WAV-03

Thin-lens equation

1/f = 1/do + 1/di.

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Geometrical opticsGauss thin-lens

Governing equation

\dfrac{1}{f}=\dfrac{1}{d_o}+\dfrac{1}{d_i},\quad m=-\dfrac{d_i}{d_o}

$d_o \in \left[1, 200\right]$

where

f: Focal length (cm)
d_o: Object distance (cm)
d_i: Image distance (cm)
m: Magnification (—)

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

\dfrac{1}{f}=\dfrac{1}{d_o}+\dfrac{1}{d_i},\quad m=-\dfrac{d_i}{d_o}

$d_o \in \left[1, 200\right]$

$d_o = 1$ $d_i = -1.111$

Along the curve

②

Domain of the variable — blue

d_o_{\min} = 1,\quad d_o_{\max} = 200

$d_o \in \left[1, 200\right]$

$d_o = 1$ $d_o = 1$

Along the curve

③

Response curve — green

\dfrac{1}{f}=\dfrac{1}{d_o}+\dfrac{1}{d_i},\quad m=-\dfrac{d_i}{d_o}

$d_o \in \left[1, 200\right]$

$d_o = 1$ $d_i = -1.111$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}d_i}{\mathrm{d}d_o}

$d_o \in \left[1, 200\right]$

$d_o = 2.252$ $d_i^{\prime} = -1.71$

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-03 — Thin-lens equation) is the form associated with Gauss thin-lens. Working symbols: $f$ , $d_o$ $\rightarrow$ $d_i$ , $m$ . Paraxial rays through a thin lens of focal length f conjugate an object at do to an image at di, Gauss's formula.

Purpose

Purpose: compute

d_i

m

from

f

d_o

in Waves & optics via

\dfrac{1}{f}=\dfrac{1}{d_o}+\dfrac{1}{d_i},\quad m=-\dfrac{d_i}{d_o}

1/f = 1/do + 1/di. 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

f = 10.000\,\mathrm{cm}

d_o = 25.000\,\mathrm{cm}

, the governing relation

\dfrac{1}{f}=\dfrac{1}{d_o}+\dfrac{1}{d_i},\quad m=-\dfrac{d_i}{d_o}

yields

d_i = 16.667\,\mathrm{cm}

m = -0.667\,\mathrm{—}

. Thin lens in air, Cartesian sign: real image positive di. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Focal length f10 cmObject distance d_o25 cm

Outputs

Image distance d_i16.667 cm
Magnification m-0.667 —

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Narration of this film

Thin lens in air, Cartesian sign: real image positive di.

Paraxial rays through a thin lens of focal length f conjugate an object at do to an image at di, Gauss's formula.

Reading speed

Watch on YouTube