For a ray of light to retrace its path after reflection from a mirror, it must strike the mirror surface normally (perpendicularly). This implies that the angle of incidence on the second face ($r_2$) is $0^\circ$.

For a prism, the refracting angle $A$ is related to the internal refraction angles $r_1$ and $r_2$ by the equation: $A = r_1 + r_2$

Given: Prism angle $A = 30^\circ$ Refractive index $\mu = \sqrt{2}$

• Condition for retracing: $r_2 = 0^\circ$

Substituting values to find $r_1$: $30^\circ = r_1 + 0^\circ \implies r_1 = 30^\circ$

Now, apply Snell's Law at the first face (refraction from air to prism): $\mu_{air} \sin i = \mu_{prism} \sin r_1$ $1 \cdot \sin i = \sqrt{2} \cdot \sin(30^\circ)$ $\sin i = \sqrt{2} \cdot \frac{1}{2} = \frac{1}{\sqrt{2}}$

Since $\sin i = \frac{1}{\sqrt{2}}$, the angle of incidence $i = 45^\circ$.