Lecture Notes: Total Internal Reflection and Critical Angle

Key Concepts

• Total Internal Reflection: Occurs when a light ray travels from a medium with a higher index of refraction to a medium with a lower index, and the incident angle exceeds the critical angle.
• Critical Angle: The incident angle at which the refracted angle is 90 degrees.

Definitions

• Incident Ray: The ray striking the boundary between two materials.
• Angle of Incidence (Theta I): The angle between the incident ray and the normal line.
• Angle of Reflection (Theta R): The angle between the reflected ray and the normal, equal to the angle of incidence.
• Angle of Refraction: The angle between the refracted ray and the normal line.
• Snell's Law: ( N_1 \sin(\theta_1) = N_2 \sin(\theta_2) )
  • ( N_1 ): Index of refraction of the first medium.
  • ( N_2 ): Index of refraction of the second medium.

Index of Refraction

• Air: Approximately 1
• Water: 1.33
• Diamond: 2.42

Important Points

• Reflection: Light bounces back into the original medium.
  • Follows the law of reflection: ( \theta_I = \theta_R )
• Refraction: Light bends when passing into another medium.
  • Bends towards the normal if moving to a higher index medium.
  • Bends away from the normal if moving to a lower index medium.

Total Internal Reflection Conditions

• Occurs when moving from a high index of refraction to a low index (e.g., water to air).
• Cannot occur when moving from a low index to a high index (e.g., water to diamond).

Calculating the Critical Angle

1. Use Snell's Law: ( N_1 \sin(\theta_1) = N_2 \sin(90^\circ) )
2. Critical angle formula: ( \theta_c = \sin^{-1}(\frac{N_2}{N_1}) )

Example Calculations

• Water to Air:

  • ( N_1 = 1.33 ), ( N_2 = 1 )
  • Critical angle: ( \theta_c \approx 48.75^\circ )

• Glass to Water:

  • ( N_1 = 1.5 ), ( N_2 = 1.33 )
  • Critical angle: ( \theta_c \approx 62.5^\circ )

Problem Solving

• For total internal reflection, ensure light travels from a higher to a lower index.
• Example: Solid to Air with a critical angle of 40°
  • Index of refraction of solid: ( N = \frac{1}{\sin(40^\circ)} \approx 1.56 )

Conclusion

• Total internal reflection converts all refracted light into reflected light when the critical angle is surpassed.
• Useful for designing optical devices like fiber optics.

• Note: Always consider the direction of light travel and the indices of refraction when applying Snell's Law and calculating critical angles.