Derivation of Lens Formula

What is Lens formula?

In optics, the relationship between the distance of an image (v), the distance of an object (u) and the focal length (f) of the lens is given by the formula known as Lens formula. Lens formula is applicable for convex as well as concave lenses. These lenses have negligible thickness. The formula is as follows:

\(\frac{1}{v}-\frac{1}{u}=\frac{1}{f}\)

Lens formula derivation

Consider a convex lens with object AB kept on the principal axis. Two rays are considered such that one ray is parallel to the principal axis and after reflection, it passes through the focus. The second ray is towards the optical centre such that it passes undeviated.

A’ is the point where the two rays intersect and also the image formed by point A. Point B image is obtained on the principal axis as the point B is on the principal axis.

As the object is perpendicular to the principal axis, even the object is perpendicular to the principal axis. To get the location of the image formed by point B, we need to draw a perpendicular from point A’ to the principal axis. Following are the things obtained after drawing the figure:

\(\frac{AB}{{A}'{B}’}=\frac{BO}{O{B}’}\) (from similar ΔABO and ΔA’B’O) (equ. 1)

\(\frac{PO}{OF}=\frac{{A}'{B}’}{F{B}’}\) (from similar ΔPOF and ΔFB’A’)

\(∴ \frac{AB}{OF}=\frac{{A}'{B}’}{F{B}’}\) (from figure PO = AB)

\(\frac{AB}{{A}'{B}’}=\frac{OF}{F{B}’}\) (equ. 2)

\(\frac{BO}{O{B}’}=\frac{OF}{F{B}’}\) (from equ. 1 and equ. 2)

\(∴ \frac{BO}{O{B}’}=\frac{OF}{O{B}’-OF}\)

\(\frac{-u}{+v}=\frac{+f}{v-f}\) (substituting optical distance values)

\(∴ -uv+uf=fv\)

\(-\frac{1}{f}+\frac{1}{v}=\frac{1}{u}\) (dividing by u,v and f on both the sides)

\(\frac{1}{v}=\frac{1}{u}+\frac{1}{f}\)

\(\frac{1}{f}=\frac{1}{v}-\frac{1}{u}\)

Therefore, this is known as Lens formula.

This was the derivation of Lens formula. Stay tuned with BYJU’S and learn various other Physics related topics.

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