Any ray entering at an angle shall be guided along AC if the angle the ray makes with the face AC(ϕ) is greater than the critical angle.
\(⇒sin\,r≥\frac{1}{μ}\)
\(⇒cos\,r≥\frac{1}{μ}\)
Or, 1 – cos2 r ≤ 1 − \(\frac{1}{μ^2}\)
i.e. sin2 r ≤ 1 − \(\frac{1}{μ^2}\)
Since sin i = μ sin r
\(\frac{1}{μ^2}sin^2\,i\) ≤ \(1-\frac{1}{μ^2}\)
Or, sin2 i ≤ μ2 − 1
The smallest angle ϕ shall be when \(i=\frac{π}{2}\). If that is greater than the critical angle then all other angle of incidence shall be more than the critical angle.
Thus 1 ≤ μ2 − 1
Or, μ2 ≥ 2
⇒ μ ≥ √2