Question

At the interface between two materials having refractive indices \( n_1 \) and \( n_2 \), the critical angle for reflection of an EM wave is \( \theta_c \). The \( n_1 \) material is replaced by another material having refractive index \( n_3 \), such that the critical angle at the interface between \( n_1 \) and \( n_3 \) materials is \( \theta_{c3} \). If \( n_1 > n_2 > n_3 \), \( \frac{n_2}{n_3} = \frac{2}{5} \), and \( \sin \theta_{c2} - \sin \theta_{c1} = \frac{1}{2} \), then \( \theta_{c1} \) is:

• A. $\sin^{-1} \left( -\frac{5}{6} \right)$
• B. \( \sin^{-1} \left( \frac{2}{3n_1} \right) \)
• C. \( \sin^{-1} \left( \frac{1}{3n_1} \right) \)
• D. \( \sin^{-1} \left( \frac{1}{6n_1} \right) \)

Hint:

For problems involving critical angles and refractive indices, use Snell's Law and the relationship between the angles to find the unknowns.

Answer 1

The Correct Option is A

To address this problem, a thorough comprehension of the critical angle and its relation to the refractive indices of the involved materials is required:

1. Critical Angle Definition:
   • The critical angle, denoted as \(\theta_c\), represents the angle of incidence at which total internal reflection occurs when light transitions from a higher refractive index medium to a lower one.
   • Snell's Law, applied at the critical angle, establishes the relationship: \[ \sin \theta_c = \frac{n_2}{n_1} \] Here, \(n_1\) signifies the refractive index of the initial medium, and \(n_2\) denotes that of the second medium.
2. Provided Problem Data:
   • Refractive indices: \( n_1 > n_2 > n_3 \)
   • The ratio of refractive indices: \(\frac{n_2}{n_3} = \frac{2}{5}\)
   • A relationship between critical angles: \(\sin \theta_{c2} - \sin \theta_{c1} = \frac{1}{2}\)
3. Critical Angle Calculations:
   • It is established that \(\sin \theta_{c2} = \frac{n_3}{n_1}\) and \(\sin \theta_{c1} = \frac{n_2}{n_1}\).
   • Applying the given condition \(\sin \theta_{c2} - \sin \theta_{c1} = \frac{1}{2}\): \[ \frac{n_3}{n_1} - \frac{n_2}{n_1} = \frac{1}{2} \] This simplifies to: \[ \frac{n_3 - n_2}{n_1} = \frac{1}{2} \] Leading to: \[ n_3 - n_2 = \frac{n_1}{2} \]
   • Given \(\frac{n_2}{n_3} = \frac{2}{5}\), it follows that \(n_2 = \frac{2}{5}n_3\).
4. Determining \(\theta_{c1}\):
   • Substitute \(n_3 = \frac{5}{2}n_2\) into the equation \(n_3 - n_2 = \frac{n_1}{2}\): \[ \frac{5}{2}n_2 - n_2 = \frac{n_1}{2} \] This yields: \[ \frac{3}{2}n_2 = \frac{n_1}{2} \] Which simplifies to: \[ n_1 = 3n_2 \]
   • Using the relation \(\sin \theta_{c1} = \frac{n_2}{n_1}\): \[ \sin \theta_{c1} = \frac{n_2}{3n_2} = \frac{1}{3} \]
5. Conclusion and Discrepancy:
   • The computed value for \(\sin \theta_{c1}\) does not align with any of the provided options for \(\theta_{c1}\). This suggests a potential error in the given options or problem conditions.
   • Upon reviewing the options, a mismatch appears evident. The calculation of the general condition reveals that the provided result \(\sin^{-1} \left( -\frac{5}{6} \right)\) is incorrect, as the sine function cannot yield negative values greater than -1. Correctly computed discrete values, such as \(\frac{1}{6}\), indicate an issue with the initial assumptions.
   • Therefore, logical inference mandates a review of the calculations or a re-application of the methodology based on standard principles, rather than accommodating inconsistent choices.