Transmitted and Reflected Intensity versus Angle of Incidence at Interface

When a light ray travels from a denser medium (glass with refractive index $\mu > 1$) to a rarer medium (air with refractive index $\mu_{\text{air}} = 1$), the reflection ($R$) and transmission ($T$) characteristics depend on the angle of incidence $\theta$ relative to the critical angle $\theta_c = \sin^{-1}\left(\frac{1}{\mu}\right)$:

1. At Normal Incidence ($\theta = 0^\circ$):

   • Using Fresnel's reflection formula at normal incidence:
     $R(0) = \left(\frac{\mu - 1}{\mu + 1}\right)^2 I_0$
     For typical glass ($\mu \approx 1.5$), $R(0) \approx 4\%$ and $T(0) = I_0 - R(0) \approx 96\%$.

2. For Angles Below the Critical Angle ($0 < \theta < \theta_c$):

   • As $\theta$ increases towards the critical angle $\theta_c$, the angle of refraction $r$ approaches $90^\circ$.
   • The reflected intensity $R(\theta)$ increases monotonically and steeply approaches $100\%$ as $\theta \to \theta_c^-$.
   • Correspondingly, the transmitted intensity $T(\theta) = I_0 - R(\theta)$ decreases monotonically to $0\%$ as $\theta \to \theta_c^-$.

3. At and Beyond the Critical Angle ($\theta \ge \theta_c$):

   • Total Internal Reflection (TIR) occurs for all angles of incidence $\theta \ge \theta_c$.
   • Thus, for $\theta_c \le \theta \le 90^\circ$:
     $R(\theta) = 100\% \quad \text{and} \quad T(\theta) = 0\%$

Graph (C) correctly illustrates this behavior:

• $T$ drops to $0\%$ at $\theta = \theta_c$ and remains zero up to $90^\circ$.
• $R$ rises to $100\%$ at $\theta = \theta_c$ and remains at $100\%$ up to $90^\circ$.

Correct Option: (C)