## What is Snell’s Law and Its Derivation

- Frank
- May 25, 2022
- read:

Snell’s law depends on the law of refraction because it can predict the amount of bend of the light ray. The law of refraction is nothing but bending of a light ray when it is traveling between two different mediums like water or glass or air etc (from one medium to another type of medium). This law gives the relation between the angle of incident ray (light) and angle of transmitted ray (light) when they interface at the two different media. The phenomenon law can be observed in all types of materials, matter especially in fiber optic cables. Willebrord Snell’s recognized law of refraction in 1621 and later named it as Snell’s law. It can calculate the speed of light and refractive index when the material or light ray interface at two different mediums through a boundary line. This article describes the complete Snell’s law worksheet.

### What is Snell’s Law?

**Definition:** Snell’s law is also called as law of refraction or Snell’s Descartes. It is defined as the ratio of sines of the angle of incidence refraction equal to the reciprocal ratio of refractive indices or phase velocities when the light ray travels from one medium to another type of medium. It gives the relation between the angle of incidence and angle of refraction when the light ray travels between two isotropic media. Also, the angle of incidence ray and the angle of refraction is constant.

### Snell’s Law Formula

The formula of Snell’s law is,

**Sin α1/ Sine α2 = V1/V2**

or

**Sin α1 / Sine α2 = n2/ n1**

or

**Sin i/sine r = constant = c**

Here constant refers to the refractive indices of two mediums

Where α1 = angle of incidence ray

α2 = angle of refraction

V1 and V2 = phase velocities of two different media

n1 and n2 = refractive indices of two different media

### Snell’s Law Equation

This equation gives the relation between the angle of incidence and angle of transmission equal to the refractive index of each medium. It is given as,

**Sin α1 / Sin α2 = n2/ n1**

Here ‘α1’ measures the angle of incidence

‘α2’ measures the angle of refraction

‘n1’ measures the refractive index of the first medium

‘n2’ measures the refractive index of the second medium.

### Derivation

Basically, **Snell’s law derivation** is derived from Fermat’s principle. Fermat’s principle is defined as the light travels in the shortest path with a small amount of time. Consider the constant light ray travels from one medium to another medium via a given normal line or boundary line as shown in the figure.

When the light ray crosses the boundary line it is refracted with a smaller or greater angle. The angles of incidence and refraction are measured with respect to the normal line.

As per this law, these angles and refractive indices can be derived from the following formula.

**Sin α1 / Sin α2 = n2/ n1**

The speed of the light depends on the refractive index of two mediums

**Sin α1/ Sin α2 = V1/V2**

Where ‘α1’ and ‘α2’ are the angles of incidence and refraction.

‘n1’ and ‘n2’ are the refractive indices of the first and second medium

‘V1’ and ‘V2’ determine the speed or velocity of the light ray.

#### Refraction

**Snell’s law of refraction** takes place when the speed of the light ray changes while passing from one medium to another medium. This law can also be called Snell’s law of refraction. It occurs when the speed of the light varies while traveling through the two different mediums.

Consider the two different mediums air and water. When the light travels from the first medium (air) to the second (water) medium, the light ray is refracted towards or away from the interface (normal line). The angle of refraction depends on the relative refractive index of the two mediums. The angle of refraction is high when the light ray propagates away from the normal. When the refractive index of the second material is higher than the refractive index of the first material, then the refracted ray propagates towards the normal and the angle of refraction is small. This gives the total internal reflection.

That means, when light ray travels from lower medium to higher medium, it bends towards the normal with respect to the interface. The refractive index of the material depends on the wavelength. If the wavelength is high, the refractive index would be low. The refractive index can be varied from one medium to another medium. For example, vacuum=1, air= 1.00029, water= 1.33, glass= 1.49, alcohol= 1.36, glycerine= 1.4729, diamond= 2.419.

The speed of the light ray propagates from one medium to other medium changes and depends on the refractive index of the material used. So, the refraction of this law can determine the speed of the refracted ray from the interface surface. Finally, it is observed that snell’s law of refraction can be applied to any type of material or medium.

#### Example

Snell’s law examples can be mostly observed in fiber optic cables, in all matters and materials. It is used in optical devices like eyeglasses, cameras, contact lenses, and rainbows.

The most important example is the refractometer instrument, which is used to calculate the refractive index of liquids.

The theory of snell’s law is used in telecommunication systems and data transmission systems with high-speed servers.

#### Snell’s Law Worksheet

Find the angle of incidence, if the refracted ray is at 14 degrees, the refractive index is 1.2.

Angle of refraction sine 1 = 14 degrees

Refractive index c = 1.2

From the snell’s law,

Sin i / sin r = c

Sin i / sin 14 = 1

Sin i = 1.2 x sin 14

Sin i = 1.2 x 0.24 = 0.24

Hence i = 16.7 degrees.

Find the refractive index of the medium if the incidence angle is 25 degrees and refraction angle is 32 degrees

Given sin i = 25 degrees

Sin r = 32 degrees

Constant refractive index = c = ?

From Snell’s law,

Sin i / sin r = c

Sin25/sin32 = c

C= 0.4226

Find the angle of refraction if the angle of incidence is 45 degrees, the refractive index of the incident ray is 1.00 and refractive index of the refracted ray is 1.33

Given sin α1= 45 degrees

n1= 1.00

n2= 1.33

Sin α2=?

From the snell’s law,

n1 sin α1 = n2 sin α2

1 x sin (45 degrees) = 1.33 x sin α2

0.707 = 1.33 x sin α2

Sin α2 = 0.53

α2 = 32.1 degrees

Thus, this is all about an overview of snell’s law – definition, formula, equation, derivation, refraction, and worksheet. Here is a question for you, “What are the advantages and disadvantages of Snell’s law of refraction?”