Snell's Law: Definition, Derivation, Formula, Examples & Application

Snell's Law

Snell's Law: Snell’s law relates the degree of refraction and the relation between the angle of incidence, and the angle of refraction in a given pair of media. As we know light experiences refraction or bending when it travels from one medium to another medium. Snell’s law anticipates the degree of the bend. It is commonly referred to as the law of refraction. Willebrord Snell formulated the concept of the law of refraction In 1621, Willebrord Snell discovered the law of refraction, hence called Snell’s law.

According to Snell's Law," The ratio of the angle of Incidence and angle of refraction is constant". This can be written as:

Sin i/Sin r = constant = µ

Where, 

Sin i is the angle of incidence

Sin r is the angle of refraction

µ is the resultant constant, that is, the refractive index

Snell's Law Formula

Snell's law is derived from Fermat Principle. The Fermat principle says, “ Light travels the smallest path and takes very less time to travel”. The formula for Snell's law is-

n1Sinϴ1 = n2Sinϴ2

Where,

Sinϴ1 is the angle in correspondence to the incidence 

Sinϴ2 is the  angle in correspondence to refraction

Snell's Law Derivation

Snell's law is derived from Fermat Principle. As per the equation, two mediums and the phase velocities of these are characterized as:

v1 = c/n1 and v2 = c/n2

 ‘c’ is the speed of light

T = [(√x2 + a2)/v1] + [(√b2 + (l-x)2)/v2]

= [(√x2 + a2)/v1] + [(√b2+l2-2lx+x2/v2]

Here, dT/dx = [x/(v1√x2 + a2)] + [-(l-x)/v2√b2 + (l-x)2] = 0

So,[x/(√x2 + a2)] = sinϴ1 and [(l-x)/(√b2 + (l-x)2)] = sinϴ2

Therefore, dT/dx = (sinϴ1/v1) – (sinϴ2/v2) = 0

(sinϴ1/v1) = (sinϴ2/v2)

By substituting the phase velocity equation, we can say

n1sinϴ1/c = n2sinϴ2/c

n1sinϴ1 = n2sinϴ2

Snell's Law Application

The application related to Snell's Law is-

• This law is applicable to optical devices like contact lenses, digital cameras, eyeglasses, kaleidoscopes, and in rainbows.
• An instrument called a refractometer uses Snell’s law to calculate the refractive index of liquids.
• Used for making candy in the industry.

Snell's Law Examples 

Example1: A parallel beam of visible light is incident at an angle of 58° on a plane glass surface. The reflected beam gets completely polarized(tan 58° =1.6). What will be the angle of refraction and the refractive index of the glass?

Solution: If the light is an incident on the surface with an angle of incidence(i) given by tan i =μ, the reflected light is completely polarized.

Thus μ=tan58°  

refractive index μ=1.6

According to snell's law, Sin i/Sin r = µ

Substituting the value, 

sin58 / 1.6 = Sin r

Sin r = 32°

Example2: If a beam of light is refracted at an angle of 30° and the refractive index is 2.5, Calculate the angle of incidence.

Solution: Given, Angle of refraction(Sin r) = 30° 

Refractive index(µ) = 2.5

According to snell's law, Sin i/Sin r = µ

Sin i/Sin 30° = 2.5

Sin i = 1.25