu_1 — and here the first medium is called 1, which leaves the second one standing for the pair."14end

playground.rb

1require_relative "../solutions/relative_refractive_index"2require_relative "../pictures/light"34RATIO = 0.25..4.056play_with :relative_refractive_index do7  title "Relative Refractive Index"89  input :mu_21, RATIO, default: 0.75, marks: { "critical" => 0.5, "alike" => 1.0 }1011  input(:mu_2) { mu_21 }1213  output :c14  output :r_r15  output :no_refracted_ray?, alarm: true1617  draw Light::Picture do18    surface across: :mu_2119    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
relative refractive index	$μ_{21}$	—	—	RelativeIndex

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

μ_{21} = \frac{μ_{2}}{μ_{1}}

relative index

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?