Reflectance — phryzby

Q. Light meets a surface between indices μ₁ and μ₂ at i. How much of it gets through?

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Q. Light meets a surface between indices $μ_{1}$ and $μ_{2}$ at $i$ . How much of it gets through?

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Inputs

angle of incidence

i

50.00°

refractive index of first medium

μ_{1}

1air

refractive index of second medium

μ_{2}

1.5glass

Outputs

angle of refraction

r_{r}

30.71°

no refracted ray?no

reflectance s polarised

r_{s}

11.2%

reflectance p polarised

r_{p}

0.3%

reflectance

r

5.8%

transmittance

τ

94.2%

The solution

Solved with Reflection, Reflectance

Given

$i$ , $μ_{1}$ , $μ_{2}$ yours to vary

Find $r_{r}$ , whether there is no refracted ray, $r_{s}$ , $r_{p}$ , $r$ , $τ$ .

Working

snells law $μ_{1} \cdot \sin i = μ_{2} \cdot \sin r_{r}$
$1 \cdot \sin 50 ° = 1.5 \cdot \sin r_{r}$
$r_{r} = 30.71°$
s polarised $r_{s} = {(\frac{μ_{1} \cdot \cos i - μ_{2} \cdot \cos r_{r}}{μ_{1} \cdot \cos i + μ_{2} \cdot \cos r_{r}})}^{2}$
$r_{s} = {(\frac{1 \cdot \cos 50 ° - 1.5 \cdot \cos 30.71 °}{1 \cdot \cos 50 ° + 1.5 \cdot \cos 30.71 °})}^{2}$
$r_{s} = 11.2%$
p polarised $r_{p} = {(\frac{μ_{1} \cdot \cos r_{r} - μ_{2} \cdot \cos i}{μ_{1} \cdot \cos r_{r} + μ_{2} \cdot \cos i})}^{2}$
$r_{p} = {(\frac{1 \cdot \cos 30.71 ° - 1.5 \cdot \cos 50 °}{1 \cdot \cos 30.71 ° + 1.5 \cdot \cos 50 °})}^{2}$
$r_{p} = 0.3%$
unpolarised $r = \frac{r_{s} + r_{p}}{2}$
$r = \frac{0.112048357 + 0.00327753215}{2}$
$r = 5.8%$
reflected and transmitted share the light $r + τ = 1$
$0.0576629447 + τ = 1$
$τ = 94.2%$
no refracted ray? $i > c$ no

Conclusion

It depends on the angle, and on polarisation: the two polarisations are reflected in quite different measure, and at one angle one of them is not reflected at all.

The Ruby

1require_relative "../physics"23question :reflectance do4  describes "Light meets a surface between indices %{first_index} and %{second_index} at %{angle}. %{transmitted}?"5  given :angle6  given :first_index7  given :second_index8  asks transmitted: "How much of it gets through"9end

1require_relative "../questions/reflectance"2require_relative "../science/ray_optics_and_optical_instruments/reflection"3require_relative "../science/ray_optics_and_optical_instruments/reflectance"45solve :reflectance, using: [ Reflection, Reflectance ] do6  take angle: :i, first_index: :mu_1, second_index: :mu_27  find_value :r_r8  find_whether :no_refracted_ray?9  find_value :r_s10  find_value :r_p11  find_value :r12  take transmitted: :tau13  find_value :transmitted1415  conclusion "It depends on the angle, and on polarisation: the two polarisations are reflected in quite different measure, and at one angle one of them is not reflected at all."16end

1require_relative "../solutions/reflectance"2require_relative "../science/ray_optics_and_optical_instruments/refractive_media"3require_relative "../pictures/light"45INDEX = 1.0..4.067play_with :reflectance do8  title "Reflectance"910  input :i, default: 50.deg11  input :mu_1, INDEX, default: 1.0, marks: REFRACTIVE_MEDIA12  input :mu_2, INDEX, default: 1.5, marks: REFRACTIVE_MEDIA1314  output :r_r15  output :no_refracted_ray?, alarm: true16  output :r_s, in: "%"17  output :r_p, in: "%"18  output :r,  in: "%"19  output :tau, in: "%"2021  draw Light::Picture do22    media :mu_1, :mu_223    ray "incident",  arriving_at: :i, angle: true24    ray "reflected", leaving_at: :i, weight: :r25    ray "refracted", crossing_at: :r_r, angle: true,26        weight: :tau, unless: :no_refracted_ray?27    mark "critical", arriving_at: :c28    note "no refracted ray — Fresnel has nothing to say", when: :no_refracted_ray?29  end30end

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_{r} = 90 °

grazing exit?

i > c

no refracted ray?

r_{r} < i

bends toward the normal?

r_{r} > i

bends away from the normal?