Question

For refraction at a spherical surface derive the relation

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

in object distance

(u)

, image distance

(v)

refractive index of media

(n_{1}, n_{2})

. and radius of curvature

(R)

. Draw the necessary ray diagram.

Accepted Answer

The following figure shows the formation of image $I$ of an object $O$ on the principal axis of a convex surface with the centre of curvature $C$ .

Since the aperture is small the length of $N M$ will be taken to be nearly equal to the length of the perpendicular from the point $N$ on the principal axis.

For small angles

$\tan (∠ N O M) \approx ∠ N O M = \frac{M N}{O M}$

$\tan (∠ N C M) \approx ∠ N C M = \frac{M N}{M C}$

$\tan (∠ N I M) \approx ∠ N I M = \frac{M N}{M I}$

Here $i = ∠ N O M + ∠ N C M = \frac{M N}{O M} + \frac{M N}{M C}$

and $r = ∠ N C M - ∠ N I M = \frac{M N}{M C} - \frac{M N}{M I}$

According to the snell's law

$n_{1} \sin (i) = n_{2} \sin (r)$

For small angles, it can be written as

$n_{1} i = n_{2} r$

$n_{1} (\frac{M N}{O M} + \frac{M N}{M C}) = n_{2} (\frac{M N}{M C} - \frac{M N}{M I})$

$\Rightarrow n_{1} (\frac{1}{O M} + \frac{1}{M C}) = n_{2} (\frac{1}{M C} - \frac{1}{M I})$

$\Rightarrow \frac{n_{1}}{O M} + \frac{n_{1}}{M C} = \frac{n_{2}}{M C} - \frac{n_{2}}{M I}$

$\Rightarrow \frac{n_{1}}{O M} + \frac{n_{2}}{M I} = \frac{n_{2}}{M C} - \frac{n_{1}}{M C}$

$\Rightarrow \frac{n_{1}}{O M} + \frac{n_{2}}{M I} = \frac{n_{2} - n_{1}}{M C}$

According to the Cartesian sign convention

$O M =$ Object distance $= - u$

$M I =$ Image distance $= + v$

$M C =$ Radius $= + R$

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

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

For refraction at a spherical surface derive the relation $\frac{n_{2}}{v} - \frac{n_{1}}{u} = \frac{n_{2} - n_{1}}{R}$ in object distance $(u)$ , image distance $(v)$ refractive index of media $(n_{1}, n_{2})$ . and radius of curvature $(R)$ . Draw the necessary ray diagram.

Important Questions on Solved Physics Paper-1

Learn and Prepare for any exam you want !

For refraction at a spherical surface derive the relation n2v-n1u=n2-n1R in object distance u, image distance v refractive index of media n1,n2. and radius of curvature R. Draw the necessary ray diagram.

Important Questions on Solved Physics Paper-1

Learn and Prepare for any exam you want !

For refraction at a spherical surface derive the relation $\frac{n_{2}}{v} - \frac{n_{1}}{u} = \frac{n_{2} - n_{1}}{R}$ in object distance $(u)$ , image distance $(v)$ refractive index of media $(n_{1}, n_{2})$ . and radius of curvature $(R)$ . Draw the necessary ray diagram.