# Lens equation

The lens equation , also known as the imaging equation , indicates the relationship between the object distance , image distance and focal length for optical imaging by means of a lens . It is: ${\ displaystyle g}$ ${\ displaystyle b}$ ${\ displaystyle f}$

${\ displaystyle {\ frac {1} {b}} + {\ frac {1} {g}} = {\ frac {1} {f}}}$.

## Geometrical derivation of the lens equation for a thin lens

Designations on the thin lens

By applying the theorem of rays of geometry to the central ray and the optical axis which crosses with it in the center of the lens, the relationship for the imaging scale A is obtained

${\ displaystyle A = {\ frac {B} {G}} = {\ frac {b} {g}},}$

where G is the size of the object to be imaged and B that of the image. The object distance or object distance , that is the distance between the main plane of the lens and the object, is denoted here by g and the image distance, ie the distance between the main plane and the image, by b .

If the theorem of rays is applied to the focal point ray on the image side and the optical axis which crosses with it at the focal point , one obtains

${\ displaystyle {\ frac {B} {G}} = {\ frac {bf} {f}}.}$

In this case, f is the focal length of the lens on the image side.

The left sides of the 1st and 2nd equations are the same, so the right sides must also be the same, this results

${\ displaystyle {\ frac {b} {g}} = {\ frac {bf} {f}} = {\ frac {b} {f}} - 1.}$

Adding 1 and dividing by b results in further

${\ displaystyle {\ frac {1} {b}} + {\ frac {1} {g}} = {\ frac {1} {f}}}$

This relationship is called the lens equation or imaging equation . An equivalent formulation is the Newtonian mapping equation .

The lens equation is also valid for thick lenses and systems made up of several lenses, the principal planes of which do not generally coincide. Then g denotes the distance between the object and the main plane on the object side and b the distance between the image and the main plane on the image side.

## Equations for desired magnification

If one looks for the image and object distances to an enlargement (ratio of image and object size), then the following applies ${\ displaystyle A}$

${\ displaystyle A = {\ frac {b} {g}}}$,
${\ displaystyle b = (A + 1) \ cdot f}$,
${\ displaystyle g = \ left ({\ frac {1} {A}} + 1 \ right) \ cdot f.}$

For example, to get four times the magnification, one has and . ${\ displaystyle b = 5f}$${\ displaystyle g = 1 {\ tfrac {1} {4}} f}$

## Equations with unknown image distance

The following equations can be used if the image distance b - for example with cameras - is not known.

In order to map a desired field of view G over the distance g onto the image sensor size B , a focal length

${\ displaystyle f = {\ frac {B} {G + B}} \ cdot g}$

needed. For a given focal length f , the field of view G at the distance g is determined with the sensor size B.

${\ displaystyle G = \ left ({\ frac {g} {f}} - 1 \ right) \ cdot B.}$

## Refractive Power and Vergence

The reciprocal value of the focal length is the refractive power  and is equal to the sum of the reciprocal values ​​of the image and object distance , as the lens equation shows in the following form derived above: ${\ displaystyle D}$

${\ displaystyle {\ frac {1} {b}} + {\ frac {1} {g}} = {\ frac {1} {f}} = D}$

The SI unit of refractive power is called the diopter . ${\ displaystyle m ^ {- 1} = 1 / m}$

Reciprocal values ​​of particular widths / lengths are called vergences in geometric optics . Just like the refractive power of a single lens, that of thin neighboring lenses can also be expressed approximately simply as the sum of vergences - the refractive powers of the individual lenses:

${\ displaystyle {\ frac {1} {f_ {1}}} + {\ frac {1} {f_ {2}}} = {\ frac {1} {f}}}$

Likewise, the refractive power of those who wear glasses is approximately the sum of that of the eye and that of the glasses.

## literature

• Douglas C. Giancoli: Physics. Volume 10, 3rd edition, Pearson Education, Munich 2006, ISBN 978-3-8273-7157-7 .

## Individual evidence

1. ^ Alfred Recknagel : Elementarphysik (Electrics Optics), PEBlank-Verlag, Weimar, 1953, p. 265.