Question

Using the relation for refraction at a single spherical refracting surface, derive the lens maker&rsquo;s formula.

Accepted Answer

Lens makers formula gives the relation between focal length, refractive index and radius of curvature of two surfaces of lens.

To find the expression of lens makers formula,

Let's take a convex lens of glass whose refractive index is $n_{2}$ . Let ${}^{'}O^{'}$ is the point object placed on the principal axis and ${}^{'}u^{'}$ is the distance of the object from $P_{1}$ .

Let a point object $O$ be placed on principal axis at a distance is $u$ from $P_{1}$ . A ray $O A$ Traveling in rarer medium (refractive index $n_{1}$ ) is incident on first surface of lens. It is refracted towards normal at the first surface of the lens.

For refraction at the first surface
$\frac{n_{2}}{v^{'}} - \frac{n_{1}}{u} = \frac{n_{2} - n_{1}}{R_{1}} . . (i)$

For refraction at the second surface
$\frac{n_{1}}{v} - \frac{n_{2}}{v^{'} - v} = \frac{n_{1} - n_{2}}{R_{2}}$

If the lens is thin $v < < v^{'},$

so $\frac{n_{1}}{v} - \frac{n_{2}}{v^{'}} = \frac{n_{2} - n_{1}}{R_{2}} . . . (ii)$

Adding $(i)$ and $(ii)$ , We get

$\frac{n_{1}}{v} - \frac{n_{1}}{u} = (\frac{n_{2} - n_{1}}{R_{1}} - \frac{n_{2} - n_{1}}{R_{2}})$

$\frac{n_{1}}{v} - \frac{n_{1}}{u} = (n_{2} - n_{1}) (\frac{1}{R_{1}} - \frac{1}{R_{2}})$

When the object is at infinity, image will be formed at focus

So, $\frac{1}{f} - \frac{1}{\infty} = (n - 1) (\frac{1}{R_{1}} - \frac{1}{R_{2}})$

$\frac{1}{f} = (n - 1) (\frac{1}{R_{1}} - \frac{1}{R_{2}}) . . . (iii)$

Equation (iii) is known as "Lens makers formula".

Using the relation for refraction at a single spherical refracting surface, derive the lens maker’s formula.

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