MHT CET · Physics · Ray Optics

A ray of light travels from a denser medium to a rarer medium. The reflected and the refracted rays are perpendicular to each other. If ' \(\mathrm{r}\) ' and ' \(\mathrm{r}_1\) ' are the angle of reflection and refraction respectively and ' \(\mathrm{C}\) ' is the critical angle, then the angle of incidence is

A \(\cot ^{-1}(\sin \mathrm{C})\)
B \(\tan ^{-1}(\sin C)\)
C \(\sin ^{-1}(\tan C)\)
D \(\cos ^{-1}(\tan C)\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\tan ^{-1}(\sin C)\)

Step-by-step Solution

Detailed explanation

Since reflected and refracted rays are perpendicular to each other,
\(
\begin{aligned}
& r+r_1=90^{\circ} \\
& \therefore r_1=90^{\circ}-r
\end{aligned}
\)
Refractive index of the denser medium
\(
\begin{aligned}
& \mu=\frac{\sin r_1}{\sin r}=\frac{\sin \left(90^{\circ}-r\right)}{\sin r}=\frac{\cos r}{\sin r}=\cot r \\
& \text { Also, } \mu=\frac{1}{\sin C} \quad \therefore \frac{1}{\sin C}=\cot r \\
& \text { or } \sin C=\tan r=\tan i \\
& \therefore i=\tan ^{-1}(\sin C)
\end{aligned}
\)

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.

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