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A point source of light is placed at a d...

A point source of light is placed at a distance h below the surface of a large and deep lake. Show that the fraction f of light that escape directly from water surface is independent of h and is given by
`f=[[1-sqrt(1-1//mu^2)])/2`

Text Solution

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Due to total internal reflection some of the light rays incident at the interface will return back into water. So, only that portion of light will escape for which the angle of incidence at the interface of the medium is less than the critical angle.
If the critical angle is `theta_c`, then the light rays that reach beyong the base of the cone whose vertical angle is `2theta_(c)` will suffer total internal reflection.
Hence, Only the light incident on the base of teh cone refracts and escapes.
Method 2:
The fraction of light escaping,
`f=(" Area of the cap ")/(" Area of the sphere ")=(2piRy)/(4piR^(2))`, i.e.,
`f=1/2[y/R]=1/2[(R-h)/R]`
Area of cap ABCD can be calculated by using mathod of integration,
i.e., `f=1/2f=1/2[1-h/R]=1/2[1-costheta_(c)]`
`=1/2[1-sqrt(1-sin^2theta_(c))]`
`=1/2[1-sqrt(1-1/n^(2))]`
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