= Bessel lens displacement method
{c}
{title2=$f=(L^2-d^2)/(4L)$}
The <Bessel lens displacement method> measures a converging <thin lens>'s <focal length> by moving only the <optical lens> between a fixed object and screen. If their separation is $L>4f$, two settings give sharp <real images>: one enlarged, the other reduced. Read their separation $d$ on an <optical bench>, then calculate
$$
d=\sqrt{L^2-4Lf},\qquad f=\frac{L^2-d^2}{4L}.
$$
The two settings exchange object and image distances, so their signed <transverse magnifications> are reciprocals. This makes the experiment a visible test of <conjugate optical planes> as well as a <focal length> measurement.<MIT 2.71 and 2.710 Optics, Spring 2014 by Nicholas X. Fang/Lecture Notes on Geometrical Optics>[p. 13, Practice problem 2 and adjacent magnification relation]{ref}
For an original stipulated example, put the object at $x=0$, the screen at $x=L=45\,\mathrm{cm}$, and take an ideal <thin lens> with $f=10\,\mathrm{cm}$. Its two settings are $x_-=15\,\mathrm{cm}$ and $x_+=30\,\mathrm{cm}$, with $d=15\,\mathrm{cm}$. The <transverse magnifications> are $m_-=-2$ and $m_+=-1/2$: both <real images> are inverted. An object $0.50\,\mathrm{cm}$ tall would give signed image heights $-1.00\,\mathrm{cm}$ and $-0.25\,\mathrm{cm}$. Conversely, stipulated readings $L=45\,\mathrm{cm}$ and $d=15\,\mathrm{cm}$ give $f=(45^2-15^2)/(4\times45)=10\,\mathrm{cm}$. These are model calculations, not measured data or a manufacturer's calibration.
\Image[/bessel-lens-conjugate-position-ray-geometry.png]
{title=Two <thin lens> settings image the same fixed object onto the same fixed screen}
{description=Original <ray tracing> for the stipulated $L=45\,\mathrm{cm}$, $f=10\,\mathrm{cm}$ example. Each panel contains one physical <thin lens>; its alternative setting is dashed. The object and screen stay fixed, while object and image distances exchange. Both panels use the same scales, with heights enlarged for visibility.}
A fixed offset between a carrier's reference mark and an ideal <thin lens> plane cancels when subtracting the two carrier readings. That useful cancellation does not make a <thick lens> equivalent to one unknown optical centre: its two <principal planes> can be separated. The <Bessel lens displacement experiment> below makes this limitation explicit.
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