Rainbows appear in the sky after rainfall, pencil dipped in water appears to be bent, people standing in a pool often appear shorter, stars appear to twinkle in the sky, and the reddish colour of the sky during sunset, all these phenomena are associated with refraction. In this topic, we will learn how refraction takes place through a glass prism.

Refraction

The bending of a ray of light as it travels from one transparent media to another is refraction. Thus during refraction, the path of a ray of light changes as it strikes at the boundary of two transparent media. Light travels fastest in a vacuum, and in any other medium, the speed of light is less than its speed in a vacuum. The difference in the speed of light in different media causes a change in the direction of light as it travels from one medium to another. The frequency of incident and refracting light remains the same, while the wavelength and speed change. 

Two laws of refraction are:

1. The incident ray, refracted ray and the normal to the surface at the point of incidence all lie in the same plane.
2. The ratio of the sine of the angle of incidence (the angle between the incident ray and the normal) to the sine of the angle of refraction( the angle between refracted ray and the normal) is a constant. The value of this constant depends on the nature of the two media and the wavelength of incident light. This law is known as Snell’s law.

A ray of light travelling from a rarer medium to a denser medium bends towards the normal. A ray of light travelling from a denser medium into a rarer medium bends away from the normal after refraction. The refractive index gives the extent of bending of light as it travels from one medium to another. It is represented by \(n\).

\(n=\frac{c}{v}\)

Where,

\(c\) is the speed of light in the vacuum

\(v\) is the speed of light in the medium

Phenomena Associated with Refraction

1. A straw dipped in water appears to be broken at the air-water interface.
2. While travelling through a desert or jungle, travellers often see water or trees that are not truly there. This phenomenon is called ‘mirage.’
3. Twinkling of stars at night or formation of a rainbow in the sky.
4. Dispersion of Light as it passes through a glass prism.

What is a Prism?

A prism is a wedge-shaped object which consists of two transparent refracting mediums bound to each other. The two plane surfaces are inclined to each other at some angle. These plane faces of a prism are called refracting faces, and the angle between the two plane faces is called the angle of the prism or refracting angle.

As shown in the figure above, a prism consists of two triangular bases and three rectangular surfaces.

Types of prism

1. Dispersive Prisms –  Dispersive prisms break up light into constituent colours because the refractive index depends on the frequency. The white light entering this prism consists of a range of wavelengths and frequencies. Each light ray of a given frequency passing through this prism bends in a different direction.  Triangular prism, Pellin-Broca prism, Grism, Amici Prism, Abbe prism, and compound prism are a few dispersive prisms. 
2. Reflective Prisms –  As their name suggests, reflective prisms are employed for reflecting light. A reflective prism is used to flip, invert, rotate, deviate, or displace a given beam of light. Binoculars or single-lens reflex cameras have these types of prisms installed in them so that the resulting image is erect. In the absence of such prisms, the image obtained will be inverted. Bauernfeind prism, Abbe-Koenig prism, Retroreflector, Porro-Abbe prism, Dove prism, Amici roof prism, Schmidt-Pechan prism, Porro prism and Pentaprism are a few examples of reflective prisms.
3. Polarizing Prisms – Polarizing prisms split a beam of light falling on their surface into components that have different polarization. These prisms are composed of birefringent crystalline material. Nomarski prism, Glan-Foucault prism, Glan-Thompson prism, Wollaston prism, Glan-Taylor prism, Rochon prism, Nicol prism and Senarmont prism are a few examples of polarizing prisms.
4. Deflecting Prisms – As their name suggests, deflecting prisms deflect a beam of light falling on their surface at a predefined angle. These prisms are quite useful for beam steering too. For this, a pair of deflecting prims are used, and on rotating these prisms, a beam of light can be deflected at a desired angle within a specific limit. Rhomboid prism and Deck prism are examples of deflecting prisms.
5. Beam-Splitting Prisms – A beam-splitting prism split a single beam into two or more beams. Beam splitter cube and Dichroic prism are examples of beam-splitting prisms.

Refraction Through a Glass Prism

Consider a ray of light entering a glass prism as shown below:

\(ABC\) represents the prism, \(PE\) represents the incident ray, \(EF\) is the refracted ray, and \(FS\) is the emergent ray. \(A\) is the angle of refraction, \(i\) is the angle of incidence, \(r\) is the angle of refraction, \(e\) is the angle of emergence, and \(D\) is the angle of deviation. \(N N^{\prime}\) is normal at the point of incidence, and \(M M^{\prime}\) is normal at the point of emergence.

1. The ray of light is moving from air into Glass. As per Snell’s law, we know that a ray of light moving from a rarer medium to a denser medium bends towards the normal; thus, a ray of light travelling from air into the prism will bend towards the normal. The incident ray \(PE\) strikes the prism and bends towards the normal \(NE\).
2. The refracted ray \(EF\), moves through the glass prism and at the glass-air interface, it bends since the ray travels from a denser medium into a rarer medium. Thus, as per Snell’s law, it bends away from the normal. Thus, we get the emergent ray \(FS\) bends away from the normal \(MF\).
3. The deviation of the emergent ray with respect to the incident ray is determined by the angle \(\angle H D S\). The angle of deviation is minimum when the angle of incidence is equal to the angle of emergence, i.e.
   When \(\angle P E N=\angle M E S, \angle H D S\) is minimum, and the refracted ray becomes parallel to the base of the prism.

Derivation of Prism Formula

Consider the prism shown below:

The angle deviation, \(\delta=i_{1}-r_{1}+i_{2}-r_{2}\)

Or, \(\delta=i_{1}+i_{2}-\left(r_{1}+r_{2}\right)\) ——–(1)

Consider the quadrilateral ALOM

Since \(\angle A L O=\angle A M O=90^{\circ}\)

\(\angle A L O+\angle A M O=180^{\circ}\)

Since the sum of four angles of a quadrilateral \(=\) four right angles,

\(\angle L A M+\angle L O M=180^{\circ}\) ———(2)

In triangle LOM,

\(\angle r_{1}+\angle r_{2}+\angle L O M=180^{\circ}\) ———-(3)

Using the equations \((2)\) and \((3)\), we get

\(\angle L A M=\angle r_{1}+\angle r_{2}\)

Since \(\angle L A M=A\)

\(A=\angle r_{1}+\angle r_{2}\) ———(4)

Substituting this value into equation \((1)\), we get:

\(\delta=i_{1}+i_{2}-A\)

Thus, \(\delta+ A = {i_1} + {i_2}\)

We can see from the expression that the angle of deviation of a ray of light depends on the nature of the material and also on the angle of incidence.

We can see from the graph above that for a certain specific angle of incidence, the angle of deviation is the least. This is called the minimum value of angle of deviation, and it is represented by the letter \(\delta_{m}\).

For minimum deviation, \(\angle i_{1}=\angle i_{2}=i\)

Thus, \(\angle r_{1}=\angle r_{2}=r\)

\(\angle A L M=\angle L M A=90^{\circ}-r\)

We can conclude, \(AL=LM\) and \(L M|| B C\)

Thus, a ray of light that undergoes minimum deviation passes symmetrically through the prism and goes parallel to the base.

In the case of a prism,

\(A=\angle r_{1}+\angle r_{2}\)

Thus, \(A=r+r=2 r\)

\(r=\frac{A}{2}\)

From above, \(\delta+A=i_{1}+i_{2}\)

Substituting the conditions for minimum deviation, we get:

\(\delta_{m}+A=i+i=2 i\)

Or, \(i=\left(\delta_{m}+A\right) / 2\) ——–(5)

According to Snell’s law, 

\(\mu = \frac{{\sin \,{i_1}}}{{\sin \,{r_1}}} = \frac{{\sin \,{i_1}}}{{\sin \,r}}\)

Substituting the values from above, 

\(\mu = \frac{{\sin \,\left[ {{\delta _m} + A} \right]/2}}{{\sin \left( {\frac{A}{2}} \right)}}\) ———-(6)

Refraction Through a Thin Prism

A thin prism is a prism that has refracting angle within the range of \(10^{\circ}\). Let us derive an expression for the refraction through a prism with a small refracting angle. For a prism with a small refracting angle, the angle of incidence and the angle of refraction will be small.

From Snell’s law, 

\(\mu = \frac{{\sin \,{i_1}}}{{\sin \,{r_1}}}\)

Since \(i_{1}\) and \(r_{1}\) are small.

We can take, \(\sin i_{1} \approx i_{1}\)

And \(\sin r_{1} \approx r_{1}\)

Thus, \(\mu=\frac{i_{1}}{r_{1}}\)

Similarly, \(\mu=\frac{i_{2}}{r_{2}}\)

From above, \(i_{1}+i_{2}=\mu r_{1}+\mu r_{2}=\mu\left(r_{1}+r_{2}\right)\)

This gives, \(i_{1}+i_{2}=\mu A\dots…(7)\)

From above, \(i_{1}+i_{2}=A+\delta \dots…(8)\)

Comparing equations \((7)\) and \((8)\)

\(\mu A=A+\delta\)

\(\delta=\mu A-A=A(\mu-1)\)

\(\delta=A(\mu-1)\)

Thus, when the prism angle is small, the angle of deviation becomes independent of the angle of incidence.

Summary

The bending of a ray of light as it travels from one transparent media to another is refraction. Thus during refraction, the path of a ray of light changes as it strikes at the boundary of two transparent media. A prism is a wedge-shaped object which consists of two transparent refracting mediums bound to each other.

The ray of light is moving from air into Glass. As per Snell’s law, we know that a ray of light moving from a rarer medium to a denser medium bends towards the normal; thus, a ray of light travelling from air into the prism will bend towards emergent ray bends away from the normal. The angle of deviation is minimum when the angle of incidence is equal to the angle of emergence, and the refracted ray becomes parallel to the prism base.

Prism formula: \(\mu = \frac{{\sin \left[ {{\delta _m} + A} \right]/2}}{{\sin \left( {\frac{A}{2}} \right)}}\)

When the prism angle is small, the angle of deviation becomes independent of the angle of incidence: \(\delta=A(\mu-1)\).

Frequently Asked Questions (FAQs)

Q.1. Name a few different types of prisms.
Ans: Types of prisms:
1. Dispersive Prisms
2. Reflective Prisms
3. Polarizing Prisms
4. Deflecting Prisms
5. Beam-Splitting Prisms

Q.2. What is refraction?
Ans: The bending of a ray of light as it travels from one transparent medium to another is called refraction.

Q.3. Give a few examples of refraction in our everyday life.
Ans: Twinkling of stars, formation of a rainbow, delayed sunset and an early sunrise are a few examples of refraction in our daily lives.

Q.4. State Snell’s law.
Ans: According to Snell’s law, for a beam of light striking a surface, the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant.

Q.5. Give the expression for the angle of deviation for a thin prism.
Ans: The angle of deviation can be given as: \(\delta=A(\mu-1)\)
Where \(A\) is the angle of prism and \(\mu\) is the refractive index.

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