Name: Refraction of Light at a Curved Surface
Uploaded: 2019-07-19
Duration: 6 min 22 s
Description: We have learned about the refraction of light at a plane surface. What happens when the light is incident on a curved surface separating two media? Watch thi...

Question

Derive lens makers formula

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

(symbols have their usual meanings) and discuss the dependence of focal-length on various factors.

Accepted Answer

The expression for refraction at a single spherical surface is given by the relation, $\frac{1}{f} = (n - 1) (\frac{1}{R_{1}} - \frac{1}{R_{2}})$ where $n_{2}$ is the refractive index of the denser medium, $n_{1}$ is the refractive index of the rarer medium, $R$ is the radius of curvature of the curved surface, $v$ is the image distance and $u$ is the object distance.

The refraction through a thin lens is as shown in the below figure:

Here, $n_{2}$ is the refractive index of the material of the lens, $n_{1}$ is the refractive index of the surrounding rarer medium.

$P_{1}$ and $P_{2}$ are the poles of the refracting surfaces.

$C_{1}$ and $C_{2}$ are the centres of curvatures as well as $R_{1} (= P_{1} C_{1})$ and $R_{2} (= P_{2} C_{2})$ are the radii of curvatures of the refracting surfaces.

$C$ is the optical centre and $t (= P_{1} P_{2})$ is the thickness of the lens.

Let, $v'$ be the image distance if the second surface were absent. Then the refraction at the first surface is given by the relation,

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

For the refraction at the second surface of the lens, $I'$ acts as a virtual object and $I$ is the real image formed of $I'$ .

The distance of $I$ from the pole $P_{2}$ is $v$ and the distance of $I'$ from the pole $P_{2}$ is $(v' - t)$ .

The refraction at the second surface from denser medium $n_{2}$ to rarer medium $n_{1}$ is given by the relation,

$\frac{n_{1}}{v} - \frac{n_{2}}{v' - t} = \frac{n_{1} - n_{2}}{R_{2}}$ .

As the lens is a thin lens $t < < v'$ and $t$ can be ignored. So, the above equation becomes,

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

Adding equation (a) and (b), we get,

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

Dividing both sides by $n_{1}$ , we get,

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

Putting $\frac{n_{2}}{n_{1}} = n$ which is also the refractive index of the material of the lens with respect to the surrounding medium, we get,

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

When the object is at infinity, the image will be formed at the principal focus of the lens, that is $u = \infty$ and $v = f$ . Thus, the above equation becomes,

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

or

$\frac{1}{f} = (n - 1) (\frac{1}{R_{1}} - \frac{1}{R_{2}})$ which is known as the Lens Maker's formula.

Derive lens makers formula

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

(symbols have their usual meanings) and discuss the dependence of focal-length on various factors.

Important Questions on Refraction of Light at Spherical Surfaces : Lenses

Learn and Prepare for any exam you want !

Derive lens makers formula 1f=(n−1)1R1−1R2 (symbols have their usual meanings) and discuss the dependence of focal-length on various factors.

Important Questions on Refraction of Light at Spherical Surfaces : Lenses

Learn and Prepare for any exam you want !

Derive lens makers formula

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

(symbols have their usual meanings) and discuss the dependence of focal-length on various factors.