Critical angle between a given transparent medium and air is `theta_( c)`. A ray of ligth in air enters the transparent medium at an angle of incidence equal to the polarising angle. Deduce a relation between angle of refraction r and critical angle `theta_(c )`.

Explanation

Let refractive index of the transparent medium `=mu` and polarising angle` = i_(p)`.
So, `mu = (sini_(p))/(sinr)`
Again, `i_(p) + r = 90^(@)`
`therefore " " sin i_(p) = sin(90^(@) - r) = cos r " " therefore mu = (cos r)/(sin r)`
According to Snell's law, for critical angle `theta_( c)`,
`mu = (1)/(sin theta_( c))` `therefore (cos r)/(sin r) = (1)/(sin theta_( c))`
or, `sin theta_( c) = (sin r)/(cos r)`, or tan r = `sin theta_( c)` or , r = `tan^(-1) (sin theta_(c))`
This is the required relation.