Obtain the equation for critical angle.

Explanation

(i) When a ray passes from an optically denser medium to an optically rarer medium, it bends away from normal.
(ii) Because of this, the angle of refraction r on the rarer medium is greater than the corresponding angle of incidence i in the denser medium.
(iii) As angle of incidence i is gradually increased, r rapidly increases and at a certain stage it becomes `90^@` or gracing the boundary.
(iv) The angle of incidence in the denser medium for which the refracted ray graces the boundary is called critical angle `i_c`
(v) If the angle of incidence in the denser medium is increased beyond the critical angle, there is no refraction possible into the rarer medium.
(vi) The entire light is reflected back into the denser medium itself. This phenomenon is called total internal reflection.
(vii) The two conditions for total internal reflection are,
(a) light must travel from denser to rarer medium,
(b) angle of incidence in the denser medium must be greater than critical angle ` (i gt i_c)`
(viii) Snell's law in the product form, equation (1) for critical angle incidence becomes,
`n_1 sin i_c = n_2 sin 90^@ " " ..(1)`
`n_1 sin i_c = n_2" " because sin 90^@ = 1`
` sin i_c = (n_2)/(n_1)`
here , ` n_1 gt n_2`
(ix) If the rarer medium is air, then its refractive index is 1 and can be taken as n itself. i.e.
`(n_2 = 1) " and " (n_1 = n)`
` sin i_c = (1)/(n) " or " i_c = sin^(-1) (1/n)`