 # 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 an optical center O. Let F  be the principle focus and f be the focal length. An object AB is held perpendicular to the principal axis at a distance beyond the focal length of the lens. A real, inverted magnified image A’B’  is formed as shown in the figure. From the given figure, we notice that △ABO and △A’B’O are similar.

Therefore,

$\frac{A’B’}{AB}=\frac{OB’}{OB}$     (1)

Similarly, △A’B’F and △OCF are similar, hence

$\frac{A’B’}{OC}=\frac{FB’}{OF}$

But, $OC=AB$

Hence,

$\frac{A’B’}{AB}=\frac{FB’}{OF}$        (2)

Equating eq (1) and (2), we get

$\frac{OB’}{OB}=\frac{FB’}{OF}$=$\frac{OB’-OF}{OF}$

Substituting the sign convention, we get

OB=-uOB’=v and OF=f

$\frac{v}{-u}=\frac{v-f}{f}$

$vf=-uv+uf\,or\,uv=uf-vf$

Dividing both the sides by uvf, we get

$\frac{uv}{uvf}=\frac{uf}{uvf}-\frac{vf}{uvf}$

$\Rightarrow \frac{1}{f}=\frac{1}{v}-\frac{1}{u}$

The above equation is known as the 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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