Name: Normal Incidence of Light Ray
Uploaded: 2019-07-19
Duration: 4 min 35 s
Description: When a light ray falls on a glass slab, the way it is refracted depends on the ray that is incident on the glass slab. Watch the video to learn more about it.

Question

Proceeding from Fermat's principle, derive the refraction formula for paraxial rays on a spherical boundary surface of radius

R

, between media with refractive indices

n_{1}

and

n_{2}

.

Accepted Answer

Given: $C P = R$ in the above diagram. For $O'$ to be the image, the optical paths of all the rays $O A O'$ must be equal up-to terms of leading order in $h$ . Thus,

$n_{1} O A + n_{2} A O' = constant . . . (1)$

But $O P = | s |, O' P = |s'|$ and so,

$∆ O A M, O A^{2} = A M^{2} + O M^{2} \Rightarrow O A = \sqrt{A M^{2} + O M^{2}} O A = \sqrt{h^{2} + (| s | + δ)^{2}} ≅ | s | + δ + \frac{h^{2}}{2 | s |} ∵ O M = | s | + δ$

Now,

$∆ O' A M, O' A^{2} = A M^{2} + O' M^{2} \Rightarrow O' A \sqrt{A M^{2} + O' M^{2}} O' A = \sqrt{h^{2} + {(|s'| - δ)}^{2}} ≅ |s'| - δ + \frac{h^{2}}{2 |s'|} ∵ O' M = |s'| - δ$

(neglecting the products $h^{2} δ$ ).

Then, put the values of $O' A & O A$ in equation $(1)$ ,

$\Rightarrow n_{1} | s | + n_{2} |s'| + n_{1} δ - n_{2} δ + \frac{h^{2}}{2} (\frac{n_{1}}{| s |} + \frac{n_{2}}{|s'|}) = constant . . . (2)$

Now,

$∆ A M C, A C^{2} = A M^{2} + M C^{2} \Rightarrow (R - δ)^{2} + h^{2} = R^{2} ∵ M C = R - δ$

or $h^{2} = 2 R δ$ or $δ = \frac{h^{2}}{2 R} . . . (3)$

Here, $R = C P$ .

Hence, from equations $(2) & (3)$ ,

$n_{1} | s | + n_{2} |s'| + \frac{h^{2}}{2} \{\frac{n_{1} - n_{2}}{R} + \frac{n_{1}}{| s |} + \frac{n_{2}}{|s'|}\} = constant$

We know that from equation $(1)$ , $n_{1} |s| + n_{2} |s'| = constant$ . Since this must hold for all $h$ , we have

$\Rightarrow \frac{n_{2}}{|s'|} + \frac{n_{1}}{| s |} + \frac{n_{1} - n_{2}}{R} = 0 \Rightarrow \frac{n_{2}}{|s'|} + \frac{n_{1}}{| s |} = + \frac{n_{2} - n_{1}}{R}$

From our sign convention, $s' > 0, s < 0$ , so we get

$\frac{n_{2}}{s'} - \frac{n_{1}}{s} = \frac{n_{2} - n_{1}}{R}$

Proceeding from Fermat's principle, derive the refraction formula for paraxial rays on a spherical boundary surface of radius $R$ , between media with refractive indices $n_{1}$ and $n_{2}$ .

Important Questions on OPTICS

Learn and Prepare for any exam you want !

Proceeding from Fermat's principle, derive the refraction formula for paraxial rays on a spherical boundary surface of radius R, between media with refractive indices n1 and n2.

Important Questions on OPTICS

Learn and Prepare for any exam you want !

Proceeding from Fermat's principle, derive the refraction formula for paraxial rays on a spherical boundary surface of radius $R$ , between media with refractive indices $n_{1}$ and $n_{2}$ .