Relative Refractive Index

Q. Light meets a medium of index μ₂ at 30°, and the two media's relative index is μ₂₁. Do both numbers matter, or only what they come to?

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The web of concepts

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The equations it used

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Check an answer

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?

The playground

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

The solution

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 Ruby

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

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

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

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?