Refraction at Curved Surface

Refraction at Curved Surface μ

Consider the point object O placed in the medium with refractive index equal to n₁ .

As n₁ sin i = n₂ sin r

For small aperture i , r → 0

⇒ n₁ i = n₂ r

In ΔOAC , i = α + γ

In  ΔIAC ,  γ =  r + β ⇒ r = γ – β

⇒  n₁(α + γ) = n₂( γ – β)

As aperture is small, α ≈ tan α , β ≈ tan β , for paraxial rays

γ ≈ tan γ

⇒ n₁(tan α + tanγ) = n₂(tan γ – tanβ)

$ \displaystyle n_1 (\frac{AM}{MO} + \frac{AM}{MC} ) = n_2 ( \frac{AM}{MC} – \frac{AM}{MI}) $

$ \displaystyle n_1 (\frac{1}{MO} + \frac{1}{MC} ) = n_2 ( \frac{1}{MC} – \frac{1}{MI}) $

As M is close to P ,

$ \displaystyle n_1 (\frac{1}{PO} + \frac{1}{PC} ) = n_2 ( \frac{1}{PC} – \frac{1}{PI}) $

Applying sign convention,

i.e., PO = -u , PI= v & PC = R

∴For a spherical surface,

$ \displaystyle \frac{n_2}{v}-\frac{n_1}{u} = \frac{n_2 -n_1}{R} $

The symbols should be carefully remembered as : n₂ → refractive index of the medium into which light rays are entering ; n₁ → refractive index of the medium from which light rays are coming. Care should also be taken while applying the sign convention to R.

Transverse Magnification for Refracting Spherical Surface

Transverse Magnification $\large m = \frac{h_I}{h_O} = \frac{-A’B’}{AB}$

$\large \frac{Sini}{Sinr} = \frac{\mu_2}{\mu_1}$

For small angle of incidence & refraction Sini = tani & Sinr = tanr

$\large \frac{tani}{tanr} = \frac{\mu_2}{\mu_1}$

In triangle ABP & A’B’P ,

$\large \frac{AB/PB}{A’B’/PB’} = \frac{\mu_2}{\mu_1}$

$\large -\frac{A’B’}{AB} = \frac{-PB’/\mu_2}{PB/\mu_1}$

Hence $\large m = \frac{v/\mu_2}{u/\mu_1} $

Solved Example : Shown in the figure is a spherical surface of radius of curvature R & R.I. (n = 1.5). Find the distance of the silvering of the plane surface so as to form an image at the pole due to a very distant object.

Solution:

Since light comes from very distant object

u = ∞

$ \displaystyle \frac{n}{v}-\frac{1}{\infty} = \frac{n-1}{R} $

$ \displaystyle v = \frac{nR}{n-1} $

$ \displaystyle v = \frac{1.5 R}{1.5-1} $

v = 3 R

Due to presence of the plane mirror, the image formed at I behaves as a virtual object for the plane mirror & a real image I’ is formed in front of the plane mirror, at the pole P.

⇒ x + x = v

⇒  x = v/2

x = 1.5R

Also Read :

Reflection of Light at a Plane Surface Reflection at Spherical Surface & Mirror Formula Lateral Magnification Refraction of Light , Laws of Refraction , Relation between real and apparent depth Refraction through Number of media Total internal reflection Refraction through a prism Angle of minimum deviation & Prism Formula Lens maker’s formula Combinations of Lenses Simple Magnifier Compound Microscope Telescope

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