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An object is placed 20 cm in front of a block of glass 10 cm thick and its farther surface is silvered. The image is formed $ 23.2\,cm $ behind the silvered face. Determine the refractive index of the glass.

Last updated date: 20th Jun 2024
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The refractive index of a medium is defined as the ratio of the apparent depth of the object to the actual depth of the object. The apparent depth of the object can be determined by subtracting the thickness of the glass from the distance of the image from the mirror.
Formula used:
 $ \mu = \dfrac{{{\text{Real depth}}}}{{{\text{Apparent depth}}}} $ where $ \mu $ is the refractive index of the block of glass

Complete step by step answer
We know that the real depth of the object is 20 cm since it is placed 20 cm from the silvered surface which acts as a mirror. Now let the apparent depth of the object be $ x $.
Since the mirrored surface is a plane mirror, to get an image $ 23.2\,cm $ behind the mirror, the object should be $ 23.2\,cm $ in front of the mirror. Hence the refraction from the glass block will shift the object position with respect to the mirror such that the distance of the object in front of the mirror is equal to the sum of the apparent depth and the thickness of the glass. So, we can write
 $ x + {\text{thickness of glass = 23}}{\text{.2}} $
 $ x = 13.2\,cm $
We can then determine the refractive index using the formula
  $ \mu = \dfrac{{{\text{Real depth}}}}{{{\text{Apparent depth}}}} $
  $ = \dfrac{{20}}{{13.2}} = 1.5 $.

The trick to solving this question is to realize that the entire glass block will shift the position of the light rays coming from the object such that the mirror sees the object a little far away than the actual distance of the object from the mirror. This shift must be counted while calculating the apparent depth of the object.