Question

The speed of light in two media '1' and '2' are \( v_1 \) and \( v_2 \) (\( v_2>v_1 \)) respectively. For a ray of light to undergo total internal reflection at the interface of these two media, it must be incident from:

• A. medium '1' and at an angle greater than \( \sin^{-1} \left( \frac{v_1}{v_2} \right) \)
• B. medium '1' and at an angle greater than \( \cos^{-1} \left( \frac{v_1}{v_2} \right) \)
• C. medium '2' and at an angle greater than \( \sin^{-1} \left( \frac{v_1}{v_2} \right) \)
• D. medium '2' and at an angle greater than \( \cos^{-1} \left( \frac{v_1}{v_2} \right) \)

Hint:

For total internal reflection to occur, the angle of incidence must be greater than the critical angle, which is \( \sin^{-1} \left( \frac{v_1}{v_2} \right) \) when light is moving from the slower medium.

Answer 1

The Correct Option is A

To determine the conditions required for total internal reflection at the interface between two media where the speed of light in medium '1' is \( v_1 \) and in medium '2' is \( v_2 \) (\( v_2>v_1 \)), consider Snell's Law: \[n_1 \sin \theta_1 = n_2 \sin \theta_2\] where \( n_1 \) and \( n_2 \) are the refractive indices of media '1' and '2'. The refractive index is inversely proportional to the speed of light in the medium: \[ n = \frac{c}{v} \] where \( c \) is the speed of light in a vacuum. Thus, \( n_1=\frac{c}{v_1} \) and \( n_2=\frac{c}{v_2} \). Total internal reflection occurs when light is incident in a denser medium and attempts to pass into a lighter medium (from medium '1' to medium '2', \( v_2>v_1 \), implying \( n_2<n_1 \)). For total internal reflection, \(\theta_2\) must reach 90°, thus: \[ n_1 \sin \theta_c = n_2 \sin 90^\circ \] where \(\theta_c\) is the critical angle, leading to: \[\sin \theta_c = \frac{n_2}{n_1}\] Since refractive indices are inversely proportional to velocity: \[\sin \theta_c = \frac{v_1}{v_2}\] To achieve total internal reflection, the incident angle \(\theta_1\) must be greater than this critical angle: \[\theta_1 > \sin^{-1} \left( \frac{v_1}{v_2} \right)\] Therefore, for total internal reflection, the ray must be incident from medium '1' at an angle greater than \(\sin^{-1} \left( \frac{v_1}{v_2} \right)\).