question_answer

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)        \[\frac{2x({{\mu }_{1}}+{{\mu }_{2}})}{{{\mu }_{1}}{{\mu }_{2}}}\]

Correct Answer: A

Solution :

[a] As refractive index, \[\mu =\frac{\text{Real}\,\text{depth}}{\text{Apparent}\,\text{depth}}\] \[\therefore \] Apparent depth of the vessel when viewed from above is \[{{d}_{apparent}}=\frac{x}{2{{\mu }_{1}}}+\frac{x}{2{{\mu }_{2}}}=\frac{x}{2}\left( \frac{1}{{{\mu }_{1}}}+\frac{1}{{{\mu }_{2}}} \right)\] \[=\frac{x}{2}\left( \frac{{{\mu }_{2}}+{{\mu }_{1}}}{{{\mu }_{1}}{{\mu }_{2}}} \right)=\frac{x({{\mu }_{1}}+{{\mu }_{2}})}{2{{\mu }_{1}}{{\mu }_{2}}}\]