Answer
$d = \frac{R}{n}$
Work Step by Step
The light ray will travel straight into the transparent material. Then it will refract out through the curved surface as long as the angle does not exceed the critical angle.
We can find the critical angle:
$\theta_c = sin^{-1}~(\frac{n_2}{n_1})$
$\theta_c = sin^{-1}~(\frac{1}{n})$
Note that $sin~\theta_c = \frac{1}{n}$
Using geometry, we can see that the angle between the normal and the light ray striking the curved surface has the following relationship:
$sin~\theta = \frac{d}{R}$
To find the maximum value of $d$, we can set this angle $\theta$ equal to $\theta_c$:
$sin~\theta = \frac{d}{R}$
$sin~\theta_c = \frac{d}{R}$
$\frac{1}{n} = \frac{d}{R}$
$d = \frac{R}{n}$
