# In the figure shown for an angle of incidence i at the top surface , What is the minimum refractive index needed for total internal reflection at vertical face ?

$(a)\;\sqrt{1+\sin i} \\ (b)\;\sqrt{1+ \cos i} \\ (c)\;\sqrt{1+\cos^2 i}\\ (d)\;\sqrt{1+\sin ^2 i}$

Applying Snell's law at top surface.
$\mu \sin r =\sin i$ -----(i)
For total internal reflection the vertical face,
$\mu \sin \theta_c=1$
Using geometry $\theta_c=90^{\circ}-r$
$\therefore \mu \sin (90-r)=1$
or $\mu \cos r =1$
On squaring and adding equation (i) and (ii) we get
$\mu^2 \sin^2 r +y^2 \cos ^2 r =1+ \sin ^2 i$
or $\mu =\sqrt {1+\sin ^2 i}$
Hence d is the correct answer.