Consider the representative rays shown in diagram (b). A ray entering the glass through surface A and passing along the inner side of the rod will be reflected by the outer side with the smallest angle $\alpha$, at which the reflected ray is tangent to the inner side. We have to consider the conditions under which the ray will undergo total internal reflection before reaching B.



If $\alpha >{\theta }_{\text{c}}$, the critical angle, at which total internal reflection occurs, all the incident beam will emerge through the surface B. Hence we require

$\text{sin}\alpha >\frac{1}{\text{n}}$

The geometry gives $\text{sin}\alpha =\frac{\text{R}}{\left(\text{R}+\text{d}\right)}$

Therefore $\frac{\text{R}}{\text{R}+\text{d}}\ge \frac{1}{\text{n}}$,

or ${\left(\frac{\text{R}}{\text{d}}\right)}_{\text{min}}=\frac{1}{\text{n}-1}=\frac{1}{\text{1.5}-1}=2$