Index of Refraction Formula: n = c/v, Snell's Law and Examples

The index of refraction formula is $n = \frac{c}{v}$ : divide the speed of light in vacuum, c = 3.00 × 10⁸ m/s, by the speed of light in the material. The result tells you how many times slower light moves inside that material. Water has n ≈ 1.33, so light crawls along at about three quarters of its vacuum speed there.

Indian books call the same number the refractive index and often write it as μ. It also appears in Snell's law, $n = \frac{\sin i}{\sin r}$ for light going from air into a medium, which is the form most numericals use. Being one speed divided by another, the refractive index carries no unit. It is the bridge between wave speed and ray optics in the waves and optics chapter.

Quick facts

Formula
$n = \frac{c}{v}$
From angles
$n = \frac{\sin i}{\sin r}$ (light entering from air or vacuum)
SI unit
None: a ratio of two speeds
Dimensional formula
[M⁰ L⁰ T⁰]
Range
n ≥ 1 for ordinary materials; air is about 1.0003
Relative index
$n_{21} = \frac{n_2}{n_1} = \frac{v_1}{v_2}$
Symbols
n in US books, μ or n in Indian books

Formula of refractive index, symbol by symbol

Quantities in n = c / v
Symbol	Name	Unit
$n$	Absolute refractive index of the medium	no unit
$c$	Speed of light in vacuum, 3.00 × 10⁸ m/s	m/s
$v$	Speed of light in the medium	m/s
$i$	Angle of incidence, measured from the normal	degree
$r$	Angle of refraction, measured from the normal	degree

A larger n means a slower speed and a stronger bend toward the normal. The word for that is optically denser. Optical density is not the same as mass density: some oils are less dense than water yet have a higher refractive index.

Absolute and relative refractive index

The absolute index compares a material with vacuum. The relative index compares two materials directly. For light going from medium 1 into medium 2:

$n_{21} = \frac{n_2}{n_1} = \frac{v_1}{v_2} = \frac{\sin i}{\sin r}$

Snell's law in its general form is $n_1 \sin i = n_2 \sin r$ . Set $n_1 = 1$ for air and you get back $n = \frac{\sin i}{\sin r}$ . The relative index of glass with respect to water, for example, is 1.50 ÷ 1.33 ≈ 1.13.

Typical refractive indices for yellow light
Material	n	Speed of light inside
Air	1.0003	about 3.00 × 10⁸ m/s
Water	1.33	2.26 × 10⁸ m/s
Crown glass	1.52	1.97 × 10⁸ m/s
Diamond	2.42	1.24 × 10⁸ m/s

Worked examples with the index of refraction formula

Example 1: speed in glass

Light travels through a glass block at 2.0 × 10⁸ m/s. Find the refractive index of the glass.

$n = \frac{c}{v} = \frac{3.00 \times 10^{8}}{2.0 \times 10^{8}} = 1.5$

No unit: the m/s cancels.

Example 2: bending at a water surface

A ray in air strikes still water at 45° to the normal. The refractive index of water is 1.33. Find the angle of refraction.

$\sin r = \frac{\sin 45^\circ}{1.33} = \frac{0.7071}{1.33} = 0.5317$

$r = \sin^{-1}(0.5317) \approx 32.1^\circ$

The ray bends toward the normal because water is optically denser.

Example 3: an exam style question with the wavelength

Yellow light of wavelength 589 nm in air enters glass with n = 1.50. Find its speed, frequency and wavelength inside the glass.

Speed: v = c / n = 3.00 × 10⁸ ÷ 1.50 = 2.00 × 10⁸ m/s.

Frequency (same in both media): f = c / λ = 3.00 × 10⁸ ÷ (589 × 10⁻⁹) = 5.09 × 10¹⁴ Hz.

Wavelength in glass: λ / n = 589 ÷ 1.50 = 393 nm.

Other ways the refractive index is found

Questions do not always give speeds or angles. Three other routes come up often.

Real and apparent depth. Looking straight down into a liquid, $n = \frac{\text{real depth}}{\text{apparent depth}}$ . A pool 2.0 m deep filled with water looks only 2.0 ÷ 1.33 ≈ 1.50 m deep.
Critical angle. For light leaving a denser medium into air, $n = \frac{1}{\sin C}$ . Glass of n = 1.5 has C ≈ 41.8°, which is why prisms in binoculars reflect light totally at 45°.
Prism at minimum deviation. $n = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$ , covered on the angle of deviation page.

The refractive index also sets the focal length of a lens through the lens maker formula. A lens made of higher index glass bends light more and so has a shorter focal length for the same shape.

Why it has no unit, and what the speed change means

Both c and v are speeds in m/s, so the units cancel and every exponent in [M⁰ L⁰ T⁰] is zero. The quantity is a pure number, just like a strain or a coefficient of friction.

When light slows down its frequency stays the same, so its wavelength must shrink by the same factor n. That is why colours do not change under water. The speed of light formula explains where the vacuum value comes from. In most materials n changes a little with colour; violet light sees a slightly larger n than red light. That small difference is dispersion, the effect that spreads white light into a spectrum.

Errors examiners see often

Measuring angles from the surface. Always measure i and r from the normal line drawn at the point where the ray strikes.
Writing n = v / c. That gives a number below 1 for every material, which should warn you straight away.
Changing the frequency on entry into the new medium. Only speed and wavelength change.
Forgetting the calculator mode. Use degrees, not radians, when you take sin 45°.

Sanity check: if your answer for n is below 1, or above about 2.5 for a common material, look again at the fraction.

Frequently asked questions

What is the formula of refractive index?

Refractive index is n = c / v, the speed of light in vacuum divided by its speed in the medium. Using angles, n = sin i / sin r for light entering from air.

What is the SI unit of refractive index?

None at all. One speed divided by another leaves the metres per second cancelled out.

Can the refractive index be less than 1?

For ordinary transparent materials and visible light, no: light is always slower in them than in vacuum, so n is at least 1. Air comes closest, at about 1.0003.

What is the difference between absolute and relative refractive index?

The absolute index compares a medium with vacuum. The relative index compares two media, n₂₁ = n₂ / n₁, and it is the number that goes into Snell's law between those two media.

Does refractive index depend on the colour of light?

Yes, slightly. In glass, violet light has a slightly higher index than red light, so it bends more. This is dispersion, which a prism uses to split white light.