At what angle should a ray of light be incident on the face of a prism of refracting angle $60^{\circ}$ so that it just suffers total internal reflection at the other face? The refractive index of the material of the prism is $1.524$.

$(29.75^{\circ})$ The incident, refracted, and emergent rays associated with a glass prism $ABC$ are shown in the figure.
Angle of prism, $A = 60^{\circ}$
Refractive index of the prism, $\mu = 1.524$
$i_1 = \text{Incident angle}$
$r_1 = \text{Refracted angle}$
$r_2 = \text{Angle of incidence at the face } AC$
$e = \text{Emergent angle} = 90^{\circ} \text{ (for grazing emergence)}$
According to Snell's law, for face $AC$, we have:
$\frac{\sin e}{\sin r_2} = \mu$
$\sin r_2 = \frac{1}{\mu} \times \sin 90^{\circ} = \frac{1}{1.524} \approx 0.6562$
$\therefore r_2 = \sin^{-1}(0.6562) \approx 41^{\circ}$
For refraction through a prism, $A = r_1 + r_2$
$\therefore r_1 = A - r_2 = 60^{\circ} - 41^{\circ} = 19^{\circ}$
According to Snell's law at the first face:
$\mu = \frac{\sin i_1}{\sin r_1}$
$\sin i_1 = \mu \sin r_1 = 1.524 \times \sin 19^{\circ} \approx 1.524 \times 0.3256 \approx 0.4962$
$\therefore i_1 = \sin^{-1}(0.4962) \approx 29.75^{\circ}$
Hence, the angle of incidence is $29.75^{\circ}$.