Refraction and total internal reflection

Light bends when it passes from one medium into another, because its speed changes. The refractive index n = c/v tells you how much the medium slows light.

Snell's law: n₁ sin i = n₂ sin r. Going into a denser medium (larger n), the ray bends towards the normal; into a rarer one, away from it.

Apparent depth: a coin under water looks raised. Apparent depth = real depth / n. In water (n = 1.33), a coin 4 m down looks only about 3 m deep.

A glass slab of thickness t makes an object behind it look nearer by t(1 – 1/n).

Total internal reflection: when light goes from a denser to a rarer medium, the refracted ray bends away from the normal. As i grows, r reaches 90°. That value of i is the critical angle i_c:

sin i_c = n₂ / n₁, or 1/n for a medium into air.

Beyond i_c, no light escapes: all of it reflects back, with no loss. Two conditions are needed: denser to rarer, and i greater than i_c.

Examples: glass (n = 1.5) has i_c ≈ 42°. Diamond (n = 2.42) has i_c ≈ 24°, so light entering it is trapped and reflected many times: that is why diamonds sparkle.

Uses:

• Optical fibres: light bounces along a thin glass core by repeated total internal reflection, even round bends. Used for internet cables and for endoscopes to see inside the body.
• Mirages: on a hot road, air near the ground is rarer, and light from the sky is totally reflected upwards, looking like water.
• Prisms turned at 45° act as perfect mirrors in binoculars.