Name: Total Internal Reflection From Prism
Uploaded: 2019-07-19
Duration: 1 min 56 s
Description: You must have seen reflectors on a bicycle. Why are they attached to the mudguards? What use do they have? Let’s watch this video on reflectors to find out.

Question

Derive the formula for the focal length of a thin lens with the help of refraction formula of spherical surface.

Accepted Answer

Let us consider the thin lens $2$ refracting surfaces having the radii of curvatures $R_{1}$ and $R_{2}$ respectively. Let the refractive indices of the surrounding medium and the lens material be $n_{1}$ and $n_{2}$ respectively.

Using the formula for refraction at a single spherical surface we can say that,
For the first surface,

$\frac{n_{2}}{v_{1}} - \frac{n_{1}}{u} = \frac{n_{2} - n_{1}}{R_{1}}$

Where radii of curvature is $R_{1}$ , and $v_{1}$ is the image distance from the first refracting surface, $u$ is the object distance.

For the second surface

$\frac{n_{1}}{v} - \frac{n_{2}}{v_{1}} = \frac{n_{2} - n_{1}}{R_{2}}$

Where radii of curvature is $R_{2}$ , and $v_{1}$ is the image distance from the first refracting surface, which is now acting as the object and $v$ is the final image distance.

Adding the two equations, $\frac{n_{1}}{v} - \frac{n_{1}}{u} = (n_{2} - n_{1}) (\frac{1}{R_{1}} - \frac{1}{R_{2}})$

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

Now, let $u = \infty$ and $v = f,$

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

But, it is also known that,

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

$∴$ It can be said that, $\frac{1}{f} = (n - 1) [\frac{1}{R_{1}} - \frac{1}{R_{2}}]$ Holds true, when $n = \frac{n_{2}}{n_{1}} .$ This is known as the Lens Makers’ Formula.

Derive the formula for the focal length of a thin lens with the help of refraction formula of spherical surface.

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