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Question
Obtain the equation for radius of illumination (or) Snell’s window.
Solution
- When light entering the water from outside is seen from inside the water, the view is restricted to a particular angle equal to the critical angle ic.
- This restricted illuminated circular area is called Snell’s window.
- The angle of view for water animals is restricted to twice the critical angle 2ic. The critical angle for water is 48.6°. The angle of view is 97.2°.
- The radius R of the circular area depends on the depth d from which it is seen and the refractive indices of the media.
- Light is seen from a point A at a depth d.
Snell’s law
n1 sin ic = n2 sin 90°
n1 sin ic = n2 ∵ sin 90° = 1
sin ic = `"n"_2/"n"_1` - From the Right angle triangle ∆ ABC,
sin ic = `"CB"/"AB" = "R"/sqrt("d"^2 + "R"^2)`
Equating the above two equations
`"R"/sqrt("d"^2 + "R"^2) = ("n"_2/"n"_1)`
Radius of Snell’s window
Squaring on both sides,
`"R"^2/("R"^2 + "d"^2) = ("n"_2/"n"_1)^2`
Taking reciprocal,
`("R"^2 + "d"^2)/"R"^2 = ("n"_1/"n"_2)^2`
On further simplifying,
`1 + "d"^2/"R"^2 = ("n"_1/"n"_2)^2; "d"^2/"R"^2 = ("n"_1/"n"_2)^2 - 1;`
`"d"^2/"R"^2 = ("n"_1^2/"n"_2^2) - 1 = ("n"_1^2 - "n"_2^2)/"n"_2^2`
Again taken reciprocal and rearranging
`"R"^2/"R"^2 = "n"_2^2/("n"_1^2 - "n"_2^2); "R"^2 = "d"^2("n"_2^2/("n"_1^2 - "n"_2^2))`
Tha radius of illumination is,
R = d`sqrt(("n"_2^2)/("n"_1^2 - "n"_2^2))`
If the rarer medium outside in air, then, n2 = 1, and we can take n1 = n
R = d`1/sqrt("n"^2 - 1)` (or) R = `"d"/sqrt("n"^2 - 1)`
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