Total Internal Reflection and Critical Angle Explained

When light tries to go from a denser medium to a rarer medium at a steep angle, it may not exit at all. Instead, it reflects completely inside the denser medium. This is total internal reflection, one of the most useful effects in optics.

It is the principle behind optical fibres that carry internet signals, the sparkle of diamonds, and mirages on hot roads. The effect occurs only when specific conditions on angle and refractive index are met.

Key Concepts

Denser and rarer medium: Higher refractive index means optically denser, light travels slower. Example: glass (n about 1.5) is denser than air (n = 1.0).

Critical angle C: The angle of incidence in denser medium for which angle of refraction in rarer medium is 90 degrees. Measured in degrees.

Total internal reflection: For incidence angle greater than C, no refraction occurs, all light reflects inside. The surface acts like a perfect mirror.

• Light must go from denser to rarer medium
• Angle of incidence must exceed critical angle
• No energy is lost into second medium in ideal case

Formulas

sin C = n_rarer divided by n_denser. If rarer is air n = 1, then sin C = 1 divided by n_denser. n is dimensionless, C in degrees.

Derivation uses Snell law n_denser sin C = n_rarer sin 90 degrees, and sin 90 degrees = 1.

For glass-air with n = 1.5, sin C = 1 divided by 1.5 = 0.667. For water-air n = 1.33, sin C = 0.752.

Worked Example

Find critical angle for glass of refractive index 1.5 in air, and state what happens at incidence 50 degrees.

1. Use sin C = 1 divided by n = 1 divided by 1.5 = 0.6667.
2. C = arcsin 0.6667 = 41.8 degrees.
3. Given incidence 50 degrees, which is greater than 41.8 degrees.
4. Therefore total internal reflection occurs, no ray exits into air.

At 30 degrees, which is less than C, partial refraction would occur with bending away from normal. Only above 41.8 degrees is reflection total.

Common Misconceptions

• Can happen from air to glass: Total internal reflection only occurs denser to rarer. Air to glass always allows entry, though bending occurs.
• Critical angle is same for all colours equally: n varies slightly with colour, so C varies slightly. Blue has slightly larger n and smaller C than red.
• Reflection is partial above C: Above C it is total, 100 percent reflects in ideal smooth surface. Below C, part refracts and part reflects.

Real-World Applications

• Optical fibres guide light over kilometres with repeated total internal reflection
• Diamond cut uses small critical angle (about 24 degrees for n = 2.42) to trap light and sparkle
• Prism binoculars use total internal reflection instead of silvered mirrors
• Mirages form when light from sky undergoes gradual total internal reflection in hot air layers

Total internal reflection needs denser to rarer travel and incidence beyond critical angle sin C = 1 divided by n. When met, light is perfectly trapped and can be guided with almost no loss.