Question

A vessel of depth $x$ is half filled with oil of refractive index $ {{\mu }_{1}} $ and the other half is filled with water of refractive index $ {{\mu }_{2}} $ The apparent depth of the vessel when viewed from above is

• A. $ \frac{x({{\mu }_{1}}+{{\mu }_{2}})}{2{{\mu }_{1}}{{\mu }_{2}}} $
• B. $ \frac{x\,{{\mu }_{1}}\,{{\mu }_{2}}}{2({{\mu }_{1}}+{{\mu }_{2}})} $
• C. $ \frac{x{{\mu }_{1}}{{\mu }_{2}}}{({{\mu }_{1}}+{{\mu }_{2}})} $
• D. $ \frac{2x({{\mu }_{1}}+{{\mu }_{2}})}{{{\mu }_{1}}{{\mu }_{2}}} $
• E. $ \frac{4({{\mu }_{1}}+{{\mu }_{2}})x}{{{\mu }_{1}}{{\mu }_{2}}} $

Answer 1

Answer: (A)

Apparent depth
$ =\frac{\text{Real}\,\text{depth}}{\mu } $
For oil apparent depth
$ =\frac{x}{2{{\mu }_{1}}} $
and for water apparent depth
$ =\frac{x}{2{{\mu }_{2}}} $
The apparent depth of the vessel
$ =\frac{x}{2{{\mu }_{1}}}+\frac{x}{2{{\mu }_{2}}} $
$ =\frac{x}{2}\left[ \frac{1}{{{\mu }_{1}}}+\frac{1}{{{\mu }_{2}}} \right]=\frac{x}{2}\left[ \frac{{{\mu }_{1}}+{{\mu }_{2}}}{{{\mu }_{1}}{{\mu }_{2}}} \right] $