Thin lens formula

$f=$

[cm]

$x_o=$

[cm]

$y_o=$

[cm]

$x_i=$

[cm]  $D=$ [m-1]

$y_i=$

[cm] $m=$

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Using the three special rays associated with at thin lens, the magnification of the lens and an equation called the thin lens formula can be derived. Consider the two red triangles in the image above. The magnification is $m=\frac{y_i}{y_o}$. Because the red triangles have the same angles,

$$m=\frac{x_i}{x_o} = \frac{y_i}{y_o}=-\frac{x_i-f}{f}.$$

To derive the thin lens formula from this expression for the magnification, give both sides a common denominator,

$$\frac{x_if}{x_of}=-\frac{x_ix_o-x_of}{x_of}.$$

Divide both sides by $x_i$,

$$\frac{x_if}{x_ox_if}=-\frac{x_ix_o-x_of}{x_ox_if}.$$

Simplify,

$$\frac{1}{x_o}=-\frac{1}{f}+\frac{1}{x_i}.$$

Rearrange,


$\hspace{0.5cm}\Large -\frac{1}{x_o}+\frac{1}{x_i}=\frac{1}{f}.\hspace{0.5cm}$

This is known as the thin lens formula. Since the object is to the left of the lens, $x_o$ is a negative number. This derivation assumed that the lens is at position $x=0$. This is an inconvenient if there are more lenses in an optical system. If the position of the lens is $x_{\text{lens}}$, the thin lens formula is,

$$\frac{1}{x_{\text{lens}}-x_o}+\frac{1}{x_i -x_{\text{lens}}}=\frac{1}{f}.$$

Notice in the simulation above that as you move the object, the image moves too.