Understanding Physics Topics is essential for solving complex problems in many fields, including engineering and medicine.

## Refraction of Light: Laws, Examples and Refractive Index

The refraction of light on going from one medium to another takes place according to two laws which are known as the laws of refraction of light. These are given below.

1. According to the first law of refraction of light : The incident ray, the refracted ray and the normal at the point of incidence, all lie in the same plane. For example, in Figure, the incident ray AO, the refracted ray OB, and the normal ON, all lie in the same plane (which is the plane of the paper here).

2. The second law of refraction gives a relationship between the angle of incidence and the angle of refraction. This relationship was discovered by Snell experimentally in 1621, so the second law of refraction is called Snell’s law of refraction. According to Snell’s law of refraction of light: The ratio of sine of angle of incidence to the sine of angle of refraction is constant for a given pair of media (such as ‘air and glass’ or ‘air and water’). That is :

This constant is called refractive index. We will now discuss the refractive index in somewhat detail.

Suppose a ray of light travelling in air enters into another medium and gets refracted. Let the angle of incidence in air be i and the angle of refraction in that medium be r (see Figure). The value of the constant \(\frac{\sin i}{\sin r}\) for a ray of light passing from air into a particular medium is called the refractive index of that medium. The refractive index is usually denoted by the symbol n. So :

Refractive index, n = \(\frac{\sin i}{\sin r}\)

where sin i = sine of the angle of incidence (in air)

and sin r = sine of the angle of refraction (in medium)

Suppose the angle of incidence (i) for a ray of light in air is 37° and the angle of refraction (r) in glass be 24°. Then:

Refractive index of glass, n = \(\frac{\sin 37^{\circ}}{\sin 24^{\circ}}\)

Now, if we look up the sine tables we will find that the value of sin 37° = 0.60 and sin 24° = 0.40.

Putting these values in the above relation, we get :

n = \(\frac{0.60}{0.40}\)

n = 1.50

Thus, the refractive index of this glass is 1.50.

Please note that since the refractive index is a ratio of two similar quantities (the sines of angles), it has no units. It is a pure number. The refractive index of a substance does not depend on the angle of incidence. When we talk of refractive index of a substance, say glass, we mean the value of \(\frac{\sin i}{\sin r}\) for light passing from air to glass. Strictly speaking, it should mean the value of \(\frac{\sin i}{\sin r}\) for light passing from vacuum to glass. But the difference in using air in place of vacuum is so small that it is ignored.

When light rays go from air (or vacuum) into another medium, then they bend (or refract) to some extent. Now, some media bend the light rays more than the others. The refractive index of a medium gives an indication of the light-bending ability of that medium. For example, the refractive index of glass is greater than the refractive index of water, therefore, the light rays bend more on passing from air into glass than from air into water. We will now solve one problem based on refractive index.

**Example Problem.**

A beam of light passes from air into a substance X. If the angle of incidence be 72° and the angle of refraction be 40°, calculate the refractive index of substance X. (Given : sin 72° = 0.951 and sin 40° = 0.642)

**Solution.**

We know that :

Refractive index \(=\frac{\text { Sine of angle of incidence }}{\text { Sine of angle of refraction }}\)

or n = \(\frac{\sin i}{\sin r}\)

Here, Angle of incidence, i = 72°

And, Angle of refraction, r = 40°

So, n = \(\frac{\sin 72^{\circ}}{\sin 40^{\circ}}\)

We are given that sin 72° = 0.951 and sin 40° = 0.642. So, putting these values of sin 72° and sin 40° in the above relation, we get :

n = \(\frac{0.951}{0.642}\)

or n = 1.48

Thus, the refractive index of substance X is 1.48.

So far we have denoted the refractive index of a substance just by the letter n. In its full form, the refractive index n has, however, two subscripts (lower words or letters) which show the two substances or media between which the light travels. For example, the refractive index for light going from air into glass is written as _{air}n_{glass} (or _{a}n_{g} where a = air and g = glass). We will discuss this in more detail after a while.

### The relation between Refractive Index and Speed of light

The material (or substance) through which light travels is called medium. Light is refracted (or bent) in going from one medium to another because its speed changes (it slows down or speeds up). Due to this, the refractive index {n) can also be written as a ratio of speeds of light in the two media.

Look at Figure in which a ray of light AO is going from medium 1 to medium 2 as OB. The speed of light in medium 2 is different from that in medium 1. Let the speed of light in medium 1 be v_{1} and that in medium 2 be v_{2}. Now, the refractive index of medium 2 with respect to medium 1 is equal to the ratio of speed of light in medium 1 to the speed of light in medium 2.

This can be written as :

Please note that the symbol _{medium 1}n_{medium 2} or _{1}n_{2} means that it is the refractive index of medium 2 with respect to medium 1 (for the light entering from medium 1 into medium 2). We have represented the refractive index of medium 2 with respect to medium 1 by the symbol _{1}n_{2}. In some books, however, the refractive index of medium 2 with respect to medium 1 is represented by the symbol n_{21} (read as n-two-one and not as n-twenty one). We have not used this notation because we find it a bit confusing.

When light is going from one medium (other than vacuum or air) to another medium, then the value of refractive index is called relative refractive index. For example, when the light is going from water into glass, then the value of refractive index will be the relative refractive index of glass with respect to water. The relative refractive index has always ‘two subscripts’ with its symbol n which indicate the two media in which the light travels. For example, for the light going from water to glass, the refractive index is written as _{Water}n_{glass} (or _{w}n_{g}). The symbol _{water}n_{glass} means that it is the refractive index of glass with respect to water, that is, it is the refractive index of glass for light entering from water into glass.

When light is going from vacuum to another medium, then the value of refractive index is called the absolute refractive index. The absolute refractive index has only one subscript with its symbol n on its right side which indicates the name of the medium (the word vacuum is not written as a subscript). For example, for the light going from vacuum into glass, the absolute refractive index of glass is represented as n_{glass} (and not as_{Vacuum}n_{glass}). The symbol n_{glass} means that it is the refractive index of glass with respect to vacuum, that is, it is the refractive index of glass for light entering from vacuum into glass. Please note that the symbol n_{glass} is also written in short as ng.

The exact speed of light in vacuum is 2.9979 × 10^{8} m/s and that in air is 2.9970 × 10^{8} m/s. We can see that the speed of light in air is almost the same as that in vacuum, so for the purpose of determining refractive index we can also treat air as if it were vacuum. So, the refractive index of a medium (or substance) with respect to air is also considered to be its absolute refractive index. Thus, _{air}n_{glass} can also be written as n_{glass}. The absolute refractive index of a medium (or substance) is just called its refractive index.

We will now write some simplified formulae for calculating the refractive index from the given values of the speed of light in the two media. These are given below.

The ratio of speed of light in vacuum to the speed of light in a medium, is called the refractive index of that medium. That is :

Refractive index (of a medium) \(=\frac{\text { Speed of light in vacuum }}{\text { Speed of light in medium }}\)

Since the speed of light in air is almost equal to the speed of light in vacuum, so for all practical purposes we can also say that : The ratio of speed of light in air to the speed of light in a medium, is called refractive index of that medium. That is :

Refractive index (of a medium) \(=\frac{\text { Speed of light in air }}{\text { Speed of light in medium }}\)

Let us take ‘glass’ as the medium and write a relation for its refractive index. Now, the speed of light in air is 3 × 10^{8} m/s and the speed of light in common glass is 2 × 10^{8} m/s. So :

Refractive index of glass, n_{g} \(=\frac{\text { Speed of light in air }(\text { or vacuum })}{\text { Speed of light in glass }}\)

or n_{g} = \(\frac{3 \times 10^8}{2 \times 10^8}\)

or n_{g} = \(\frac{3}{2}\)

or n_{g} = 1.5

Thus, the refractive index of this glass is 1.5. By saying that the refractive index of glass is 1.5 we mean that the ratio of the speed of light in air (or vacuum) to the speed of light in glass is equal to 1.5.

Let us solve one problem now.

**Example Problem.**

Light enters from air into a glass plate having refractive index 1.50. What is the speed of light in glass ? (The speed of light in vacuum is 3 × 10^{8} m s^{-1}). (NCERT Book Question)

**Solution.**

We know that :

Refraction index of glass \(=\frac{\text { Speed of light in air (or vacuum) }}{\text { Speed of light in glass }}\)

So, 1.50 = \(\frac{3 \times 10^8}{\text { Speed of light in glass }}\)

or Speed of light in glass = \(\frac{3 \times 10^8}{1.50}\) ms^{-1}

= 2 × 10^{8} m s^{-1}

Thus, the speed of light in glass is 2 × 10^{8} m s^{-1} (or 2 × 10^{8} m s^{-1}).

The refractive index depends on the nature of the material of the medium and on the wavelength (or colour) of the light used. The value of refractive index of a substance is a characteristic property of that substance which can be used to identify it. The refractive indices of some of the common substances are given below (indices is the plural of index). These refractive index values have been obtained by using yellow sodium light of wavelength 589 run (or 5.89 × 10^{-7} m).

Refractive Index of Some Common Substances

(with respect to air or vacuum)

Please note that different types of glass have different chemical compositions due to which they have somewhat different values of refractive indices. Because of this reason no single value can be given for the refractive index of all types of glass. The refractive index of glass usually varies from 1.5 to 1.9. Another point to be noted is that diamond has a very high refractive index of 2.42.

The construction of lenses of the optical instruments like cameras, microscopes and telescopes, etc., depends on an accurate knowledge of the refractive index of glass used for making lenses. Please note that if any two media are optically exactly the same, then no bending occurs when light passes from one medium to another. In other words, if the refractive indices of two media are equal, then there will be no bending of light rays when they pass from one medium to another. The ability of a substance to refract light is also expressed in terms of its optical density.

The optical density of a substance (or medium) is the degree to which it retards (or slows down) the rays of light passing through it. A substance having higher refractive index is optically denser than another substance having lower refractive index. For example, the refractive index of one type of glass is 1.52 and that of water is 1.33. Since glass has a higher refractive index than water, therefore, glass is optically denser than water, and more bending of light rays takes place in glass than in water. From this we conclude that higher the refractive index of a substance, more it will change the direction of a beam of light passing through it.

Please note that the optical density of a substance is different from its mass density. A substance may have a higher optical density than another substance but its mass density may be less. For example, kerosene having a higher refractive index than water is optically denser than water though its mass density is less than that of water. We have been using the terms ‘rarer medium’ and ‘denser medium’ in our discussions. It actually means ‘optically rarer medium’ and ‘optically denser medium’.

We will now show that the refractive index for light going from medium 1 to medium 2 is equal to the reciprocal of the refractive index for light going from medium 2 to medium 1. Suppose we have two media : medium 1 and medium 2. We can find out the refractive index in two ways : one for the light going from medium 1 to medium 2 (as shown in Figure) and the other for light going in the reverse direction, from medium 2 to medium 1 (as shown in Figure).

Suppose the speed of light in medium 1 is υ_{1} and that in medium 2 is υ_{2}. Now :

(i) For the light going from medium 1 to medium 2 (Figure), the refractive index is given by :

\({ }_1 n_2\) = \(\frac{v_1}{v_2}\) …. (1)

(ii) And for the light going from medium 2 to medium 1 (Figure), the refractive index is given by :

\({ }_2 n_1\) = \(\frac{v_2}{v_1}\)

Let us take the reciprocal of this equation. This will give us :

\(\frac{1}{{ }_2 n_1}\) = \(\frac{v_1}{v_2}\) ….. (2)

Now, if we compare equations (1) and (2), we find that their right hand sides are equal, so their left hand sides should also be equal. Thus,

\({ }_1 n_2\) = \(\frac{1}{{ }_2 n_1}\)

This can also be written as :

_{medium 1}n_{medium 2} = \(=\frac{1}{\text { medium } 2 n_{\text {medium } 1}}\)

This means that the refractive index for light going from medium 1 to medium 2 is equal to the reciprocal of refractive index for light going from medium 2 to medium 1.

If medium 1 is air and medium 2 is glass, then the above relation can be written as :

_{air}n_{glass} = \(\frac{1}{\text { glass } n_{\text {air }}}\)

or _{a}n_{g} = \(\frac{1}{\mathrm{~g} n_{\mathrm{a}}}\)

Thus, the refractive index of glass for light going from air to glass is the reciprocal of the refractive index for light going from glass to air. Let us solve some problems now.

**Example Problem 1.**

If the refractive index of water for light going from air to water be 1.33, what will be the refractive index for light going from water to air ?

**Solution.**

Here, _{a}n_{w} = 1.33

NOW, _{w}n_{a} = \(\frac{1}{{ }_a n_w}\)

= \(\frac{1}{1.33}\)

= 0.75

**Example Problem 2.**

The refractive indices of kerosene, turpentine and water are 1.44, 1.47 and 1.33, respectively. In which of these materials does light travel fastest ? (NCERT Book Question)

**Solution.**

We know that:

Refractive index \(=\frac{\text { Speed of light in air }}{\text { Speed of light in medium }}\)

So, Speed of light m medium \(=\frac{\text { Speed of light in air }}{\text { Refractive index }}\)

It is obvious from the above relation that the speed of light will be the maximum in that medium (or substance) which has the lowest refractive index. Now, out of kerosene, turpentine and water, water has the lowest refractive index of 1.33. So, the light will have maximum speed in water or light will travel fastest in water.