With the help of ray diagram, show the formation of image of a point object due to refraction of light at a spherical surface separation two media of refractive indices $n_{1}$ and $n_{2} (n_{2} > n_{1})$ respectively. Using the diagram, derive refraction formula. Write the sign conventions used. What happens to the focal length of a convex lens when it is immersed in water.?

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Solution

When light ray enters a medium of different refractive index from another medium, it undergoes change of velocity. This is known as refraction of light. The change in velocity happens both in direction and magnitude of the light wave.
Assumption:
$i$ and $r$ are small
$⟹ x, y, z$ are small
$⟹ M N \approx$ length of arc $(l)$
$O M \approx u :$ Object distance
$I M \approx u :$ Image distance
$C m \approx R :$ Radius
Using snell's law,
$n_{1} sin i = n_{2} sin r$
$n_{1} sin (x + y) = n_{2} sin (y - z)$ ......(iii) using and (ii)
Using approximation, $sin (x + y) \approx x + y \approx sin x + sin y$
$sin (y - z) \approx y - z \approx sin y - sin z$
Using definition of sinusoid,
$sin x \approx \frac{M N}{u}$
$sin z \approx \frac{M N}{u}$
$sin y \approx \frac{M N}{R}$
Using above approximations, equation (iii) can be rewritten as:
$n_{1} (\frac{M N}{u} + \frac{M N}{R}) = n_{2} (\frac{M N}{R} - \frac{M N}{u})$
Rearranging the terms, we get:
$\frac{n_{1}}{u} + \frac{n_{2}}{v} = \frac{n_{2} - n_{1}}{R}$
Applying sign convention, $u \to - u$
$v \to + v$
$R \to + R$
$\frac{n_{2}}{v} - \frac{n_{1}}{u} = \frac{n_{2} - n_{1}}{R}$

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Similar questions

Q. Draw a ray diagram showing the formation of the image by a point object on the principal axis of a spherical convex surface separating two media of refractive indices

n_{1}

and

n_{2}

, when a point source is kept in rarer medium of refractive index

n_{1}

. Derive the relation between object and image distance in terms of refractive index of the medium and radius of curvature of the surface.
Hence obtain the expression for lens-maker's formula in the case of thin convex lens.

Q. Define refraction of light waves.
Draw a ray diagram for refraction at a spherical separating two media. 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)

Q. (a) Derive the mathematical relation between refractive indices

n_{1}

and

n_{2}

of two media and radius of curvature R for refraction at a convex spherical surface. Consider the object to be a point since lying on the principle axis in rarer medium of refractive index

n_{1}

and a real image formed in the denser medium of refractive index

n_{2}

. Hence, derive Lens Maker's formula.

(b) Light from a point source in air falls on a convex spherical glass surface of refractive index 1.5 and radius of curvature 20 cm. The distance of light source from the glass surface is 100 cm. At what position is the image formed ?

Q. (a) A point object 'O' is kept in a medium of refractive index n in front of a convex spherical surface of radius of curvature R which separate the second medium of refractive index

n_{2}

from the first one as shown in the figure.
Draw the ray diagram showing the image formation and deduce the relationship between the object distance and the image distance in terms of

n_{1}, n_{2}

and

R

.
(b) When the image formed above acts as a virtual object for a concave spherical surface separating the medium

n 2

and

n 1 (n 2 > n 1)

draw this ray diagram and write the similar (similar to (a)) relation. Hence obtain the expression for the lens maker's formula

In the adjoining figure, SS is a spherical surface separating two media of refractive indices n₁ and n₂ where n₁ > n₂. C is the centre of curvature of the spherical surface. An observer, keeping his eye beyond C in the medium of refractive index n₂ views the refracted image of an object AB placed as shown in the medium of refractive index n₁. The image will be:

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