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. $\frac{2x({{\mu }_{1}}+{{\mu }_{2}})}{{{\mu }_{1}}{{\mu }_{2}}}$

Answer 1

Answer: (A)

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

Answer 2

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
$=\frac{x}{2{{\mu }_{1}}}+\frac{x}{2{{\mu }_{2}}}$
$=\frac{x}{2}\left[ \frac{1}{{{\mu }_{1}}}+\frac{1}{{{\mu }_{2}}} \right]=\frac{x}{2}\left[ \frac{{{\mu }_{1}}+{{\mu }_{2}}}{{{\mu }_{1}}{{\mu }_{2}}} \right]$