An Introduction to Relation Between Critical Angle and Refractive Index
Relation between critical angle and refractive index can be formed because both of them are inversely proportional. But, before going into this detail, you must understand these topics separately.
What is Critical Angle?
In optics as a topic of Physics, the critical angle makes reference to a particular angle of incidence, which gives an angle of refraction of 90 degrees. Additionally, a water–air boundary has a critical value of 48.6 degrees. while the critical angle for crown water and glass boundary is 61.8 degrees.
However, the value of critical angle is always dependent on the mediums situated on both sides of a boundary.
Additionally, the equation of critical angle is:
\[\theta _c\] = sin-1 nr / ni
Here, \[\theta _c\] is the critical angle of refraction, and nr and ni are refraction index and incident index, respectively.
What is Refractive Index?
The extent at which rays of light bend when it enters from one medium source to another is known as its refractive index. The refractive index is represented using the alphabet 'n.' Moreover, it can be written as n = c/v, where ‘c’ is light speed or velocity of a specific wavelength in the medium air. On the other hand, v is light’s velocity or speed in other media.
Furthermore, three factors determine this refractive index - medium nature, light colour, and physical conditions.
Fact: An optically rarer medium is one where light travels faster through it. Whereas, an optically denser medium is one where light travels slower through it.
Refractive Index and the Critical Angle Relationship
The mathematical representation of their relationship is:
sin C= 1/ µab
Here, C = critical angle, µ = refractive index, and a and b are two mediums within which light passes.
Furthermore, take a look at this derivation below!
Snell's Law (also known as the Second Law of refraction) is applied to derive the relation between critical angle and refractive index.
Hence, take a light ray having an incident angle i, refractive angle r = 90 degrees, critical angle = C, and refractive index of rarer and denser medium be µa and µb, respectively.
So, by applying the second Law of refraction or Snell's Law:
sin i / sin r = µa / µb
Therefore, µb sin C = µa sin90o
Therefore, µb / µa = 1 / sin C
Thus, with the help of this equation, critical angle and refractive index relation can be stated as:
µab = 1/ sinC
Solved Numericals
(i) Find out the ratio between sine of incident angle and the sine of reflected angle where their refractive indices are provided. In medium 1 it is 2.33, while in medium 2 it is 1.66.
Solution: Snell's Law gives \[n_1sin\theta _i=n_2sin\theta _r\]
In order to get \[\frac{sin\theta _i}{sin\theta _r}\] , you must note that this ratio is \[\frac{n_2}{n_1}\]
After substituting for n1= 2.33 along with n2 = 1.66
=> 1.66/2.33 = 0.71
(ii) Find the ratio between refractive index of two mediums, 1 and 2. Here, the reflected angle of medium 1 is 300, while that of medium 2 is 450.
Solution: Snell's Law gives \[n_1sin\theta _i=n_2sin\theta _r\]
So, for getting \[\frac{n_2}{n_1}\], this ratio is \[sin\theta _i=sin\theta _r\]
On putting values for θi= 30 and θr= 45
Therefore, \[\frac{sin45}{sin30}\] = 2
Applications of Critical Angle
The principle of critical angle is used in several practical ways in our day-to-day life. The most widely used is fibre optic cables. The concept is the base for the construction of fibre optic cables and how they work.
Applications of total internal reflection are multi-touch screens, spatial filtering of lights, prismatic binoculars, automotive rain sensors, fluorescence microscopes, and ubiquitous fibre optics communications.
Do It Yourself
1. Angle of incidence is equal to the angle of reflection for perfect reflection. Answer true or false.
(a) False
(b) True
Ans: (b) True
2. The higher the value of the refractive index of a given medium, the bending of light will be
(a) zero
(b) smaller
(c) higher
(d) negative
Ans: (c) Higher
3. The refractive index of a medium is the relation between light’s speed in vacuum or air, and
(a) Light’s speed in a medium
(b) Can be a or c
(c) Speed of sound in a medium
(d) none
Ans: (a) Light’s speed in a medium
FAQs on Relation Between Critical Angle And Refractive Index
1.Why is light used instead of electricity in signal transmission in fibre cables?
Laser light is used as the form of signal in fibre cables because laser light is a single wavelength light source. Other light signals like bulbs or sunlight have multiple wavelengths. Multiple wavelengths produce a weaker beam. The principle of critical angle and total internal reflection works in optic fibre cables. Light as the mode of transmission also causes less dispersion and takes less time and more signals are transmitted at one time, making it an ideal choice.
2.What are prismatic binoculars?
Prismatic binoculars use the principle of critical angle and total internal reflection to work. It magnifies light rays and creates an image that is the right way up and provides a clear image of objects far away. Prisms are used inside the binoculars. On entering the binocular, lights refract through the prism. As the angle of incidence is higher than the critical angle, it reflects off the prism. The shape of the prism makes the light undergo total internal reflection many times before finally exiting the prism.
3.What is the numerical formula of critical angle?
The formula for calculating critical angle is θcric = sin-1 nr/ni.
4.Define critical angle and write its conditions?
Critical angle happens when an incident ray causes refraction to make an angle of 90 degrees. Moreover, the critical angle value depends upon the two mediums.
5.How do you calculate the critical angle of refraction?
With the help of Snell's Law and setting the angle of refraction to 90 degrees, the critical angle is calculated.