WAV-09

Snell's law

n1 sin θ1 = n2 sin θ2. The conserved transverse wavevector.

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WavesSnell

Governing equation

n_1\sin\theta_1=n_2\sin\theta_2

$\theta_1 \in \left[0, 85\right]$

where

n_1: Index 1 (—)
n_2: Index 2 (—)
\theta_1: Angle of incidence (°)
\theta_2: Refracted angle (°)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

n_1\sin\theta_1=n_2\sin\theta_2

$\theta_1 \in \left[0, 85\right]$

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

Along the curve

②

Domain of the variable — blue

\theta_1_{\min} = 0,\quad \theta_1_{\max} = 85

$\theta_1 \in \left[0, 85\right]$

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

Along the curve

③

Response curve — green

n_1\sin\theta_1=n_2\sin\theta_2

$\theta_1 \in \left[0, 85\right]$

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

Along the curve

④

Rate of change — amber

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

$\theta_1 \in \left[0, 85\right]$

$\theta_1 = 0.535$ $\theta_2^{\prime} = 0.667$

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-09 — Snell's law) is the form associated with Snell. Working symbols: $n_1$ , $n_2$ , $\theta_1$ $\rightarrow$ $\theta_2$ . Fermat's least time, or matching of phase along the interface.

Purpose

Purpose: compute

\theta_2

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. The conserved transverse wavevector. 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 = 19.47\,\mathrm{^{\circ}}

. Two media, a kinked ray, an interface. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Index 1 n_11 —Index 2 n_21.5 —Angle of incidence \theta_130 °

Outputs

Refracted angle \theta_219.47 °

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Narration of this film

Two media, a kinked ray, an interface.

Fermat's least time, or matching of phase along the interface.

Reading speed

Watch on YouTube