Understanding Critical Angle
The critical angle is the angle of incidence above which total internal reflection occurs when light travels from a denser medium to a less dense medium. This phenomenon is vital in optics and is defined by the relationship between the refractive indices of the two media.
Formula for Critical Angle
The critical angle (\(C\)) can be calculated using Snell's Law:
\[ C = \sin^{-1}\left(\frac{n_2}{n_1}\right) \]
Where:
- \( n_1 \) = refractive index of the denser medium
- \( n_2 \) = refractive index of the less dense medium (usually air, \( n \approx 1 \))
Given Information
- The critical angle is \( 45^\circ \).
Calculating the Refractive Index
From the formula, we can rearrange it as follows:
\[ n_1 = \frac{n_2}{\sin(C)} \]
Substituting the known values:
- \( n_2 = 1 \) (for air)
- \( C = 45^\circ \)
\[ n_1 = \frac{1}{\sin(45^\circ)} \]
Since \( \sin(45^\circ) = \frac{\sqrt{2}}{2} \):
\[ n_1 = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} \]
Thus,
\[ n_1 \approx 1.414 \]
Conclusion
The refractive index of the medium, when the critical angle of incidence is equal to \( 45^\circ \), is approximately \( 1.414 \). This indicates that the medium is denser than air but not as dense as many other materials like glass or water.

Answer :--1.414