Refractive Index and Snell's Law
Refraction is one topic where the definition and the calculation are easy to keep separate in your head but hard to connect on paper. This guide gives both formulas — refractive index in terms of speed, and Snell's law in terms of angles — and works four numericals: finding an angle of refraction, finding the speed of light inside a medium, finding a critical angle, and comparing two media directly.
What refractive index actually measures
where c is the speed of light in vacuum (3 × 10⁸ m/s) and v is the speed of light inside the medium. A higher n means light travels slower in that medium — glass (n ≈ 1.5) slows light down more than water (n ≈ 1.33), which is why glass bends light more sharply.
Snell's law
θ₁ is the angle of incidence and θ₂ is the angle of refraction, both measured from the normal — the line perpendicular to the surface, never from the surface itself. n₁ and n₂ are the refractive indices of the medium the light is leaving and the medium it is entering. Rearranged for the refraction angle:
Critical angle and total internal reflection
When light travels from a denser medium into a rarer one (glass into air, for instance), there is one particular angle of incidence — the critical angle C — at which the refracted ray grazes along the surface (θ₂ = 90°). Beyond C, no light refracts out at all; it reflects entirely back into the denser medium.
Worked example 1 — angle of refraction, air into glass
Light hits a glass surface (n = 1.5) from air (n = 1) at an angle of incidence of 30°. Find the angle of refraction.
sinθ₂ = (n₁/n₂) × sinθ₁ = (1/1.5) × sin30° = (1/1.5) × 0.5
1 ÷ 1.5 = 0.6667, so sinθ₂ = 0.6667 × 0.5 = 0.3333
Using a sine table, sin19° = 0.3256 and sin20° = 0.3420. Interpolating: (0.3333 − 0.3256) ÷ (0.3420 − 0.3256) = 0.0077 ÷ 0.0164 ≈ 0.47, so θ₂ ≈ 19° + 0.47° = 19.5°. The ray bends toward the normal, exactly as expected going from a rarer medium (air) into a denser one (glass).
Worked example 2 — speed of light in water, and its refraction angle
Water has n = 1.33. First, its speed of light: v = c/n = (3 × 10⁸) ÷ 1.33.
3 ÷ 1.33 = 2.2556, so v = 2.256 × 10⁸ m/s — about 75% of light's speed in vacuum.
Now find the angle of refraction for light entering water at 45° from air: sinθ₂ = (1/1.33) × sin45° = 0.7519 × 0.7071 = 0.5317
From a sine table, sin32° = 0.5299 and sin33° = 0.5446. Interpolating: (0.5317 − 0.5299) ÷ (0.5446 − 0.5299) ≈ 0.12, so θ₂ ≈ 32° + 0.12° = ≈ 32.1°. A 45° incidence angle refracts down to about 32° inside water.
Worked example 3 — critical angle for a glass-air surface
Glass has n = 1.5. Find the critical angle for light travelling from glass into air.
sinC = nair / nglass = 1 / 1.5 = 0.6667
From a sine table, sin41° = 0.6561 and sin42° = 0.6691. Interpolating: (0.6667 − 0.6561) ÷ (0.6691 − 0.6561) ≈ 0.82, so C ≈ 41° + 0.82° = ≈ 41.8°. Any ray inside the glass hitting the surface at more than about 41.8° reflects entirely back in — this is the principle behind optical fibres and the sparkle of a cut diamond (diamond's much higher n gives it a far smaller critical angle, around 24°, trapping far more light inside).
Worked example 4 — relative refractive index between two media
Find the refractive index of glass (n = 1.5) relative to water (n = 1.33) — i.e. how much glass bends light compared to water, not compared to vacuum.
wng = nglass / nwater = 1.5 ÷ 1.33 = 1.128
Check by multiplying back: 1.33 × 1.128 = 1.500 ✓. A relative index greater than 1 means glass is optically denser than water — light entering glass from water still bends toward the normal, just by a smaller amount than it would coming from air.
Common mistakes that cost marks
- Measuring the angle from the surface instead of the normal. Snell's law and the critical-angle formula both use the angle from the perpendicular — a question that states "the ray grazes the surface at 20°" means the angle from the normal is 70°.
- Inverting the ratio in Snell's law. sinθ₂ = (n₁/n₂)sinθ₁, not (n₂/n₁)sinθ₁ — writing it upside down gives an angle larger than the incidence angle when entering a denser medium, which is physically wrong.
- Using sinC = 1/n when the second medium is not air. The "1/n" shortcut only works because nair ≈ 1; between two other media use nrarer/ndenser.
- Thinking a higher n means light moves faster. It is the opposite — n = c/v, so a larger n means a smaller v.
- Forgetting total internal reflection only happens going from denser to rarer. Light travelling from a rarer into a denser medium always refracts; there is no critical angle in that direction.
Where refraction appears in exams
| Exam | Typical use |
|---|---|
| CBSE Class 10 | Light — reflection and refraction: Snell's law, refractive index of common media |
| CBSE Class 12 | Ray optics — critical angle, total internal reflection, optical fibres, prisms |
| JEE Main & Advanced | Multi-surface refraction, apparent depth, and combined lens-refraction numericals |
| NEET | Direct single-surface Snell's law and critical-angle substitutions |
Refraction numericals are mostly ratio and sine-table work. Use the Scientific Calculator's trigonometric keys to check a division like 1.5 ÷ 1.33 or a sine value quickly while you practise.
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