Refraction of Light Spherical Surfaces, Lenses and Prism - Physics

What is Refraction? 

• Refraction is the bending of a wave when it passes from one medium to another. The bending is caused due to the differences in density between the two substances.
  Refraction
• Refraction of light is one of the most commonly observed phenomena, but other waves like sound waves and water waves also experience refraction. 
• Refraction makes it possible for us to have optical instruments such as magnifying glasses, lenses and prisms. It is also because of the refraction of light that we are able to focus light on to our retina.

Why do stars twinkle?
Did you know that the twinkling effect of stars is due to atmospheric refraction? The starlight undergoes several refractions while reaching the Earth. This atmospheric refraction occurs in a medium of gradually changing refractive index.

Causes of Refraction

• A light ray refracts whenever it travels at an angle into a medium of different refractive index. This change in speed results in a change in direction. As an example, consider air travelling into water. The speed of light decreases as it continues to travel at a different angle.
  
• The refraction of light in glass is shown in the figure above. When light travels from air into glass, the light slows down and changes direction slightly. When light travels from a less dense substance to a denser substance, the refracted light bends more towards the normal line. 
• If the light wave approaches the boundary in a direction that is perpendicular to it, the light ray doesn’t refract in spite of the change in speed.

Laws of Refraction of Light

Laws of refraction state that:

1. The incident ray refracted ray, and the normal to the interface of two media at the point of incidence all lie on the same plane.
2. The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant. This is also known as Snell’s law of refraction.
   

What is Refractive Index?

• The Refractive index, also called the index of refraction describes how fast light travels through the material.
• Refractive Index is dimensionless. For a given material, the refractive index is the ratio between the speed of light in a vacuum (c) and the speed of light in the medium (v). If the refractive index for a medium is represented by n, then it is given by the following formula:
  
• Based on the refractive index of the medium, the light ray changes its direction, or it bends at the junction separating the two media. If the light ray travels from a medium to another of a higher refractive index, it bends towards the normal, else it bends away from the normal.

Refraction of Light in Real Life

• Mirage and looming are optical illusions which are a result of refraction of light.
• A swimming pool always looks shallower than it really is because the light coming from the bottom of the pool bends at the surface due to refraction of light.
• Formation of a rainbow is an example of refraction as the sun rays bend through the raindrops resulting in the rainbow.
• When white light passes through a prism it is split into its component colours – red, orange, yellow, green, blue and violet due to refraction of light.

Applications of Refraction of Light

Refraction has many applications in optics and technology. A few of the prominent applications are listed below:

1. A lens uses refraction to form an image of an object for various purposes, such as magnification.
2. Spectacles worn by people with defective vision use the principle of refraction.
3. Refraction is used in peepholes of house doors, cameras, movie projectors and telescopes.

Refraction at a Spherical Surface

• Let us now see the refraction of light at the spherical surface. Now, the change in direction or bending of a light wave passing from one transparent medium to another caused by the change in wave’s speed is the Refraction. 
• Suppose the above figure is a spherical surface. There is one medium with refractive index n₁ and the second medium with refractive index n₂.
• There is an object O and a ray of light from the object O is incident on the spherical mirror. Since it is moving from a rarer medium to a denser medium, the ray bends towards the normal.  
• An image is formed and the radius of curvature of a spherical surface is R with the center C of the spherical surface.

Now as we know that:

• n₁ is the refractive index of a medium from which rays are incident.
• n₂ is the refractive index of another medium

We get:
► tanα = MN / OM
► tanγ = MN / MC
► tanβ = MN / MI
Now, for Δ NOC, i is the exterior angle.
i = ∠ NOM + ∠ NCM
i= MN / OM + MN / MC   ...(1)
Similarly,
r =  MN / MC – MN / MI  ...(2)
Now by using Snell’s law we get:
n₁ sin i = n₂sin r
Substituting i and r from Eq. (1) and (2), we get
n₁ / OM + n₂ / MI = (n₂ − n₁) / MC
As, OM = -u, MI = +v, MC = +R
Hence, the equation becomes:

Lens

A lens is a uniform transparent medium bounded between two spherical or one spherical and one plane surface.
Here are some terms related to Lens:

• Centre of Curvature: The centre of the actual glass sphere, of which your lens forms a part
• Principal Axis: When two spheres are part of your lens, it is the imaginary line joining the centres of curvatures of both spheres.
• Principal Focus: It is point on the principal axis, where light rays parallel to principal axis meet in case of a convex lens (or appear to meet after extrapolation in case of a concave lens).
• Optical Centre: It is a point within the lens where the diameter of the lens and the principal axis meet
• Focal Length: The distance between the focus and the optical centre

Types of Lens

1. Convex Lens
   A lens that is thinner at the edges and thicker at the middle is called a convex or converging lens. A convex lens is also referred to as a converging lens since it “converges” light rays that are incident on it. 
2. Concave Lens  
   A lens which is thicker at edges and thinner at middle, is called a concave or diverging lens. A concave lens is also referred to as a diverging lens since it “diverges” light rays that are incident on it.
   
   Convex and Concave Lens

➢ Lens Formula
1/f = 1/v – 1/u
where, f = focal length of the lens, U = distance of object, U = distance of image.
Lens Maker’s formula
1/f=(μ – 1) (1/R₁ – 1/R₂)
where, μ = refractive index of the material of the lens and R₁ and R₂ are radii of curvature of the lens.
➢ Power of a Lens
The reciprocal of the focal length of a lens, when it is measured in metre, is called power of a lens.
Power of a lens, (P)= 1/f(metre)
Its unit is dioptre (D).
The power of a convex (converging) lens is positive and for a concave (diverging) lens it is negative.

➢ Focal Length of a Lens Combination
(i) When lenses are in contact 1/F – 1/f₁ + 1/f₂
Power of the combination P = P₁ + P₂

(ii) When lenses are separated by a distance d
1/F = 1/f₁ + 1/f₂ – d/f₁f₁
Power of the combination:
P = P₁ + P₂ – dP₁P₂

➢ Linear Magnification
m = I/O = v/u
For a small sized object placed linearly along the principal axis, its axial (longitudinal) magnification is given by
Axial magnification = – dv/du = (v/u)²
= (f / f+u)² = (f - v/f)²

➢ Focal Length of a Convex Lens by Displacement Method
Focal length of the convex lens f = (a² – d²) / 4a
where, a = distance between the image pin and object pin and
d = distance between two positions of lens.
The distance between the two pins should be greater than four times the focal length of the convex lens, i.e., a > 4f.
Height of the object O = √I₁I₂

➢ Cutting of a Lens

• If a symmetrical convex lens of focal length f is cut into two parts along its optic axis, then focal length of each part (a plane convex lens) is 2f. However, if the two parts are joined as shown in figure, the focal length of combination is again f.
  
• If a symmetrical convex lens of focal length f is cut into two parts along the principal axis, then focal length of each part remains unchanged as f. If these two parts are joined with curved ends an one side, focal length of the combination is f/2. But on joining two 2 parts in opposite sense the net focal length becomes
  

Refraction Of Light Through Prism

• Prism is a uniform transparent medium bounded between two refracting surfaces, inclined at an angle. you can see it in figure.

• When light travels from one medium to another, the speed of its propagation changes, as a result, it ‘bends’ or is ‘refracted’. Now when light passes through a prism, it is refracted towards the base of the triangle.
• The different colours in the spectrum of light have different wavelengths. Therefore, the speed with which they all bend varies depending on this wavelength, where violet bends the most, having the shortest wavelength and red bends the least, having the longest wavelength.
• Because of this, the dispersion of white light into its spectrum of colours takes place when refracted through a prism.
• Angle of Deviation: The angle subtended between the direction of incident light ray and emergent light fay from a prism is called angle of deviation (δ).
  

• Let A, B, C be the glass of the prism. Suppose BC is the base and AB and AC are its two refracting surfaces. From the above figure, we can say that OP is the incident. The ray traveling through the rarer medium and than the refractive index of the prism is the incident ray. As the ray PQ strikes the surface of the and it is called as the refracted ray. OR is the emergent ray which comes out.

• When the ray light enters the glass, it bends towards normal and when ray comes out, it bends away from the normal. Now the angle between the emergent ray and incident ray is the angle of deviation. For a single refracting surface,  δ = |i – r|
  In this case, δ = (i1 + i2) – (r1 + r2)
  δ = i1 + i2 – A, A is the angle between the prism between two lateral surfaces. 

• We know that ∠A and ∠Q is 180º and Angle of the prism of (A) is: r1 + r2
  r1  is the angle of refraction inside the prism and r2 is the angle of refraction outside it. For an angle of minimum deviation, δ is minimum and i1 = i2 = i
  δmin = 2i – A
  For small A, δ = (µ – 1) A

• Minimum Angle of Deviation for a Prism: At the minimum deviation, Dm the refracted ray inside the prism becomes parallel to its base, i.e. i = e ⇒ r1 = r2 = r, then r = A/2 and Dm = 2i – A, where i is the angle of emergence, r1 and r2 are the angles of refraction and A is the angle of the prism.

• Prism Formula
  The refractive index of the material of prism:
  
  

• Angular Dispersion: The angle subtended between the direction of emergent violet and red rays of light from a prism is called angular dispersion.
  Angular dispersion
  (θ) δ_v – δ_R = (μ_v – μ_R)A
  where δ_v and δ_R are angle of deviation.

• Dispersive Power
  W = θ/δ_Y = (μ_v – μ _R) / (μ_Y – 1)
  where μ_Y = (μ _v + μ _R ) / 2, is mean refractive index.

Dispersion of Light Through Prism

• The splitting of white light into seven constituent colours takes place when passed through the prism. This is known as the dispersion of light by a prism.
• With the help of a narrow beam of light, a glass prism, and a white wall it is possible to produce the band of seven colors using white light. Keep this arrangement near the window. Place the glass prism in such manner that the sunlight through the window falls on one side of the prism and then on the white wall.
• You can see that the light reflected on the wall has several colors. The prism splits the white light into seven different colors. This splitting of white light into many colors is called as a dispersion of light. Dispersion is nothing but splitting of white light into its constituents colors.i.e into seven different colors.
• The seven colors are violet, indigo, blue, green, yellow, orange, and red (VIBGYOR). The pattern of color which is obtained is called as a spectrum.  Sometimes in the rainbow, you may not see all the seven colors. This is because of the colors overlap each other.