Name: Thin Lenses in Contact
Uploaded: 2019-07-19
Duration: 4 min 6 s
Description: If two or more lenses are in contact, we can treat them as one lens by calculating the effective focal length of the lenses. Watch and learn the formula for...

Question

Derive an expression for refraction at a single(convex) spherical surface, i.e. a relation between $u, v, R, n_{1}$ (rarer medium) and $n_{2}$ (denser medium), where the terms have their usual meaning.

Accepted Answer

The figure drawn above shows the formation of image I of an object O on the principal axis of a convex surface with centre of curvature C, and radius of curvature R. The rays are incident from a medium of refractive index $n_{1}$ to another medium of refractive index $n_{2}$ . Let the aperture of the surface be small compared to other distances involved. Hence, MN will be taken to be nearly equal to the length of the perpendicular from the point N on the principal axis. Let the aperture of the surface be small compared to other distances involved. Hence, NM will be taken to be nearly equal to the length of the perpendicular from the point N on the principal axis. Let another medium is of refractive index $n_{2}$ . For small angles:
$\tan$ $∠ NOM = \frac{MN}{OM}$ and
$\tan ∠ NCM = \frac{MN}{MC}$
$\tan ∠ NIM = \frac{MN}{MI}$

In $△ NOC$ :

$i = ∠ NOM + ∠ NCM = \frac{MN}{OM} + \frac{MN}{MC}$ ......(i)

Similarly:

$r = \frac{MN}{MC} - \frac{MN}{MI}$ .....(ii)

From(i) and(ii), using snell's law:

$n_{1} \sin i = n_{2} \sin r$
For small angles:

$n_{1} i = n_{2} r$ ........(iii)

Substituting values from (i)and(ii) in (iii) we get:

$\frac{- n_{1}}{u} + \frac{n_{1}}{R} = \frac{n_{2}}{R} - \frac{n_{2}}{v}$

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

(iv) is the required expression.

Derive an expression for refraction at a single(convex) spherical surface, i.e. a relation between $u, v, R, n_{1}$ (rarer medium) and $n_{2}$ (denser medium), where the terms have their usual meaning.

Important Questions on Refraction of Light at Spherical Surfaces : Lenses

Learn and Prepare for any exam you want !

Derive an expression for refraction at a single(convex) spherical surface, i.e. a relation between u, v, R, n1(rarer medium) and n2(denser medium), where the terms have their usual meaning.

Important Questions on Refraction of Light at Spherical Surfaces : Lenses

Learn and Prepare for any exam you want !

Derive an expression for refraction at a single(convex) spherical surface, i.e. a relation between $u, v, R, n_{1}$ (rarer medium) and $n_{2}$ (denser medium), where the terms have their usual meaning.