Calculate refractive index from light speeds
Define the incident-to-transmitted speed ratio and preserve which region is the reference.
Section P3.2.2 · Refraction of light
Cross-referenced to the 2028–2029 0653 section using original teaching from the related single-science courses. A 0653-specific objective-by-objective and Core/Supplement review is in progress. Written practice is self-checked and does not demonstrate supervised laboratory competence or award an official Cambridge grade.
Before you begin
- Separate speed from frequency at an interface.
- Rearrange a ratio and cancel consistent units.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which ratio defines a vacuum-reference material index?
What you will learn
- Define refractive index as a ratio of wave speeds.
- Calculate speed and index for stated transparent regions.
Identify the reference region
For the index of a material relative to vacuum, n = c₀/v, where c₀ is the vacuum light speed and v is the material light speed for the light considered. An index is a dimensionless ratio because both speeds use the same unit. Ordinary glass with supplied speed 2.0 × 10⁸ m/s and vacuum speed 3.0 × 10⁸ m/s has n = 1.50.
More generally the refractive index of the second region relative to the first is v₁/v₂. Name that order: reversing it gives the reciprocal relative index. 'Optically denser' refers to higher index and lower light speed, not a claim that mass density alone determines the speed.
Rearrange and check the interpretation
For a vacuum-reference index, v = c₀/n. With n = 1.25 and c₀ = 3.0 × 10⁸ m/s, v = 2.4 × 10⁸ m/s. A larger index gives a smaller speed at a specified frequency. At a stationary interface frequency stays unchanged; wavelength changes with speed rather than the source acquiring a new frequency.
Use the supplied reference speed and consistent units before dividing. If the incident and transmitted speeds are 2.4 × 10⁸ and 1.8 × 10⁸ m/s, the second-relative-to-first index is 4/3, about 1.33. It is not automatically the second material's vacuum-reference index. Material index can depend on frequency, so a supplied value applies to the stated light.
Sources for this lesson
Pause and explain
For c₀ = 3.0 × 10⁸ m/s and v = 2.4 × 10⁸ m/s, what is n?
Worked example
Light speed is 2.0 × 10⁸ m/s in a material. Use c₀ = 3.0 × 10⁸ m/s to find its vacuum-reference index.
Show the worked solution
- Use n = c₀/v.
- Cancel the common factor 10⁸ and the speed units.
- n = 3.0/2.0 = 1.50.
Answer n = 1.50, with no unit.
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find n for v = 2.0 × 10⁸ m/s and c₀ = 3.0 × 10⁸ m/s.
Give me a hint
Keep the vacuum reference in the numerator.
Compare my reasoning
- n = c₀/v.
- n = 3.0/2.0.
- The speed units cancel.
n = 1.50.
Look for these in your work
- Use the correct ratio.
- Give a dimensionless answer.
Find material speed for n = 1.60 and c₀ = 3.0 × 10⁸ m/s.
Give me a hint
Rearrange n = c₀/v before substituting.
Compare my reasoning
- v = c₀/n.
- v = 3.0 × 10⁸/1.60.
- v = 1.875 × 10⁸ m/s.
1.875 × 10⁸ m/s, approximately 1.88 × 10⁸ m/s.
Look for these in your work
- Rearrange correctly.
- Retain the speed unit.
Speeds in regions 1 and 2 are 2.4 × 10⁸ and 1.8 × 10⁸ m/s. Find the second-relative-to-first index and its reciprocal.
Give me a hint
State the region order before forming each ratio.
Compare my reasoning
- n₂ relative to 1 = v₁/v₂ = 2.4/1.8 = 4/3.
- The reverse ratio is v₂/v₁ = 3/4.
- Neither ratio alone gives both vacuum-reference indices.
Second relative to first 4/3; reverse 3/4.
Look for these in your work
- Name both reference directions.
- Avoid an unstated vacuum assumption.
Common mistakes
- Name the index's reference region.
- Index is dimensionless.
- Higher index means lower speed for the stated light.
What can you explain now?
Is a second-relative-to-first index always a vacuum-reference index?
Compare with the explanation
No.
That occurs only when the first reference is vacuum or a stated air approximation.
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Make it stick
Define refractive index with its reference, rearrange the speed ratio and explain its units.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.