Question

The Apparent vertical shift of the image of a coin placed at the bottom of a water tank having constant depth of water is proportional to (given refractive index of water ${{ = \mu }}$)
(A) ${{\mu }}$
(B) $\dfrac{{{1}}}{{{\mu }}}$
(C) ${{\mu - 1}}$
(D) ${{\mu + 1}}$

Answer 1

Hint: By knowing the relation between the apparent vertical shift and Real Depth, we can solve the above question. ${{Apparent Shift = }}\dfrac{{{{RealDepth}}}}{{{\mu }}}$ . There is a shift in the height of an object placed in water or any other fluid. This change is due to the difference in refractive indices of air and water.

Formula Used:
\[{\text{Apparent Depth(AD) = }}\dfrac{{{\text{Real Depth(RD)}}}}{{{\text{Refractive index(\mu )}}}}\]
$AD = \dfrac{{RD}}{\mu }$

Complete step by step answer:
The Apparent vertical Shift is given by the equation $AD = \dfrac{{RD}}{\mu }$.
That is $\mu = \dfrac{{RD}}{{AD}}$ or $\mu = \dfrac{{RD}}{{AD}}$
Where ${{AD}}$ is the Apparent Depth (or vertical shift) ${{RD}}$is the Real Depth of the tank and ${{\mu }}$ is the refractive index of the denser medium, that is water.
As Real Depth and Refractive Index is going to be constant always, the Apparent Shift or Depth will be independent of viewing angle.
From the above equation, we can see that the apparent vertical shift is inversely proportional to the Refractive Index. That is, $\dfrac{{{1}}}{{{\mu }}}$.

Hence, the correct answer is option (B).

Additional information:
Real Depth is the actual distance of an object beneath the surface, as would be measured by submerging a perfect ruler along it.
Apparent Depth in a medium is the depth of an object in a denser medium as seen from the Rarer medium. Its value is smaller than the real depth.
\[{D_{\text{Apparent}}} = \dfrac{{{D_{\text{real}}}}}{\mu }\]

Note: As the Real Depth ${{(RD)}}$ is different from the Apparent Depth ${{(AD)}}$. By knowing the relation between Apparent Depth, Real Depth and refractive index.
The phenomenon of apparent bending of a straight stick in water occurs due to refraction of light. When immersing a stick in water, the rays of light pass from rarer medium to denser medium and they move towards the normal. So the part of the stick immersed in water appears to be broken towards the normal drawn at the surface and gives an apparent bending, it also appears short like being raised up in the water.