Q.A monochromatic ray of light is incident at 37° on an equilateral prism of refractive index 3/2. Determine angle of emergence and angle of deviation. If angle of prism is adjustable, what should its value be for emergent ray to be just possible for the same angle of incidence.

A prism has two refracting surfaces (faces $AB$, $AC$) inclined at the PRISM ANGLE $A$, with the third face $BC$ as the (uninvolved) base. For a ray entering at incidence $i$, refracting to $r_1$ at the first face, crossing to internal incidence $r_2$ at the second face, and finally emerging at angle $e$: geometry gives $A=r_1+r_2$, and since the total deviation $\delta$ is the exterior angle of the triangle formed by the incident and emergent ray extensions, $\delta=i+e-A$, i.e. $i+e=A+\delta$.

As $i$ increases from the minimum value at which any ray can emerge at all (below this, $r_2$ would exceed the critical angle and TIR occurs at the second face instead), $\delta$ first DECREASES to a single minimum $\delta_m$ and then increases again, tracing an asymmetric curve. Except exactly at the minimum, every deviation corresponds to two possible incidence angles, interchangeable (by reversibility) as $i$ and $e$. At $\delta=\delta_m$ exactly, $i=e$, $r_1=r_2=A/2$, and the internal ray runs parallel to the base; substitut …