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

Deduce the relation between

n, u, v, R

for refraction at a spherical surface, where the symbols have their usual meaning.

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 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

$O M = - u, M I = + v, M C = + R$

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

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

Deduce the relation between $n, u, v, R$ for refraction at a spherical surface, where the symbols have their usual meaning.

Important Questions on Ray Optics and Optical Instruments

Learn and Prepare for any exam you want !

Deduce the relation between n, u,v, R for refraction at a spherical surface, where the symbols have their usual meaning.

Important Questions on Ray Optics and Optical Instruments

Learn and Prepare for any exam you want !

Deduce the relation between $n, u, v, R$ for refraction at a spherical surface, where the symbols have their usual meaning.