Question

The critical angle for a ray of light from glass to air is \( \theta \) and refractive index of glass with respect to air is \(n\). If a ray of light is incident from air to glass at an angle \( \theta \), then corresponding angle of refraction is

• A. \( \cos^{-1}\!\left(\dfrac{1}{n^2}\right) \)
• B. \( \sin^{-1}\!\left(\dfrac{1}{n}\right) \)
• C. \( \sin^{-1}\!\left(\dfrac{1}{n^2}\right) \)
• D. \( \cos^{-1}\!\left(\dfrac{1}{n}\right) \)

Hint:

Always relate critical angle with refractive index using \( \sin C = \dfrac{1}{n} \).

Answer 1

The Correct Option is C

Step 1: Using definition of critical angle.
For light going from glass to air,
\[ \sin \theta = \frac{1}{n} \]
Step 2: Applying Snell’s law for air to glass.
\[ \sin i = n \sin r \]
Step 3: Substituting angle of incidence.
Here \( i = \theta \), so
\[ \sin \theta = n \sin r \]
Step 4: Solving for angle of refraction.
\[ \frac{1}{n} = n \sin r \Rightarrow \sin r = \frac{1}{n^2} \]
Step 5: Conclusion.
\[ r = \sin^{-1}\!\left(\frac{1}{n^2}\right) \]