Relative refractive index

The idea

The refractive index n21 of medium 2 with respect to medium 1, the constant sin i / sin r, equals v1/v2, the ratio of the speeds of light in medium 1 and medium 2; likewise n12 = v2/v1.

A number for the bend

The speed of light is different in different media. A scientist wants one number that says how much a ray bends when it goes from one medium into another.

What could this number be built from?

• The speeds of light in the two media
• The masses of the two media
• The colours of the two media

Pairs of media

The speeds are worked out from Table 9.3. Pick a pair, then compare n₂₁ = v₁ ÷ v₂ with sin i ÷ sin r. Choose water to air and then glass to water.

A ratio of speeds

For every pair, sin i ÷ sin r came out equal to v₁ ÷ v₂. Reversing the trip turned the ratio upside down.

The refractive index of medium 2 with respect to medium 1, n₂₁, is the constant sin i ÷ sin r. It equals the ratio of the speeds of light: n₂₁ = v₁ ÷ v₂, where v₁ is the speed in medium 1 and v₂ the speed in medium 2. In the same way, n₁₂ = v₂ ÷ v₁.

Notes

n₂₁ = sin i ÷ sin r = v₁ ÷ v₂, the ratio of the speeds of light in medium 1 and medium 2. Likewise n₁₂ = v₂ ÷ v₁.

Check yourself

Light goes from medium 1 into medium 2. What is n₂₁?

In medium 1 light travels at 3×10⁸ m s⁻¹ and in medium 2 at 2×10⁸ m s⁻¹ (an example). What is n₂₁?

Answer: 1.5

n₂₁ = v₁ ÷ v₂ = 3×10⁸ ÷ 2×10⁸ = 1.5.

For air to water n₂₁ = 1.33. What is n₁₂, for water to air? Give your answer to two decimal places.

Answer: 0.75

n₁₂ = v₂ ÷ v₁ = 1 ÷ n₂₁ = 1 ÷ 1.33 = 0.75.

The constant sin i ÷ sin r for light going from medium 1 into medium 2 equals

• The speed of light in medium 2 divided by its speed in medium 1. That is n₁₂, the refractive index of medium 1 with respect to medium 2.
• The speed of light in medium 1 divided by its speed in medium 2 — correct. Yes!
• The speed in medium 1 multiplied by the speed in medium 2. The refractive index is a ratio of the two speeds, found by dividing, not multiplying.
• v₂ ÷ v₁, the speed in medium 2 over the speed in medium 1. That ratio is n₁₂. The constant sin i ÷ sin r for medium 1 to medium 2 is n₂₁ = v₁ ÷ v₂.
• i ÷ r, the ratio of the angles. The constant is the ratio of the sines of the angles, and it equals the ratio of the speeds.
• v₁ ÷ v₂, the ratio of the speeds in medium 1 and medium 2 — correct. Yes!