One side of an equilateral prism is painted by a transparent material of refractive index $$n_2$$. The refractive index of prism is 1.6. The minimum value of $$n_2$$ required for total internal reflection from painted face is ________ .

$$\text{Angle with bottom surface} = 180^\circ - (90^\circ + 60^\circ) = 30^\circ \quad \text{(From the right-angled triangle formed at entry)}$$

$$\theta = 90^\circ - 30^\circ = 60^\circ \quad \text{(Angle measured relative to the surface normal)}$$

$$\sin(60^\circ) = \frac{n_2}{n_1} \quad \text{(Limiting threshold condition for internal reflection)}$$

$$\frac{\sqrt{3}}{2} = \frac{n_2}{1.6}$$

$$n_2 = 1.6 \times \frac{\sqrt{3}}{2}$$

$$n_2 = 0.8\sqrt{3}$$

$$n_2 = \frac{4\sqrt{3}}{5}$$