(i) When a light source like electric bulb is kept inside a water tank, the light from the source travels in all direction inside the water. The light that is incident on the water surface at an angle less than the critical angle will undergo refraction and emerge out from the water. (ii) The light incident at an angle greater than critical angle will undergo total internal reflection. (iii) The light falling particularly at critical angle graces the surface. Thus, the entire surface of water appears illuminated when seen from outside as shown in Figure. (iv) On the other hand, 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`. (v) The restricted illuminated circular area is called Snell's window. (vi) The angle of view for water animals is restricted to twice the critical angle `2i_c` . The critical angle for water is `48.6^@`. Thus the angle of view is `97.2^@` . (vii) The radius R of the circular area depends on the depth d from which it is seen and also the refractive indices of the media. The radius of Snell's window can be deduced with the illustration.
(viii) Light is seen from a point A at a depth d. The Snell's law in product form, equation (`n_1 sin i = n_2`) sin r for the refraction happening at the point B on the boundary between the two media is,
` n_1 sin i_c = n_2 sin90^@ " " ...(2)`
`n_1 sin i_c = n_2" " because sin 90^@ = 1`
` sin i_c = (n_2)/(n_1) " " ..(3)`
From the right angle triangle `triangleABC, `
`sin i_c = (CB)/(AB) = (R)/(sqrt(d^2 + R^2)) " " ...(4)`
Equating the above two equation (4) and equation (5),
` (R)/(sqrt(d^2 + R_2)) = n_2/n_1`
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 taking reciprocal and rearranging.
` (R^2)/(d^2) = (n_2^2)/(n_1^2 - n_2^2) , R^2 = d^2 ((n_2^2)/(n_1^2 - n_2^2))`
The radius of illumination is,
` R = d sqrt((n_2^2)/(n_1^2 - n_2^2)) " " ....(5)`
if the rarer medium outside is air, then `n_2 = 1` and we can take `n_1 = n`
` R =d ((1)/(sqrt(n_2 - 1))) "or" R = (d)/(sqrt(n^2 - 1))`
