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Refraction at Spherical Surfaces is the basic principle behind the design and working of lenses. Refraction is the bending of the light wave when it moves from one transparent medium to another. This phenomenon occurs as a result of change in speed of the incident light wave. Twinkling of stars, an early sunrise, delayed sunsets, are all examples of refraction of light occuring at spherical surfaces. A pencil in a beaker appears to be broken when viewed from outside, this is another classic example of refraction in daily life.
Spherical surfaces are part of a sphere. Spherical mirrors and lenses are good examples of spherical surfaces. These spherical surfaces can be classified as: concave and convex. When a ray of light passes from air to the spherical surfaces and spherical lenses, the refracted ray gets converged or diverged. This refraction by spherical lenses that makes the incident light ray to bend is used for a wide scale of applications in the field of Ray Optics and Optical Instruments.
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Refraction of Light: Principle
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- Light travels with enormous speed and in a straight line.
- A wavelength of light can be constructed when the light travels from one medium to another (or one point to another) in a straight line.
- The path of the light wave is known as a ray of light. A cluster of such rays is known to be a beam of light.
- A spherical lens is a transparent glass bounded by two spherical surfaces.
- In refraction, light changes its direction and moves away from the normal lens.
- This occurs as light travels from a denser medium to a rarer medium.
- Refraction by spherical lenses makes the incident ray of light to switch direction and move closer to the normal lens.
Refraction of Light at Spherical Surface
Explanation
With the concurrence into another transparent medium, a part of the beam of light gets reflected back into the source medium with the rest entering the other medium. This change in direction of the incident ray of light at the interface of the two media is represented as the phenomenon of refraction of light. Snell’s law provides the basic tool to develop the theory of spherical lenses.
A medium is said to be optically denser, when the speed of the light is more in that medium. Similarly when the speed of a light is less in a medium, it is said to be an optically rarer medium.
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Laws of Refraction by Spherical Lenses
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The laws of refraction of light on spherical lenses or surfaces are listed below.:
- The incident ray, the refracted ray and the normal, to the interface at the point of incidence, lies in the same plane.
- The ratio of sine of both the angles, that is, the angle of incidence to the angle of refraction is constant for the two media, the light is travelling through.
The angles of incidence (i) and refraction (r) are the angles that the incident and its refracted ray make with the normal respectively.
The refractive index of a medium is the speed of the light in first medium through which the light is travelling to the speed of the light in the second medium, where the light gets refracted.
Refractive Index = sin i/sin r = Constant
The refractive index of the air is 1.0003.
Refraction by Spherical Lenses
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A lens is a formation of two transparent glass surfaces on the same side. The portion having an inward curve is known as Concave Surface and the portion which is bulged outwards is known as Convex Surface. These two surfaces combine together to form a lens. Refraction of light occurs at this curved surface when incident light ray passes from air to the spherical lens.
Refraction by Spherical Lenses
Types of Spherical Lenses
- Convex or Convergent Lenses: A convex lens is thicker in the middle and thinner at the edges. A convex lens is also known as a “biconvex lens” as the two spherical surfaces appear to be bulging outwards.
- Concave or Divergent Lenses: A concave lens is thicker at the edges and thinner in the middle. A concave lens is also known as a “biconcave lens” as the two spherical surfaces appear to be bulging inwards.
Types of Spherical Lenses
The point from which the light rays appear to diverge is known as the Focus or Focal point.
Image Formation after Refraction of Light
The following set of diagrams shows the image formation by the spherical lenses when light is incident on them.
Image Formation By Spherical Lenses
Optical Centre
The centre point of a lens is called optical centre. The ray of light that passes through the optical centre moves in a straight direction and does not deviate.
Principal Axis
A line that passes straight through the optical centre of the spherical lens in such a way that it is perpendicular to its sides from the centre is known as the principal axis.
Focal Length of a Convex Lens
The distance between the optical centre and principal focus of a spherical lens is called focal length of convex lens.
Determining Focal Length
Focal length is determined by the refractive index of the glass and its curvature. With higher refractive index focal length will be short. Similarly, if the curvature of the spherical lens is more, the focal length will be short.
Images Formed by a Convex Lens
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In convex lens, the image is always formed at a point where the two rays meet after refraction of light. Listed below are some of the rules for image formation by Convex lenses:
Rule 1: A ray of light which is originally parallel to the principal axis passes through the focus after refraction through lens.
Rule 2: A ray of light passing through the optical centre of the convex lens does not bent after refraction by spherical lens but goes straight. Also, a ray of light going along the path of principal axis of a convex lens also goes straight and does not deviate.
Rule 3: When a ray of light passes through the focus of the convex lens then it becomes parallel to the principal axis after refraction through lens.
The image formation by convex lens can be explained in 5 cases as listed below:
Case 1
When the object is placed between optical centre and focus (between C and F’) then the first ray of light starting from the top of the object is parallel to the principal axis and passes through another focus after refraction through the lens. The second ray from the object passes through the optical centre of the lens and goes straight after refraction of light through the lens. Thus, after refraction by spherical lens through the spherical lens both the light rays diverge and does not meet. Therefore, both the refracted rays are produced backwards to make contact at a point to form an image.
Hence, the image that will be formed will appear behind the object, virtual, erect and larger than the object.
Case 2
When the object is placed at the focus of the convex lens (at F’) then it implies that the object is placed at the distance equal to the focal length of the spherical lens.
One ray of light becomes parallel to the principal axis of the lens and thus, passes through another focus after refracting the lens. Another ray of light passes through the optical centre of the spherical lens and goes straight.
Hence, the image that is formed is at infinity, real and inverted and is highly enlarged.
Case 3
When the image is placed between focus and distance less than twice the focal length (F’ and 2F’) then a ray of light parallel to the principal axis of the spherical lens passes through another focus (F) after refraction through lens. Another ray of light passes through optical centre of the lens and goes straight.
Hence, the image will appear real and inverted, larger than object and beyond 2F
Case 4
When the object is paced at a distance equal to twice the focal length (at 2F’) of the convex lens then one ray of light, after having refraction through lens, becomes parallel to the principal axis and passes through another focus of the lens. Another ray of light passes though optical centre and goes straight after refraction. Both the refracted light rays coincide at 2F` on another side.
Hence, the image obtained will be real, inverted and of the same size as the object.
Case 5
When the object is placed at a distance greater than twice the focus (beyond 2F’) one ray of light becomes parallel to principal axis and passes through focus after refraction through lens and another light ray passes through optical centre and goes straight after refraction by spherical lens.
Hence, the image will be obtained between F and 2F and will be real and inverted and smaller than object.
Case 6
When the object is placed at infinity, the light rays become parallel after being in contact with the spherical lens.
Hence, the image will be formed at the focus on another side and will be real and inverted being highly diminished.
The following set of figures summarises the above cases for better understanding.
Image Formation by Convex Lens
Focal length of concave lens
The distance between optical centre and principal focus is called focal length of a concave lens.
Images Formed by Concave Lens
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Listed below are the rules for images formed by concave lens after refraction of light:
Rule 1: A ray of light parallel to the principal axis of the concave lens appears to be coming from focus after refraction through the lens.
Rule 2: A ray of light passing through the optical centre of the concave lens goes straight after refraction through the lens.
Rule 3: A ray of light going towards the focus on another side of the concave lens becomes parallel to the principal axis after refraction through the lens.
Images Formed by a Concave Lens
Image Formation by Concave Lens
Concave lenses will always form an image having virtual effect being erect and diminished.
Case 1: When an object is placed anywhere between optical centre and infinity, the image formed is between optical centre and focus after refraction.
Case 2: When an object is placed at infinity, the image formed by concave lens will be at focus after refraction by spherical lens.
Spherical Lens Formula
Spherical Lens formula is relatable with the image distance (v), object distance (u) and the focal length (f) of the lens.
1/image distance (v) – 1/object distance (u) = 1/focal length (f)
Ray Optics Handwritten Notes
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Things to Remember
- Refraction of light occurs with bending of incident light ray as it travels from one medium to another.
- Light changes its direction and moves away from the normal lens while travelling from a denser medium to a rarer medium.
- There are two types of spherical Lens employed to understand the prinsiple of refraction of light: Convex Lens and Concave Lens.
- A convex lens is thicker in the middle and thinner at the edges.
- A convex lens is also known as a “biconvex lens” as the two spherical surfaces appear to be bulging outwards.
- A concave lens is thicker at the edges and thinner in the middle.
- A concave lens is also known as a “biconcave lens” as the two spherical surfaces appear to be bulging inwards.
- The distance between optical centre and principal focus is called focal length of a concave lens.
- Spherical Lens formula is relatable with the image distance, object distance and the focal length of the spherical lens.
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Previous Year Questions
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- A bulb is located on a wall. Its image is to be obtained on a parallel wall with the help of convex lens… [NEET 2002]
- The size of the image of an object, which is at infinity, as formed by a convex lens of focal length… [JEE Advanced 2003]
- A bubble in glass slab (μ=1.5) when viewed from one side appears at 5 cm… [NEET 2000]
- The frequency of a light ray is 6×104Hz. Its frequency when it propagates in a… [DUET 2003]
- A convex lens is dipped in a liquid whose refractive index is equal to the… [NEET 2003]
- When a light ray enters from oil to glass on oilglass interface. Velocity of light… [JIPMER 2019]
- A point object O is placed in front of a glass rod having spherical end of radius of… [BITSAT 2008]
- A and B are two parallel sided transparent slabs of refractive indices n1 and n2 respectively… [WBJEE 2016]
- An isosceles prism of angle 120∘ has a refractive index 1.44. Two parallel rays of… [JEE Advanced 1995]
- Two beams of red and violet colour are made to pass separately through a prism with angle of prism… [UPSEE 2019]
- A rectangular glass slab ABCD of refractive index n1 is A rectangular glass slab ABCD of refractive index… [JEE Advanced 2000]
- A vessel of depth 2d cm is half filled with a liquid of refractive index… [VITEEE 2017]
Sample Questions
Ques 1) A biconvex lens made up of a transparent material of refractive index 1.25 is immersed in the water of refractive index of 1.33. Will the spherical lens behave as a converging or a diverging lens? Give reasons. (All India 2014)
Ans. The spherical lens will behave as a diverging lens when light rays enter from air to lens. The refractive index of the lens is lesser than the refractive index of the medium that is water. Therefore, the spherical lens will act as a converging lens when the light rays travel from lens to water.
Ques 2) A ray of light falls on a transparent sphere with center C as displayed by the figure below. The ray emerges from the sphere parallel to the line AB. find the angle of refraction at A if the refractive index of the material of the sphere is √3. (Foreign 2014)
Ans.
Ques 3) Under what condition of refraction by spherical lens does a biconvex lens of a glass having a certain refractive index when immersed in a liquid, acts as a plane glass sheet? (Delhi 2012)
Ans. When immersed in a liquid, biconvex lens of a glass will act as a plane glass sheet when μL = μg ; where μL is the refractive index of the liquid and μg is the refractive index of the glass.
Ques 4) A glass spherical lens of refractive index 1.45 disappears when it is immersed in a liquid. Find out the value of the refractive index of the liquid. (Delhi 2010)
Ans. A glass lens when immersed in a liquid, for its disappearance in the liquid, the value of the refractive index of the liquid
= Refractive index of the lens = 1.45.
Ques 5) Is it possible for the lenses to undergo refraction of light?(1 mark)
Ans. When a ray of light enters a lens, it gets refracted, and similarly when it exits the lens, it is refracted back.
Ques 6) Does refractive index has a unit?(1 mark)
Ans. Refractive index does not have any unit as units tend to cancel out while calculating, but it is calculated as the ratio of unit of light in vacuum to the speed of light in the medium.
Ques 7) What are some examples of concave lens working on the principle of refraction of light?(1 mark)
Ans. Cameras, Lasers, Flashlights, Binoculars are some of the examples of concave lens.
Ques 8) What is the use of refractive index?(1 mark)
Ans. Refractive index is mostly applied for identification of a particular substance, to confirm purity of a substance or to measure its concentration.
Ques 9) An equiconvex spherical lens of focal length ‘f’ is cut into two identical plane convex lenses. How will the power of each part be linked with the focal length of the original lens? A double convex lens of +5 D is made of a glass of refractive index 1.55 with both faces of equal radii of curvature. Find out the value of its radius of curvature. (Foreign 2015)
Ans. The focal length of the original equiconvex lens is f,
After cutting, let the focal length of each part be F,
Here,
1/f = 1/F +1/F
Or, 1/f = 2/F
Or, f = F/2
Or, F = 2f
Therefore, power of each part will be given by,
P = 1/F
Or, P = 1/2f.
From the lens maker formula, we have,
Where, P is the power of spherical lens = +5D
Μ is the refractive index of the spherical lens = 1.55
R1 is the radius of curvature of first face (positive)
R2 is the radius of curvature of second face (negative)
Given,
The radius of the curvature of the spherical lens is 22 cm.
Ques 10) A convex lens of focal length 25 cm is placed coaxially in contact with a concave lens of focal length 20 cm. Determine the power of the combination and say whether the system will converging or diverging in nature? (Delhi 2013)
Ans. As shown in the figure below, the convex lens and the concave lens are in contact,
The system of the lenses is diverging in nature.
Ques 11) A convex lens of focal length f1 is kept in contact with a concave lens of focal length f2. Find out the focal length of the combination. (All India 2013)
Ans. The focal length of a convex lens is +f1
Now, by using spherical lens formula we have,
Again, for concave lens of formula, the focal length is -f2
Now, adding equations 1 and 2, we get
Using the lens formula, for the combined lens, we get
Hence, from the equations 3 and 4 we get
This is the required focal length of the combination of spherical lenses.
Ques 12) (i) Obtain the mathematical relation between refractive indices n1 and n2 of two radii and radius of curvature R for refraction at a convex spherical surface. Consider the object to be a point since it lies on the principle axis in the rarer medium of refractive index n1 and a real image formed in the denser medium of the refractive index n2. Thus, derive the spherical lens maker’s formula for the refraction of light.
(ii) Light from a point source in air falls on a convex spherical glass surface of refractive index 1.5 and radius of curvature 25 cm. The distance of the light source from the glass surface is 100 cm. Then, at what position the image is formed? (AI 2016)
Ans.
The figure above shows the geometrical formation of real image I of an object O and the principle axis of a spherical surface with the center of curvature c and the radius of curvature R.
Assumptions,
- The aperture of the surface is small in comparison to other distances involved.
- NM is taken to be nearly equal to the length of the perpendicular from the point N on the principle axis.
Considering the incident ray to be very close to the principle axis, all the angles are very small. Thus, for small angles,
tan x = x = sin x
Therefore,
By putting the values of i and r from equations 1 and 2 we get,
Applying new cartesian sign conventions,
OM = -u, MI = +v, MC = +R
Putting these values in equation 3 we get,
(ii) As per the question,
Thus, the image is formed at a distance of 100 cm in the denser medium.
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