First slide

Total internal reflection - Ray optics and optical instruments

Question

There is a small source of light at some depth below the surface water (refractive index $= \frac{4}{3}$ ) in a tank of large cross sectional surface area. Neglecting any reflection from the bottom and absorption by water, percentage of light that emerges out of surface is (nearly): [use the fact that surface area of a spherical cap of height and radius of curvature $r$ is $2 π r h$ ]

Difficult

A

17%
B

50%
C

34%
D

21%

Solution

When angle of incidence is equal to critical angle then light ray grazes the water surface.
If angle of incidence is greater than critical angle light is reflected in water only
$\sin c = \frac{1}{μ} = \frac{3}{4}$
Light source emits the light in three dimensional solid angle subtended by this portion is

$Ω = 2 π (1 - \cos c) = 2 π (1 - \frac{\sqrt{7}}{4})$
For solid angle $4 π$ ----------100% light comes out
For solid angle $Ω$ ………?
$x = \frac{Ω \times 100}{4 π} = \frac{2 π (1 - \frac{\sqrt{7}}{4})}{4 π} \times 100 = \frac{1 - \frac{\sqrt{7}}{4}}{2} \times 100$ =17%

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