Refractive Index: Formula, Laws & Refractive Gradient

Refractive index can be defined as the ratio of the speed of light in a vacuum to the speed in a particular medium. The Refractive index is also called the Refraction Index or Index of Refraction. The formula of Refractive Index is:

Here,

• n = refractive index
• c = velocity of light in a vacuum ( 3 × 10⁸ m/s)
• v = velocity of light in a substance

The refractive index is a dimensionless quantity. It can be expressed as a number which denotes the number of times a light wave in a material would be slower than when it is in a vacuum. Represented by the symbol n, it is the velocity of light in a vacuum divided by the velocity of light in a medium. 

Read Also: Light-Reflection and Refraction Revision Notes

Key Terms: Refraction, Rarefraction, Velocity of Light, Vacuum, Refractive Gradient, Electromagnetic Energy, Vector Point Function

---

What is Refraction?

[Click Here for Sample Questions]

Refractive Index is:

• The Refractive index is also known as the Refraction Index or Index of Refraction. 
• The speed of light in a medium depends on the medium’s properties.
• The speed depends on the optical density of the medium in the case of electromagnetic waves.
• Optical density can be expressed as the tendency of atoms in a material to revive the absorbed electromagnetic energy.
• Thus, the more optically dense a material is going to be, the slower will be its speed of light.
• An indicator of the optical density of a medium is the refractive index.

Experiment of Refractive Index

In an experiment, put a pencil in a glass of water, and look from different angles. The pencil will seem to be broken or curved. This happens because a ray of light changes its direction when passing from one medium to another.

The shape of the pencil dipped in water will look different if you look from the top of the glass, directly into the water versus the side of the glass. This phenomenon is known as Refraction.

The phenomenon of Refraction follows two aspects:

• Speed of the light in the medium
• Angle of Refraction

Also Read:

---

Speed of Light Calculation

[Click Here for Sample Questions]

In order to calculate the refractive index, the speed of light can be considered in two mediums.

Let,

• Speed of Light in the 1st medium = v₁
• Speed of light in the 2nd Medium = v₂

Thus, the refractive index (n) of 2nd medium with respect to the 1st medium can be expressed as:

= \(n_{21} = \frac {Speed\ of\ Light\ in\ 1st\ Medium}{Speed\ of\ Light\ in\ 2nd\ Medium}\)

Or, we can also state,

n₂₁ = v₁/v₂                                   

Thus, the refractive index (n) of 1st medium with respect to 2nd medium can be expressed as:

= \(n_{12} = \frac {Speed\ of\ Light\ in\ 2nd\ Medium}{Speed\ of\ Light\ in\ 1st\ Medium}\)            

Or, we can also state,

n₁₂ = v₂ / v₁

Read More: Transmission, Absorption, and Reflection of Light

---

Refractive Index Formula

[Click Here for Sample Questions]

Refraction occurs because of the difference in the Speed of light in every medium. The ratio between the Speed of light in medium to speed in a vacuum is called the refractive index.

• Light travels the fastest in a vacuum with the highest Speed of 3×10⁸ m/s.
• In the air, the Speed of light is only marginally less than that in a vacuum; it reduces significantly in glass or water.
• The refractive index indicates the number of times slower a light wave would be in the material than in a vacuum. 
• The refractive index, represented by the symbol n, is the velocity of light in a vacuum divided by the velocity of light in a medium.

The formula for calculating the Absolute Refractive Index is as follows:

Where,

• n denotes the refractive index
• c denotes the velocity of light in a vacuum (3 × 10⁸ m/s)
• v denotes phase velocity of light in a medium

Refractive Index

Refractive Index Gradient

The refractive index gradient can be expressed as the rate of change of the refractive index with respect to distance in the material. Herein, Distance is the slope of the refractive index profile at any given point.

• The refractive index gradient can be represented in terms of the reciprocal of a unit of distance.
• The Refractive index gradient example includes the rate of change of the refractive index at any point with respect to distance.
• The refractive index gradient is known to be a vector point function.

---

Relative Refractive Index

[Click Here for Sample Questions]

The vacuum has a refractive index of 1. The higher the refractive index, the higher the optical density, and the slower is the Speed of light. Thus, n = Speed of the light in medium 1/Speed of light in medium 2.

Below are the refractive index of different mediums and materials:

Optical Density

The optical density is used to express the ability of a medium to refract light. The optical density is directly proportional to the refractive index and inversely proportional to the Speed of light for a given medium.

• Hence, the denser the medium, the lesser the Speed of light as a denser medium has a more considerable refractive index value.
• Similarly, the Larger the refractive index is, the optically denser the medium than the other.
• The other medium of lower refractive index is optically rarer.
• We should keep in mind that the Speed of light is always higher in a rarer medium than in a denser medium.

Note: Optical density is not the same as mass density. For example, if you look at the table above, Kerosene has a higher refractive index than water; hence it is optically denser than water. However, its mass density is less than water.

Also check:

---

Laws of Refraction

[Click Here for Sample Questions]

According to Laws of Refraction:

• The incident ray, the refracted ray, and the normal interface of two transparent media at the point of incidence all lie in the same plane.
• The ratio of the sine of the angle of incidence to the sine of the angle of Refraction is constant for the light of a given colour and the given pair of media. (This law is also known as Snell's Law)

In the above figure, the angle of incidence is the angle between the incident ray and the normal, denoted as 'i'. The angle between the refracted ray and the normal is the Refraction, denoted as 'r' is.

If i is the angle of incidence and r is the angle of Refraction, then,

Note: This constant value "n" is the refractive index of the second medium for the first.

Snell’s Law

As per Snell’s Law:

• Whenever a ray of light or waves passes through the air, water, or glass, then it can be used to find the relationship between the angle of incidence and the angle of refraction.
• Snell’s law also states that for normal to the boundary between the two transparent objects, then the refracted way and the ray of incidence will lie on the same plane.

Thus, it is also similar to the ratio of indices of refraction. It can be shown as: \(\frac{Sin \theta_2}{Sin \theta_1} = \frac{v_2}{v_1} = \frac{n_1}{n_2}\).

Where,

• θ = Calculated as a line perpendicular to the boundary of the surface (normal)
• v - Velocity of light with respect to the medium
• n - Refractive index with respect to the medium

Read More: Refraction by Spherical Lenses

---

Things to Remember

• The Refractive index can be expressed as the ratio of the speed of light in a vacuum to the speed in a particular medium.
• The Refractive index can also be called the Refraction Index or Index of Refraction.
• The Refractive Index formula is \(\begin{array}{l}n=\frac{c}{v}\end{array}\).
• The refractive index gradient is the rate of change of the refractive index with respect to distance in the material.
• The refractive index gradient is a vector point function.

Read More: Refraction of Light