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Problem 6.92: A point sized  object is buried in a spherical glass pebble of diameter $6.0\:cm$. The object appears at a depth of $2.0\:cm$ when viewed normal to the surface of the pebble. Find out how deep it actually is. Glass has refractive index of $1.5$.

Solution:

In case of refraction at spherical surfaces, The object distance $u$, the image distance $v$,and the radius of curvature $R$ of the surface are related by

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

where $n_1$ is the index of refraction of the material where the object is located and $n_2$ is the index of refraction on the other side of the surface.

In the application of the above equation, we follow the following convention

• If the surface faced by the object is convex, $R$ is positive,

and if it is concave, $R$ is negative.

• Images on the object’s side of the surface are virtual, and

images on the opposite side are real.

In the given problem  the object distance $u$ muse be calculated.

The object and its image are on the same side of the refracting surface. So the  image is virtual and Image distance $v\:=\:-2.0 cm$.

Refractive index of the object medium is glass and surrounding medium is air, So $n_1\:=\:1.5$ and $n_2\:=\:1$

The object is inside the glass sphere and faces a concave refracting surface.So the radius of curvature is negative, and so $R\:=\:6.0\: cm$.

Now using these,

$\frac{1.5}{u}\:+\:\frac{1}{-2}\:=\:\frac{1\:-\:1.5}{-6}$

Solving we get object distance $u\:=\:2.57\:cm$.