Question

A point source of light is placed at a distance $h$ below the surface of a large deep lake. Neglecting the partial reflection of light during refraction, the percentage of light energy that escapes from the water surface is $[{\mu }_{water}=4/3]$

• A. $50\%$
• B. $25\%$
• C. $20\%$
• D. $17\%$

Answer 1

Answer: (D)

Solid angle $\left(\varphi \right)=2\pi \left(1-cos\theta \right)$
where $\theta =$ plane angle
$\text{escape energy}=\frac{\text{total energy}}{4\pi }\times \text{solid angle}$
$\frac{\text{escape energy}}{\text{total energy}}=\frac{1}{4\pi }\times 2\pi \left(1-\text{cosC}\right)$
$=\frac{1}{2}\left(1-\text{cosC}\right)....(1)$
from law of refraction
${\mu }_{w}sini={\mu }_{a}sinr$
$\frac{4}{3}\times \text{sin C}=1\times {\text{sin 90}}^{\text{o}}$
$\sin C=\frac{3}{4}\Rightarrow \cos C=\sqrt{1-{\sin }^{2}C}\Rightarrow \cos C=\frac{\sqrt{7}}{4}$
from (1) and (2)
${\mu }_{w}sini={\mu }_{a}sinr$
$\frac{\text{escaped energy}}{\text{total energy}}=\text{0.17}$
total $\%$ energy enscaped $=17\%$