Question

Derive the formula lens maker with full explanation !

Hint :- See the attachment .

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Answer 1

$\\ \underbrace{ \large \sf{Concept :- }}$

★ It is a relation between the focal length of a lens to the refractive index of its material and the radii of curvature of its two surfaces. It is called because it is used by lens manufacturers to make lenses of particular power from the glass of given refractive index.

$\\ \underbrace{ \large \sf{Assumption :- }}$

★ The lens is thin so that the distance measured from the poles of the two surfaces of the lens can be taken to be equal to the distance measured from the optical center.

★ The aperture of the lens is small.

$\\ \underbrace{ \large \sf{Derivation :- }}$

Consider a thin convex lens made of a material of absolute refractive index $\sf n_2$ , placed in a rarer medium of absolute refractive index $\sf n_1$

Also, $\sf \: n = \dfrac{n_2}{n_1}$ be the refractive index of the material of the lens with respect to the medium surrounding it.

R1 and R2 are the radii of curvature of surfaces $\sf XP_1 Y \: and \: XP_2 Y$ respectively.

For refraction at surface $\sf XP_1 Y$ :

O is the object and $\sf l^1$ it's real image.

Using formula ,

$\\ \: \: \: \: \: \sf \: \: \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_1 - n_2}{R_2}$

We get ,

$\\ \: \: \: \: \: \sf \: \: \frac{n_2}{v ^{1} } - \frac{n_1}{u} = \frac{n_1 - n_2}{R_2} \dots \: (1)$

For refraction at surface $\sf XP_2Y$ :

$\sf l^1$ is the virtual object and l is it's real image which is final image.

Using formula,

$\\ \: \: \: \: \: \sf \: \: \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_1 - n_2}{R_2}$

We get,

$\\ \: \: \: \: \: \sf \: \: \frac{n_2}{v} - \frac{n_1}{ {v}^{1} } = \frac{n_1 - n_2}{R_2} \dots(2)$

Now , Adding equation (1) and (2),we get

$\\ \: \: \: \: \: \sf \: \: \frac{n_2}{v} - \frac{n_1}{ u } = (n_2 - n_1) \bigg[ \frac{1}{R_1} - \frac{1}{R_2} \bigg]$

Dividing both side by $\sf n_1$, we get

$\\ \: \: \: \: \: \sf \: \: \longmapsto \: \: \frac{1}{v} - \frac{1}{ u } = \bigg( \frac{n_2 - n_1} {n_1 } \bigg) \bigg[ \frac{1}{R_1} - \frac{1}{R_2} \bigg]$

$\\ \: \: \: \: \: \sf \: \: \longmapsto \: \: \frac{1}{v} - \frac{1}{ u } = \bigg( \frac{n_2 } {n_1 } - 1\bigg) \bigg[ \frac{1}{R_1} - \frac{1}{R_2} \bigg]$

$\\ \: \: \: \: \: \sf \: \: \longmapsto \: \: \frac{1}{v} - \frac{1}{ u } = ( {n } - {1 } ) \bigg[ \frac{1}{R_1} - \frac{1}{R_2} \bigg]$

As we know, $\sf \dfrac{1}{f} = \sf\dfrac{1}{v} - \dfrac{1}{ u }$

Therefore,

$\\ \: \: \: \: \overline{\underline{ {\boxed{\: \sf \: \frac{1}{f} \: = \: \: \frac{1}{v} - \frac{1}{ u } = ( {n } - {1 } ) \bigg[ \frac{1}{R_1} - \frac{1}{R_2} \bigg]}} }}$

This is called lens makers formula.