A glass prism of refractive index $$1.5$$ is immersed in water (refractive index $$\frac{4}{3}$$ ) as shown in figure. A light beam incident normally on the face $$AB$$ is totally reflected to reach the face $$BC$$, if

Solution & Explanation

Let us analyze the path of the light beam inside the glass prism immersed in water. We are given:

• Refractive index of the glass prism: $$\mu_g = 1.5 = \frac{3}{2}$$
• Refractive index of water: $$\mu_w = \frac{4}{3}$$

A light beam is incident normally on the face $$AB$$. Because it strikes perpendicular to the surface ($$\angle i = 0^\circ$$), it passes straight into the prism without any deviation and travels directly toward the inclined face $$AC$$.

Let us determine the angle of incidence ($$i_c$$) at the face $$AC$$ using geometry:

• The ray is parallel to the face $$BC$$ because it travels straight down perpendicular to $$AB$$.
• From the geometry of the triangle, the angle that this ray makes with the normal to the face $$AC$$ is exactly equal to the prism apex angle $$\theta$$.

Therefore, the angle of incidence at the glass-water interface on face $$AC$$ is:

$$i = \theta$$

For the light beam to be totally reflected at the face $$AC$$, the angle of incidence must be strictly greater than the critical angle ($$\theta_c$$) for the glass-water boundary:

$$i > \theta_c \implies \theta > \theta_c$$

Taking the sine of both sides preserves the inequality:

$$\sin\theta > \sin\theta_c$$

According to Snell's Law, the sine of the critical angle when traveling from a denser medium (glass) to a rarer medium (water) is given by the ratio of their refractive indices:

$$\sin\theta_c = \frac{\mu_{\text{rarer}}}{\mu_{\text{denser}}} = \frac{\mu_w}{\mu_g}$$

Substitute the given numerical values of the refractive indices:

$$\sin\theta_c = \frac{\frac{4}{3}}{\frac{3}{2}} = \frac{4}{3} \times \frac{2}{3} = \frac{8}{9}$$

Substituting this back into our total internal reflection condition inequality gives:

$$\sin\theta > \frac{8}{9}$$

Correct Option Key: Option C ($$\sin\theta > \frac{8}{9}$$)