Refractive index

Dip a pencil into a glass of water and it looks broken at the surface — the part underwater seems to jump sideways. That bend is refraction, and refractive index is the single number that tells you exactly how much a transparent medium will bend a ray of light passing through it. It appears the moment light crosses from one transparent substance into another: air to glass, air to water, glass to diamond. Once you know a medium's refractive index you can predict the angle a ray will take, compare how "optically dense" different substances are, and understand why a diamond sparkles far more than a piece of ordinary glass. This section builds the two equivalent definitions of refractive index — as a ratio of sines (Snell's law) and as a ratio of speeds — and uses them to solve the same kind of one-surface numericals your ICSE exam expects.

The core insight

Refractive index $\mu = c/v$ measures how much slower light travels in a medium than in vacuum, and this same number fixes how sharply a ray bends at that medium's surface through Snell's law, $\sin i/\sin r = \mu$ .

The core idea

When light crosses the boundary between two transparent media at any angle other than straight along the normal, it changes direction — it refracts. Three laws govern this bending, and ICSE expects you to state all three.

Law 1. The incident ray, the refracted ray, and the normal at the point of incidence all lie in the same plane.

Law 2 — Snell's law. For a given pair of media, the ratio of the sine of the angle of incidence to the sine of the angle of refraction is always the same constant, called the refractive index of the second medium with respect to the first:

\frac{\sin i}{\sin r} = {}_1\mu_2

where: $i$ is the angle of incidence (measured from the normal in the first medium), $r$ is the angle of refraction (measured from the normal in the second medium), and ${}_1\mu_2$ is a pure number with no units, since it is a ratio of two sines.

Law 3 — the speed definition. The same refractive index can be found from how fast light travels in each medium:

\mu = \frac{c}{v}

where: $c$ is the speed of light in vacuum (or air, to a very good approximation), and $v$ is the speed of light inside the medium. Since light always slows down on entering a denser transparent medium, $\mu$ is always greater than $1$ for light passing from air into glass, water, or any denser transparent solid or liquid. A medium in which light slows down more is called optically denser; one where light travels closer to $c$ is optically rarer. This is a statement about the speed of light, not about mass — a substance can be lighter in weight yet optically denser. Typical values: water $\mu \approx 1.33$ , ordinary glass $\mu \approx 1.5$ , diamond $\mu \approx 2.42$ — which is why diamond bends and internally reflects light so dramatically. The rule for the direction of bending follows directly: going from a rarer medium to a denser one, the ray bends towards the normal ( $r < i$ ); going from denser to rarer, it bends away from the normal ( $r > i$ ); and a ray travelling exactly along the normal ( $i = 0°$ ) passes straight through, undeviated.

Figure

A ray of light bending toward the normal as it crosses from air into glass, with the angle of incidence larger than the angle of refraction

Derivation Aof the two-medium Snell's law

The goal is to show that the two definitions of refractive index — the ratio of sines and the ratio of speeds — are not two separate facts to memorise but the same physical statement written two ways, and to derive the general form $n_1 \sin i = n_2 \sin r$ that links the absolute refractive indices of both media rather than just one relative index.

Step 1. Define the absolute refractive index of each medium using the speed definition: $n_1 = c/v_1$ for the first medium (where light travels at $v_1$ ) and $n_2 = c/v_2$ for the second medium (where light travels at $v_2$ ).

Step 2. Write Snell's law for light passing from medium 1 into medium 2, using the relative index: $\dfrac{\sin i}{\sin r} = {}_1\mu_2$ . It is a fact (not proved here, but usable) that the relative index equals the ratio of speeds in the two media, ${}_1\mu_2 = v_1/v_2$ — light travels faster in the medium it is leaving if it bends toward the normal on entry.

Step 3. Substitute the speed definitions from Step 1 into ${}_1\mu_2 = v_1/v_2$ . Since $v_1 = c/n_1$ and $v_2 = c/n_2$ , we get ${}_1\mu_2 = \dfrac{c/n_1}{c/n_2} = \dfrac{n_2}{n_1}$ .

Step 4. Combine Steps 2 and 3: $\dfrac{\sin i}{\sin r} = \dfrac{n_2}{n_1}$ , which rearranges to the symmetric, general form

n_1 \sin i = n_2 \sin r

This single equation contains the special case ${}_1\mu_2 = 1/{}_2\mu_1$ (swap which medium is "1" and the roles of $i$ and $r$ swap too, flipping the ratio), and it reduces to the everyday $\sin i/\sin r = \mu$ whenever medium 1 is air or vacuum, since then $n_1 \approx 1$ .

Worked example

A ray of light strikes the flat surface of a glass slab at an angle of incidence of $45°$ , and the refractive index of the glass is $1.5$ . Find the angle of refraction, and the speed of light inside the glass.

Step 1. Apply Snell's law for the air-to-glass surface, treating air's refractive index as $1$ : $\dfrac{\sin i}{\sin r} = \mu$ , so $\sin r = \dfrac{\sin i}{\mu} = \dfrac{\sin 45°}{1.5} = \dfrac{0.707}{1.5} = 0.471$ .

Step 2. Take the inverse sine: $r = \sin^{-1}(0.471) \approx 28.1°$ . Since air is rarer than glass, the ray bends towards the normal, and indeed $28.1° < 45°$ , as the bending rule predicts.

Step 3. Find the speed of light inside the glass using $\mu = c/v$ , with $c = 3 \times 10^8\ \text{m/s}$ : $v = \dfrac{c}{\mu} = \dfrac{3 \times 10^8}{1.5} = 2.0 \times 10^8\ \text{m/s}$ .

Answer: the angle of refraction is about $28.1°$ , and light travels at $2.0 \times 10^8\ \text{m/s}$ inside this glass — noticeably slower than its speed of $3 \times 10^8\ \text{m/s}$ in air, which is exactly why the ray bends toward the normal on entry. (This is the same speed found for a liquid in the reference article's second example, whose $\mu = 1.5$ makes it optically denser than water, $\mu_{water} = 1.33$ .)

Common traps

Confusing "optically denser" with "physically denser" (higher mass density) — the two need not match; optical density is about the speed of light, not weight.
Writing Snell's law upside down: $\sin i/\sin r = \mu$ means $\mu$ multiplies $\sin r$ , i.e. $\sin i = \mu \sin r$ . Dividing the wrong way gives $\mu < 1$ for a rarer-to-denser transition, which is impossible.
Forgetting that refractive index depends on the pair of media, and that ${}_1\mu_2 = 1/{}_2\mu_1$ — glass-with-respect-to-air is not the same number as air-with-respect-to-glass.
Thinking a ray along the normal still "refracts" — at $i = 0°$ , $r = 0°$ too, so there is no bending, only a small amount of slowing down.

Ask the tutor: Want to see how refractive index also explains total internal reflection and why a diamond sparkles so much more than glass?

Why does refractive index have no units at all?

Because it is defined as a ratio of two quantities of the same kind — either two speeds ( $c/v$ ) or two sines ( $\sin i/\sin r$ ). Any units in the numerator and denominator cancel exactly, leaving a pure number. That is also why refractive index is such a convenient way to compare media: a value of $1.5$ means the same thing regardless of which unit system you measured the speeds in.

Is refractive index the same thing as optical density?

They describe the same underlying idea from two angles. A medium with a higher refractive index is said to be optically denser, meaning light travels more slowly inside it and bends more sharply on entering it from a rarer medium. But "optically denser" is not the same as physically denser (higher mass density) — kerosene, for instance, can have a mass density lower than water while still bending light comparably or more strongly, so never assume the two densities move together.

Why does a pencil look bent when I dip it halfway into water?

Light from the submerged part of the pencil refracts as it leaves the water and enters your eye through air, bending away from the normal at the water surface because it is going from a denser medium (water) to a rarer one (air). Your eye and brain assume light always travels in straight lines, so they trace the bent ray backward along its final straight path and perceive the submerged part of the pencil as displaced from where it actually is. This same refraction is what makes a pool look shallower than it really is.

Refractive index

The core idea

Worked example

Common traps

Before this

Where this leads