Method of Bessel

The method of Bessel is a focusing method of experimental determination of the focal distance of a convergent lens.

Principle

One considers a convergent thin lens of focal distance f', center O, hearths F' image and object F.

Are D, the distance between the object has (on the optical axis) and the screen (where one visualizes a' image), and D, the distance between the two positions of the lens which ensure the conjugation of has and A', (i.e. the clearness of the image on the screen). One can deduce the value from the focal distance f' by the formula:

$f\text{'}=\backslash frac\{D^2-d^2\}\{4.D\}$

Explanation

Formulas of conjugation

The Formules of conjugation of Descartes give a relation between the positions on the optical axis of an object has and of its A' image compared to the optical center O. They are expressed with algebraic distances.

That is to say has a point of the optical axis and A' its image by the lens:

$\backslash \; frac\; \{1\}\; \{\backslash \; overline\; \{OA\text{'}\}\}\; -\; \backslash \; frac\; \{1\}\; \{\backslash \; overline\; \{OA\}\}\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; overline\; \{OF\text{'}\}\}\; =\; \backslash \; frac\; \{1\}\; \{f\text{'}\}$

It is supposed that A' is real (i.e. projectable on a screen): $\backslash overline\{OA\text{'}\}>0$.

It is necessary for that has is placed on the optical axis at a distance .

Formation of a real image starting from a real object

One fixes $D=\; \backslash \; overline\; \{AA\text{'}\}$, the distance between the (A) object and the screen (A') and one poses $x=\; \backslash \; overline\; \{OA\}$ and $y=\; \backslash \; overline\; \{OA\text{'}\}$, therefore

$D=\; \backslash \; overline\; \{AA\text{'}\}\; =\; \backslash \; overline\; \{OA\text{'}\}\; -\; \backslash \; overline\; \{OA\}\; \backslash \; Rightarrow\; y=D+x$.

The Relations of conjugation are rewritten:

$\backslash \; frac\; \{1\}\; \{there\}\; =\; \backslash \; frac\; \{1\}\; \{f\text{'}\}\; +\; \backslash \; frac\; \{1\}\; \{X\}\; \backslash \; Rightarrow\; y=\; \backslash \; frac\; \{f\text{'}.x\}\; \{f\text{'}+x\}$.

The combination of the two preceding equations gives well an equation of the second order in X: $x^2+D.x+f\text{'}.D=0$

This equation has real solution only if $\backslash \; Delta\; =\; D^2-4.f^\; \{“\}.\; D=D.\; \backslash \; left\; (D-4.f”\; \backslash \; right)\; \backslash \; geq\; 0$

As, it is necessary as $D\; \backslash \; geq\; D\_\; \{min\}\; =\; 4.f\text{'}$

Respective positions of the image and the object

If $D>\; D\_\; \{min\}$, then $\backslash \; Delta>\; 0$: there are two real solutions (there then exist two positions of the lens which allow to combine C and A').

The solutions are: $x\_\; \{\backslash \; pm\}\; =\; \backslash \; frac\; \{-\; D\; \backslash \; pm\; \backslash \; sqrt\; \{D^2-4.f^\; \{\text{'}\}.\; D\}\}\; \{2\}$. Also, these two possible positions of the object are distant from $\backslash \; left|\; x\_\; \{+\}\; -\; x\_\; \{-\}\; \backslash \; right|=\; \backslash \; sqrt\; \{D^2-4.f^\; \{\text{'}\}.\; D\}$.

This distance is also the distance between the two positions of the lens which ensure the conjugation of has and A': $d=\; \backslash \; left|\; x\_\; \{+\}\; -\; x\_\; \{-\}\; \backslash \; right|=\; \backslash \; sqrt\; \{D^2-4.f^\; \{\text{'}\}.\; D\}$.

While raising squared, one finds the formula: $f\text{'}=\backslash frac\{D^2-d^2\}\{4.D\}$

Notice

The Méthode of Silbermann seems a particular case of the method of Bessel, that where the position of the lens is single (either d=0).

See too

Related articles

• Friedrich Wilhelm Bessel

• optical Focométrie
• Lens
• thin Lens

External bonds

• a video explanatory on the method of Bessel

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