A monochromatic light is travelling in a medium of refractive index $n=1.6$ . It enters a stack of glass layers from the bottom side at an angle $\theta =30°.$ The interfaces of the glass layers are parallel to each other. The refractive indices of different glass layers are monotonically decreasing as $n_m=n–m\mathrm{\Delta }n$ where $n_m$ is the refractive index of the $m^{th}$ slab and $\mathrm{\Delta }n=0.1$ (see the figure). The ray is refracted out parallel to the interface between the ${\left(m–1\right)}^{th}$ and $m^{th}$ slabs from the right side of the stack. What is the value of m?

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.