WAV-02

Snell's law

n1 sin θ1 = n2 sin θ2.

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RefractionSnell / Descartes

Governing equation

n_1\sin\theta_1=n_2\sin\theta_2

$\theta_1 \in \left[0, 89\right]$

where

n_1: Index 1 (—)
n_2: Index 2 (—)
\theta_1: Incidence (°)
\theta_2: Refraction angle (°)
\theta_c: Critical angle (°)

The figure, part by part

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①

Governing law — red

n_1\sin\theta_1=n_2\sin\theta_2

$\theta_1 \in \left[0, 89\right]$

$\theta_1 = 0$ $\theta_2 = 0$

Along the curve

②

Domain of the variable — blue

\theta_1_{\min} = 0,\quad \theta_1_{\max} = 89

$\theta_1 \in \left[0, 89\right]$

$\theta_1 = 0$ $\theta_1 = 0$

Along the curve

③

Response curve — green

n_1\sin\theta_1=n_2\sin\theta_2

$\theta_1 \in \left[0, 89\right]$

$\theta_1 = 0$ $\theta_2 = 0$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\theta_2}{\mathrm{d}\theta_1}

$\theta_1 \in \left[0, 89\right]$

$\theta_1 = 0.56$ $\theta_2^{\prime} = 0.667$

Along the curve

Lecture brief

Historical brief

d’Alembert’s wave equation, Snell, the thin-lens maker, Doppler and Bragg interference are the classical optics-and-sound toolkit. The lab is propagation, image and shift. This sheet (WAV-02 — Snell's law) is the form associated with Snell / Descartes. Working symbols: $n_1$ , $n_2$ , $\theta_1$ $\rightarrow$ $\theta_2$ , $\theta_c$ . Continuity of the wave's tangential k across an interface, or Fermat's least time, yields Snell's law of refraction.

Purpose

Purpose: compute

\theta_2

\theta_c

from

n_1

n_2

\theta_1

in Waves & optics via

n_1\sin\theta_1=n_2\sin\theta_2

n1 sin θ1 = n2 sin θ2. Use it when a real waves & optics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

n_1 = 1.000\,\mathrm{—}

n_2 = 1.500\,\mathrm{—}

\theta_1 = 30.000\,\mathrm{^{\circ}}

, the governing relation

n_1\sin\theta_1=n_2\sin\theta_2

yields

\theta_2 = 19.471\,\mathrm{^{\circ}}

\theta_c = 90.000\,\mathrm{^{\circ}}

. Planar interface, isotropic dielectrics. Returns critical flag via NaN-safe arcsin. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Index 1 n_11 —Index 2 n_21.5 —Incidence \theta_130 °

Outputs

Refraction angle \theta_219.471 °
Critical angle \theta_c90.000 °

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Narration of this film

Planar interface, isotropic dielectrics. Returns critical flag via NaN-safe arcsin.

Continuity of the wave's tangential k across an interface, or Fermat's least time, yields Snell's law of refraction.

Reading speed

Watch on YouTube