In this article, we shall study refraction through prism.

Prism:

A prism is a portion of a transparent medium which is bounded by non-parallel plane surfaces. The two plane surfaces ABCD and AEFD intersecting each other in line AD are called the refracting faces of the prism.  The angle between the two refracting faces ABCD and AECD is called the refracting angle of the prism.  The face BEFC is called the base of the prism.

Terminology of Prism:

Prism Formula:

From figure, ∠ NQP = i and ∠  RQT = r₁

∴   ∠  FQR = i – r₁

From figure, ∠ MRS = e and ∠ QRT = r₂

∴  ∠ FRQ = e – r₂

Now ∠ GFR is external angle of ΔFQR by property at external angle

∠ GFR = ∠ FRG + ∠ FQR

∴  ∠ GFR = e – r₂ + i – r₁

∴  ∠  GFR = i + e – r₁ – r₂

∴   ∠ GFR = i + e – (r₁ + r₂)

∴   δ = I + e – (r₁ + r₂ ) ………………(1)

Consider ΔQTR

∴  ∠ RQT + ∠  QTR + ∠ TRQ = 180°

∴   r₁ + ∠ QTR+ r₂ = 180°

∴   ∠ QTR = 180° – r₁ – r₂

∴   ∠ QTR = 180° – (r₁ + r₂) …………….(2)

□ AQRT is cyclic quadrilateral, by property of cyclic quadrilateral

∴   ∠ QAR + ∠ QTR = 180°

∴   A + ∠ QTR = 180°

∴   ∠ QTR = 180° – A …………… (3)

From (2) & (3)

∴  180° – A = 180°  – (r₁ + r₂)

A =  –  (r₁ + r₂)

∴  A =  r₁  +  r₂  …………..(4)

from (1) and (4)

= i + e – A ……………(5)

As angle of incidence increases angle of deviation (ð) decreases up to certain minimum value ð_m called angle of minimum deviation then it starts increasing

At minimum deviation ð =  ð_m and i = e.

Substituting these values in equation (5)

ð_m = i + i – A

∴ ð_m + A = 2 i

∴ i = (A + ð_m) /2   ………….(6)

similarly at minimum deviation, r₁ = r₂ = r

substituting this value in equation (4)

r₁ + r₂ = A

r + r = A

2r = A

∴ r =  A/2  ………(7)

by Snell’s Law

where μ = refractive index of the material of the prism

This formula is known as prism formula.

Terminology:

Deviation:

When a ray of light passes from one medium to another it changes its path, this phenomenon of light is called the deviation of light. When a ray of light passes through a prism it suffers refraction twice. Once while it enters the prism and again while it emerges from it.

When light rays pass from optically rarer medium to optically denser medium, the light rays bend towards the normal. When light rays pass from optically denser medium to optically rarer medium, the light rays bend away from the normal.

Angle of deviation:

The angle between the direction of the incident ray and the direction of the emergent ray is called the angle of deviation.

Angle of Minimum Deviation:

For a given prism the value of the angle deviation depends on the value of the angle of incidence. As the angle of incidence is increase the angle of deviation steadily decreases up to a certain minimum value of the angle of deviation. This minimum value of the angle of deviation is called angle of minimum deviation.  It can be shown graphically as follows.

For a particular angle of incidence, the angle of deviation is the least. This least angle of deviation is called the angle of minimum deviation.

At minimum deviation angle of incidence and angle of emergence are equal i.e. i = e. Similarly, the angle of refraction are equal i.e. r₁ = r₂

The angle of minimum deviation as produced by a prism depends upon (i) the refracting angle of the prism and (ii) Refractive index of the material of the prism.

Thin Prism:

The prism whose refracting angle is less than 5° is called thin prism.

By prism formula we have

∴   μ A = A + δ

∴  δ = μ A – A

∴  δ = A(μ  – 1)

This is an expression for the angle of deviation in case of a thin prism.

Science > Physics > Refraction of Light > Refraction Through Prism

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When a ray of light approaches a smooth polished surface and the light ray bounces back, it is called the reflection of light. The surface which reflects the light is called a reflecting surface. The ray of light approaching the reflecting surface is called the incident ray. The ray bounced back and moving away from the reflecting surface is called the reflected ray.

Diffusion Reflection:

In this type of reflection, the incident ray of light strikes a surface and the light is scattered after reflection. Reflected rays move in many directions. The angle of incidence and angles of reflection are different. A perfectly diffusing surface is one for which luminance is independent of the direction of observation. e.g. reflection from paper, unpolished clay pot.

Regular Reflection:

In regular reflection, a parallel beam of incident light is reflected as a parallel beam in one direction. In this case, parallel incident rays remain parallel even after reflection and go only in one direction. Surfaces showing regular reflection appear bright only in one direction only for the given position of the eye. A surface which reflects regularly nearly all the light incident on it is called a mirror.

The objects having a shiny or polished surface reflects more light compared to the objects having a dull or unpolished surface. Silver metal is the best reflector of light. Hence plane mirror is made by depositing a thin layer of silver metal on one side of a plane glass sheet. The silver coating is protected by red paint. e.g. mirror, polished metal surface, polished pots

Water at glazing incidence (i = 90°) is a good reflector. Hence we can see images of the sun and nearby trees in the river in the morning and evening but not at noon. If the sea is observed from an airplane the part of the sea exactly below appears dark but that part at the horizon appears shiny.

Terminology of Reflection:

• Incident Ray: The ray of light falling on the surface of a mirror is called incident ray.
• Point of Incidence: The point at which the incident ray touches the mirror surface is called the point of incidence.
• Reflected Ray: The ray of light which is sent back by the mirror from the point of incidence is called reflected ray.
• Normal: A line perpendicular to the mirror surface at the point of incidence is called normal.
• Angle of incidence: 
• The angle made by the incident ray with the normal is called the angle of incidence.
• Angle of Reflection: 
• The angle made by the reflected ray with the normal at the point of incidence is called the angle of reflection.
• Object: Anything which gives out light with off its own of reflected by it is called an object.
• Image: When the light rays coming from an object are reflected from a mirror then an optical appearance which is produced by the mirror is called an image. Images are of two types, real image, and virtual image.
• Real Image: The image which can be seen on the screen is called a real image.
• Virtual Image: The image which cannot be obtained on a screen is called a virtual image.
• Lateral Inversion: The change of sides of an object in its mirror image is called lateral inversion. It happens due to the reflection of light. In this case, the right side of the object becomes left side of the image and the left side of the object becomes the right side of the image

Laws of Reflection:

• The angle of incidence is equal to the angle of reflection
• The incident ray and the reflected ray lie on either side of the normal at the point of incidence
• The incident ray, reflected ray and the normal at the point of incidence lie in the same plane.

Image of Point in a Plane Mirror:

Theory:

XY is a reflecting surface and A is a point source of light(object) emitting light in all the directions. AO is normal to the reflecting surface XY. Ray AP is incident on the surface XY at an angle of inclination i, and reflected ray is ray PB. PN is normal to surface XY at P.

∠ APN = ∠ PAO = i  (alternate angles)

∠ OA’P = ∠ NPB = i (corresponding angles)

In Δ AOP and Δ A’OP

∠ AOP = ∠ A’OP  (Each 90°)

∠ OA’P = ∠ OAP  (each equal to i)

OP is a common side

∴ Δ AOP ≅ Δ A’OP

∴ OA’ = OA

Conclusions:

• The image of a point in a plane mirror lies behind the mirror along the normal produced from the object and is as far behind the mirror as the object is in front.
• The image is virtual and erect. The image is virtual because the rays of light are not coming from it but appear to come from it.

Image of an Extended Object in a Plane Mirror:

Let us consider an object AB in front of the mirror as shown. The plane of the mirror is perpendicular to the plane of the paper (screen). As per the image of a point in a plane mirror, point A’ is an image of point A and point B’ is an image of point B. Hence A’B’ is the image of the object AB.

Generally, plane mirrors are put in the vertical plane, there is a lateral inversion of the image. Thus the image formed is virtual, erect,  as the same size of object, laterally inverted, and is as far behind the mirror as the object is in front.

Rotation of Mirror:

A ray of light is incident on a mirror at the angle of incidence i and is reflected back at the angle of reflection r = i. Thus there is a deviation of the ray is (180° – 2i). Let the mirror be rotated through angle θ keeping the direction of incident ray same. Then angles of incidence and deviation  become (i + θ) and (180° – 2(i + θ)).

The change in direction of incident ray = (180° – 2i) – (180° – 2(i + θ))

The change in direction of incident ray = 180° – 2i – 180° + 2i + 2θ = 2θ

Thus if the mirror itself is rotated by θ, then reflected ray turns through angle 2θ

Inclined Mirrors:

Mirrors Perpendicular to Each Other:

If the two mirrors are perpendicular to each other, then the ray suffers one reflection each at the two mirrors and the final reflected ray (emergent ray) is antiparallel to the incident ray.

The object O will give image I₁ in mirror M₁ and image I₂ in mirror M₂. Now image I₁ is an object for extended mirror M2 its image in M2 is I3. Now image I2 is an object for extended mirror M1 its image in M1 is I4. The images I3 and I4 coincide. Thus three images are formed I1, I2, and I₃(or I₄).

Mirrors at angle 60° With each Other:

Number of Images in Inclined  Mirrors:

If θ is the angle between the mirrors, then find quantity 360/θ = n (say). If the value of n is even then the number of the image is (n -1) (it is true for both symmetrical and unsymmetrical position of the object)

If the value of n is even then the number of the image is n for the symmetrical position of the object and (n – 1) for the unsymmetrical position of the object If n is fraction number of images are equal to integer part.

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Refraction of Light:

When light ray travelling in one optically active medium enters another optically active medium, then the light ray deviates from its path. This phenomenon is known as the refraction of light.

Laws  of Refraction:

• The ratio of the sine of the angle of incidence to the sine of the angle of refraction is always constant and is equal to the refractive index of the medium.  This law is known as Snell’s law.
• The incident ray and the refracted ray lie on the opposite side of the normal at the point of incidence.
• The incident ray, refracted ray and the normal at the point of incidence lie in the same plane.

Terminology of Refraction:

• Optical Medium: The medium capable of transmitting light is called as an optical medium.
• Isotropic medium:
• The homogeneous medium, which has the same properties in all directions is called an isotropic medium.
• Incident ray:
• The light ray, which is falling on reflecting or refracting surface is called the incident ray.
• Point of incidence: The point at which the incident ray cuts the reflecting or refracting surface is called the point of incidence.
• Normal: A perpendicular drawn to the surface at the point of incidence is called the normal.
• Angle of incidence:
• The angle made by the incident ray with the normal at the point of incidence is called the angle of incidence. It is denoted by ‘i’.
• Angle of Refraction:
• The angle made by the refracted ray with the normal at the point of incidence is called the angle of refraction. It is denoted by ‘r’.
• Angle of Deviation:
• The angle between the direction of the incident ray to the direction of the refracted ray is called the angle of deviation. it is denoted by ‘δ’.
• Glancing Angle:
• The angle made by incident ray with a refracting surface or interface is called a glancing angle.

Refractive Index of a Medium:

According to Huygens, the cause of the refraction of light is that the velocity of light is different in different media. The velocity of light is maximum in vacuum and air. The velocity of light in vacuum or air is denoted by letter ‘c’.

The relative refractive index of the second medium with respect to the first medium (₁μ₂)is defined as the ratio of the velocity of light in the first medium (v₁) to the velocity of light in the second medium (v₂). Mathematically,

When the first medium is vacuum or air the relative refractive index is known as the absolute refractive index.

Snell’s Law:

The ratio of the sine of the angle of incidence to the sine of the angle of refraction is always constant and is equal to the refractive index of the medium.  This law is known as Snell’s law.

Mathematically,

Principle of Reversibility of Light:

If, after undergoing a number of reflections and refractions, the direction of a ray of light is reversed, the ray traces its original path in the reverse direction. This principle is applicable to prisms and lens also.

Property:

If ₁μ₂ is the  refractive index of the second medium with respect to the first medium  and  ₂μ₁ is  the refractive index of the first medium with respect to the second medium, then

Proof:

To show that ₁μ₂ = μ₂/μ₁

The absolute refractive index of the medium is defined as the ratio of the velocity of light in the vacuum to the velocity of light in the medium.

The absolute refractive index of the first medium = μ₁ = v₁ / c …….. (1)

The absolute refractive index of the second medium = μ₂ = v₂ / c ………. (2)

Dividing equation (2) by equation (1) we have

μ₂/μ₁ =   v₂/v₁    …………………. (3)

The relative refractive index of the second medium with respect to the first medium (₁μ₂)is defined as the ratio of the velocity of light in the first medium (v₁) to the velocity of light in the second medium (v₂). Mathematically,

₁μ₂ = v₂/v₁     …………………. (4)

From equation (3) and (4) we have

₁μ₂ = μ₂/μ₁      (Proved as required)

Numerical Problems on Refractive Index:

Example – 01:

A ray of light in air is incident on a glass surface making an angle of 30° with the surface. the angle of refraction in a glass is 35°16′. Find the angle of deviation of the ray.

Given: Glancing angle = i_g = 30°. Hence angle of incidence = 90° – 30° = 60°

To find: Angle of deviation = δ = ?

Solution:

The angle of deviation = Angle of incidence – Angle of refraction

Angle of deviation = δ = 60° –   35°16′ = 24°44′

Ans:  The angle of deviation of the ray is 24°44′

Example – 02:

Find the refractive index of water. Given velocities of light in air and water as 3 × 10⁸ m/s and 2.25 × 10⁸ m/s respectively.

Given: Velocity of light in air = c = 3 × 10⁸ m/s and velocity of light in water = v = 2.25 × 10⁸ m/s

Fo find: Refractive index of water = μ =?

Solution:

μ = c /ν= 3 × 10⁸ /2.25 × 10⁸ = 1,333

Ans:  The refractive index of water is 1.33

Example – 03:

The refractive index of glass with respect to air is 1.5. Calculate the velocity of light in the glass. Given the velocity of light in the air as 3 × 10⁸ m/s.

Given: Velocity of light in air = c = 3 × 10⁸ m/s and refractive index of glass = μ = 1.5

Fo find: velocity of light in glass = v =?

Solution:

μ = c /v

∴   v = c/μ

∴   v = 3 × 10⁸ /1.5 = 2 × 10⁸ m/s

Ans:  The velocity of light in glass is 2 × 10⁸ m/s

Example – 04:

The refractive index of water w.r.t. air is 4/3. Find the relative refractive index in case when light travels from water into air.

Given: refractive index of water w.r.t. air = _aμ_w = 4/3

Fo find: refractive index of air w.r.t. water = _wμ_a =?

Solution:

_wμ_a ==  1/_aμ_w = 3/4 = 0.75

Ans:  The refractive index of air w.r.t. water is 0.75

Science > Physics > Refraction of Light > Introduction to Refraction of Light

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