Question #43234

A transparent right-angled prism ABC has $\angle B = 90^\circ$ and $\angle A = 75^\circ$. Face AB is vertical, face BC is horizontal, and the refractive index of the prism is $\mu_p = \sqrt{2}$. Face AB is covered by an opaque screen containing a slit of width $2w = 6 \text{ cm}$ (measured along AB). A converging beam of light in air is directed towards face AB such that its central ray is horizontal, passes exactly through the center of the slit, and is perpendicular to AB. In the absence of any other optical elements, the beam would converge to a virtual object $O$ located at a distance $L_0$ behind face AB.

A transparent dielectric slab of thickness $t = 6 \text{ cm}$ and refractive index $\mu_s = 1.5$ is now placed in the path of the beam, parallel to face AB and in front of the slit. Determine the minimum value of $L_0$ such that all rays passing through the slit undergo total internal reflection (TIR) at face AC.

(Assume the longitudinal shift produced by the dielectric slab can be calculated using the paraxial approximation formula).