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Question

There is a small source of light at some depth below the surface of water (refractive index=43), in a tank of large cross-sectional surface area. Neglecting any reflection from the bottom and absorption by water, the percentage of light that emerges out of the surface is (nearly):
[use the fact that surface area of a spherical cap of height h and radius of curvature r is 2πrh]

A
21%
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B
34%
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C
17%
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D
50%
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Solution

The correct option is C 17%

Given,
Refractive index, μ=43
Consider the cone of light formed on the surface of water, as shown in the figure. C is the critical angle.

Now, sinC=1μ=34

cosC=74 [cosC=1sin2C]

The solid angle subtended by the cone of light, at the source is,

Ω=2π(1cosC)=2π(174)

Fractional energy transmitted is,

=solid angle subtended by the cone of lighttotal solid angle

=2π(1cosθ)4π=(174)2=0.17

So, percentage of light emerging out of the water surface is,
=0.17×100=17%

Hence, (C) is the correct answer.

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