A glass rod of the refractive index $1.50$of rectangular cross-section, $\left\{\mathrm{d}\times l\right\}$ is bent into a $“\mathrm{U}”$ shape see figure (A). The cross-sectional view of this rod is shown in figure (B).

The bent portion of the rod is semi-circular with inner and outer radii $R$and $R+d$, respectively. A parallel monochromatic beam of light is incident normally on face $ABCD$.

(a) Consider two monochromatic rays $r_1$ and $r_2$ in figure(B). State whether the following statements are True or False.

(b) Consider the ray $r_1$ whose point of incidence is very close to the edge $\mathrm{BC}$. Assume it undergoes total internal reflection at $p_1$. In cross-sectional view below, draw the trajectory of this reflected ray beyond the next glass-air boundary that it encounters.

(c) Obtain the minimum value of the ratio $\frac{R}{d}$ for which any light ray entering the glass normally through the face $ABCD$ undergoes at least one total internal reflection.

(d) A glass rod with the above computed minimum ratio of $\frac{R}{d}$, is fully immersed water of refractive index $1.33$. What fraction of light flux entering the glass through the plane surface $ABCD$ undergoes at least one total internal reflection?