Question

Obtain the equation for radius of illumination (or) Snell’s window.

Answer 1

1. 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 i_c.
2. This restricted illuminated circular area is called Snell’s window.
3. The angle of view for water animals is restricted to twice the critical angle 2i_c. The critical angle for water is 48.6°. The angle of view is 97.2°.
4. The radius R of the circular area depends on the depth d from which it is seen and the refractive indices of the media.
5. Light is seen from a point A at a depth d.
   Snell’s law
   n₁ sin i_c = n₂ sin 90°
   n₁ sin i_c = n₂ ∵ sin 90° = 1
   sin i_c = `"n"_2/"n"_1`
6. From the Right angle triangle ∆ ABC,

sin i_c = `"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)`