A ray of light travelling through a denser medium is incident at an angle I in a rarer medium. If the angle between the reflected ray and refracted ray is `90^(@)` , show that the critical angle of the two media, `theta_(c) = sin^(-1) (tan i)`

Explanation

Suppose the angle of refraction in the rarer medium = r .
From Fig. 2.29, we get,
`I + 90^(@) + r = 180^(@)`
`or, " " r = 90^(@) - i`
According to Snell,s law,
`(sini)/(sinr) = 1^(mu)2 = (mu^(2))/(mu_(1))`
`or, " " (sini)/(sin(90^(@) - i)) = (mu^(2))/(mu_(1))`
`or, " " (sini)/(cosi) = (mu_(2))/(mu_(1)) or, tani = (mu_(2))/(mu_(1))`
If the critical angle for the two media is `theta_(c)`, then
`sintheta_(c) = (1)/(2^(mu)1) = (1)/((mu_(1))/(mu_(2))) = tani or, theta_(c) = sin ^(-1) (tani)`