In this blog post, we will derive the prism
formula derivation for Class 12 students. You will also get definition of prism, diagram and numerical questions.
We will provide step-by-step instructions for Prism Formula derivation.
Definition of Prism Formula
Prism is a three-dimensional transparent optical device. The shape of the prism is a special type, which has a very special feature: when the white light ray of the sun passes through the prism, it splits into its seven basic colors, it is called a rainbow.
When white light is refracted by a prism, all the colors present in the white light diverge at different angles. This is what scientists call "dispersion of light" due to which the rainbow appears.
“The angle between two surfaces of a prism is known as refracting angle or angle of prism.”
In the below figure, ABC represents the principal section of a glass prism having ∠A as its refracting angle.
Derivation of Prism Formula
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| Prism Formula |
Derivation of Prism Formula
`r_1 + r_2 + \angle N = 180^0` ....eq (1)
Again in quadrilateral AQNR
`\angle A + 90^0 + \angle N + 90^0 = 360^0`
`\angle A + \angle N = 180^0` .....eq(2)
By eq. 1 and 2
`r_1 + r_2 + \angle N =``\angle A + \angle N`
`r_1 + r_2 =``\angle A`
`r_1 + r_2 = A` ...eq (3) `\because \angle A = A`
In tringle MQR
`\delta = i_1 - r_1 + i_2 - r_2`
`\delta = i_1 + i_2 -( r_1 - r_2)`
`\delta = i_1 + i_2 - A` .. eq. (4) { from eq. 3 }
The angle of deviation of a ray of light passing through a prism depends not only on its material but also on the angle of incidence. If the graph of deviation angle is obtained with the angle of incidence, then first the angle of deviation decreases with the increase of angle of incidence, then the minimum value of deviation angle δm is obtained, and after that the angle of deviation increases with the increase of angle of incidence. In this way, we get the minimum value of deviation angle δm.
“The minimum value of the angle of deviation suffered by a ray on passing through a prism is called the angle of minimum deviation and is denoted by `\delta m`”
In the minimum deviation position
`i_1 = i_2 = i` , `r_1 = r_2 = r` and `\delta = \delta_m`
Thus from equation (3)
`r + r = A`
`2r = A`
`r = \frac{A}{2}`
and from equation (4)
`\delta_m = i + i - A`
`\delta_m = 2i - A`
`\delta_m + A = 2i `
`\frac{\delta_m + A}{2} = i`
Thus
`i = \frac{\delta_m + A}{2}`
Snell's rule
`\mu = \frac {sin i}{sin r}`
`\mu = \frac {sin \frac{\delta_m + A}{2}}{sin \frac{A}{2}}`
`\mu = \frac ({\frac{\delta_m + A}{2}})({\frac{A}{2}})`
`\mu = \frac{\delta_m + A}{A}`
`\mu A = \delta_m + A`
`\mu A - A = \delta_m`
`(\mu - 1)A = \delta_m`
`\delta_m = (\mu - 1)A`
This is the Relation Between the Refractive Index and the Angle of Minimum Deviation. Where
`delta_m =` minimum deviation
`\mu =` Refractive index
A = angle of the prism
It is clear, for a thin prism, the value of deviation produced depends upon the angle of the prism and the refractive index of the substance of the prism.
Watch the full video for Prism Formula Derivation Class 12
Prism Definition
"A prism is an object made of transparent material such as glass or plastic, which has at least two flat surfaces that create a sharp angle (less than 90 degrees). It includes all colors of the rainbow in white light."
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Numerical Questions Related to Prism Formula
1. The minimum deviation angle of the glass prism of the refractive index `n = \sqrt {3}` is equal to the angle of the prism. What is the angle of the prism?
Sol.
Given, A = `\delta m` and `n = \sqrt {3}`
`n = \frac {sin (\frac{A+\delta m}{2})}{sin frac{A}{2}}`
`n = \frac {sin (\frac{A + A}{2})}{sin frac{A}{2}}`
`n = \frac {sin (\frac{2A}{2})}{sin frac{A}{2}}`
`n = \frac {sin A}{sin frac{A}{2}}`
`n = \frac {2sin \frac{A}{2}cos\frac{A}{2}}{sin frac{A}{2}}`
`n = 2cos\frac{A}{2}`
`\sqrt {3} = 2cos\frac{A}{2}`
`2cos\frac{A}{2} = \sqrt {3}`
`cos\frac{A}{2} = \frac{\sqrt {3}}{2}`
`cos\frac{A}{2} = cos \30^\circ`
`\frac{A}{2} = 30^\circ`
`A = 60^\circ`
Hence angle of prism = `60^\circ`
2. On a surface of a prism for small angle A, the angle of incidence of light is i and emerges perpendicularly from its opposite face. If the refractive index of the prism is n, then find the incident angle.
Sol.
As the ray is emerging perpendicularly from opposite face. This is possible only when
`r_2 = 0`
`r_1 + r_2 = A`
`r_1 = A`
From Snell's law
`n = \frac{sin i}{sin r_1} `
`n = \frac{ i}{ r_1} ` { for small angle}
`i = n r_1`
`i = n A`
Q. What is an example of refraction?
Ans. There is a lot of examples of refraction in our daily life. Refraction is everywhere. Our atmosphere, our eyes, surface, etc. You just need to observe it in your life.
1. Take a glass cup of water and put a pencil inside it & u will see the pencil look like it bends also due to refraction.
2. Put torch light in water and u see refraction.
3. Formation of the rainbow is also an example of refraction.
4. Eye lens also refracts light.
5. light dispersing in the prism is also due to the refraction of light.
6. When you look inside water. You see some object are look larger than their actual size. This is also due to refraction of light.
7. When u visit in desert, you will see an illusion of light called Mirage. This is also due to the refraction of light.
8. You see the sun before sunrise and after sunset is also due to the refraction of light.
9. Sometimes when driving a car, you will see on a highway that a swath of water looks ahead of the road like water on road. Actually, this is also an illusion due to refraction.
10. Twinkling of stars is also an example of the refraction of light.
11. When you will visit the bank of the river, You will see inverted tree in the water in the river is also the refraction of light is everywhere, the only thing is to observe it.
