In the figure shown, for an angle of incidence 45° at the top surface, what is the minimum refractive index needed for total internal reflection at vertical face

At point A, by Snell’s law μ=sin45sinr⇒sinr=1μ2----------(i) From figure, i1=90−r ∴sin90∘−r=1μ ⇒cosr=1μ-------(ii)  Now cosr=1−sin2r=1−12μ2 =2μ2−12μ2----(iii) From equation (ii) and (iii)   1μ=2μ2−12μ2 Squaring both side and then solving we get  μ=32