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 $1.5$.
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
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,
Solving we get object distance $u\:=\:2.57\:cm$.