Question

One face of a rectangular plate 6 cm thick is silvered. An object held 8 cm in front of the first face, forms an image 12 cm behind the silvered face. The refractive index of the glass is
(A) $1.2$
(B) $1.6$
(C) $1.5$
(D) $1.3$

Answer 1

Hint : In an isolated silver surface, the object distance from the surface will be equal to the image distance from the surface. It is different for a thick glass because the depth (position) as “seen” by the light (apparent depth) of the mirror changes.

Formula used: In this solution we will be using the following formula;
$\Rightarrow n = \dfrac{d}{{d'}}$ where $n$ is the refractive index of transparent material, $d$ is the real depth of an object placed in or behind such an object, and $d'$ is the apparent depth of such object.

Complete step by step answer
If an object is placed behind a thick glass, the depth of the object as seen through the glass is different from the depth (which should be the thickness of the glass) as seen through air. In general, the glass appears thinner than it really is.
In our question, one of the faces is silvered, and an object is placed 8 cm from the un-silvered face. When the light ray from the object travels through the glass to the silver surface, the depth travelled by the light waves before reflection will be the apparent depth. Hence, the distance of the object from the silvered surface is actually,
 $\Rightarrow s = d' + 8 $ where $ d' $ is the apparent depth of the glass.
Now in a flat reflecting surface, the image distance from mirror is equal to object distance from mirror, and, the image distance is actually,
 $\Rightarrow y = 12 + (6 - d') $
Hence,
 $\Rightarrow d' + 8 = 12 + 6 - d' $
 $ \Rightarrow d' = 5cm $
Now refractive index is given as
 $\Rightarrow n = \dfrac{d}{{d'}} $ where $ d $ is the real depth which here is 6 cm.
Hence, $ n = \dfrac{6}{5} = 1.2 $
The correct option is A.

Note
The reason why the depth appears less deep is because of refraction at the boundary. When the light waves from the object, coming up into our eyes gets to the boundary, they refract away from the normal. In entering your eyes, the light appears to have been coming straight from a depth lower than the actual depth.