Image formation in lenses

A convex lens can make real or virtual image of an object depending upon the position of the object from optical centre i.e., when object lies between the principal focus F and optical centre O of the lens, the image formed is virtual and magnified.

Concave lens always produces an imaginary or virtual image independent of the position of the object from optical centre O.

Relative Position, Nature and Size of the images formed by lenses when object is moved from the surface of the lens to infinity

(a) THIN CONVEX LENS :

1. Object Between Lens and Focus F : If an object is placed between a convex lens and one of its principal focus, th rays from a point on the object will be diverging when they emerge out from the other side of the lens as  shown in fig.. 
              
      The diverging rays can then be focused by the eye to form a sharp magnified image on the retina. The lens used in this way is called a magnifying glass.
        
The image is :
1. On the same side as the object
2. Virtual and erect (upright)
3. Larger than object (magnified)

2. Object at focus F : If the object is at the principal focus, the rays from each point of the object become parallel on the other side after passing through the lens as in fig.. 
                 
The image is :
1. Formed at infinity
2. Image is infinitely large 

3. Object between F and 2F :
The image is :
1. Beyond 2F on the other side
2. Real and inverted
3. Larger than object (magnified).(fig. below)
                

4. Object at 2F :

The image is :
1. At 2F on the other side, 
2. Real and inverted,
3. Same size as object
                

5. Object beyond 2F :
The image is :
1. Between F and 2F on the other side
2. Real and inverted
3. Smaller than object (diminished)
           

6. Object at infinity :
The image is :
1. At F on the other side
2. Real and inverted
3. Much smaller than object (point image)
        

(b) A CONCAVE LENS :
The course of rays (see fig.) shows that the image formed by a concave lens is always virtual, erect, on that same side as the object and smaller in size. 

            
As the object is moved from infinity to the surface of the lens, the image moves from the principal focus to the lens.


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Lens in Optics (Physics)

Lens : 
A lens is a portion of a transparent medium bounded by two co-axial spherical refracting surfaces or one spherical and the other plane surface.

Types Of Lenses : 
Lenses are divided into two main classes i.e.,
(i)  Convex lenses
(ii) Concave lenses

Convex lenses : Convex lenses are thick at the middle and thin at the edges. These lenses make the rays passing through them convergent and are also known as converging lenses.

Concave lenses : Concave lenses are thin at the middle and thick at the edges. These lenses make the rays passing through them divergent and are also known as diverging lenses.
             

Important Terms Used In Lenses :

1. Centre Of Curvature : The centre of the sphere of which the spherical surface forms a part is called the centre of curvature of that spherical surface.

2. Radius Of Curvature : The radius of the sphere of which the surface forms a part is called the radius of curvature of that spherical surface.

            

3. Principal axis : The straight line joining the two centres of curvature of a lens is called the principal axis.

                    
In fig. The line joining C1 and C2, the centre of curvature of two spherical surfaces respectively, is the prinicipal axis.

4. Optical centre : Optical centre of a lens is a point on the principal axis, inside or outside the lens, such that any ray passing through it suffers no deviation. It is represented by O in above fig..

5. Principal focus : A point on the principal axis, where, a number of rays coming parallel to the principal axis after suffering refraction converge to or appear to diverge from is called Principal focus of the lens. It is denoted by F.
           
6. Focal Length : The distance between the optical centre and the principal focus of lens is called focal length. It is denoted by f. In above fig., the distance CF is the focal length, where O is the optical centre.
                                                              The focal length of convex lens is taken as positive whereas the focal length of a concave lens is taken as negative.

7. Aperture : The diameter of the circular boundary of a lens is called the aperture of the lens.

8. Image : If a number of rays starting from a point after suffering refraction through a lens, converge to or appear to diverge from a second point, then this second point is called the image of the first point. 
                                                                       The image formed is real if the rays actually meet at the second point whereas, the image formed is virtual if the rays appear to diverge from the second point.


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Real depth and Apparent Depth

Real depth and Apparent Depth :

When an object is placed in a denser medium, it appears to be raised. 

         

To find the apparent depth, consider an object O placed in denser medium, say, water at a distance u from the surface XY. A ray of light OA incident normally on the surface of separation passes into air along AA' without suffering refraction.

Another ray OB incident at B at an angle i is refracted along BC away from the normal making an angle r with the normal NB. The ray CB when produced backward will meet OA in I.

                                       To an eye looking into the medium the rays BC and AA' will appear to diverge from I. The object O will thus appear to be raised to the point I, which is, therefore, the virtual image to the object O. 

If µ is the refractive index of water with respect to air, then 1/µ will be the refractive index of air with respect to water. Thus

            sin i / sin r = 1/µ

Now 

             ∠AOB = i     and      ∠AIB  = r

 or               1/µ  = sin i/sin r

                                          =  AB/OB   X    IB/AB

                                          = IB/OB

If B lies very close to A, then

OB = OA         and          IB = IA

1/µ = IA/OA

therefore,         µ = OA/IA = Real depth/ Apparent depth

                          = u/v

now                 µ > 1

∴                     u > v

Hence, the image of the object O appears to be raised to the position I.

Thus when an object is placed in a denser medium and is viewed through a rarer medium it appears to be raised. It is for this reason that the bottom of a tank filled with water appears to be raised.

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Numerical Problems Optics ( Refractive Index)

Question

1. Refractive index of glass is 1.5. If the speed of light in vacuum is 3 X 10⁸

  m/s, find velocity of light in medium.

Solution: Refractive index,   µ = C / v

                                            = Velocity of light in vacuum / Velocity of light in medium

                                              v = C / µ

                                                 = 3 X 10⁸ / 1.5

                                                 = 2 X 10  m/s. (Ans.)

Question

2. Speed of light in glass is 2 X 10⁸ m/s. Find the refractive index of glass.

Solution: Refractive Index of glass (µ) = C / v

                                                        = 3 X 10⁸ / 2 X 10⁸

                                                        = 1.5  (Ans.)

Question

3. If active index of water is 4/3 and that of glass is 3/2. Find the refractive index of glass w.r.t. water.

Solution:             µ_w =  4/3,       µ_g = 3/2

Refractive index of glass w.r.t. water ( ^wµ_g  ) = ^aµ_{g   /}  ^aµ_w

                                                                 = 3/2 / 4/3

                                                                 = 3/2 X 3/4

                                                                 = 9/8      (Ans.)

Question

4. An object at the bottom of a beaker filled with a liquid up to height of 10 cm. If the refractive index of liquid w.r.t. air ( ^aµ_w ) is 4/3, find the apparent depth of the object.

Solution:              Real depth = 10 cm

                                ^aµ_w      =  4/3

                                µ         =  Real depth / Apparent depth

                                4/3      =          10    /  Apparent depth

                 Apparent depth    =  10 X 3 / 4

                                           =   30 / 4    

                                           = 7. 5 cm (Ans.)  

Multiple Choice Questions:  

1. The refractive index of a material depends upon

(a) Temperature                  (b) Wavelength of light

(c) Nature of the material    (d) All the above

2. When light travels from glass to water separated by a sharp boundary then it 

(a) proceeds undeviated               (b) bends towards the normal

(c) bends away from the normal    (d) is reflected back

3. A cut diamond sparkles because of

(a) it's hardness

(b) it's high refractive index and small value of  the critical angle

(c) it's high refractive index and high value of  the critical angle

(d) it's very low refractive index

4.Velocity of light is maximum in 
(a) diamond              (b) water
(c) glass                   (d) vacuum

5. Refractive index of glass with respect to air is 1.9. What is the refractive index of air with respect to glass
(a) 1                        (b) 1.9
(c) 0.523                  (d) 0.776

6. The cause of twinkling of stars is :
(a) periodic bursts of light from the stars
(b) interference of sunlight with the star light
(c) partial absorption of light in the atmosphere
(d) refractive index fluctuation in the atmosphere

7. When a light ray travels from air to water separated by a sharp boundary it:
(a) proceeds undeviated
(b) bends towards the normal
(c) bends away from the normal
(d) is totally internally reflected

8.The sun becomes visible before the actual sunrise and remains visible even after the actual sunset. It is because of
(a) scattering of light
(b) diffraction of light
(c) refraction of light
(d) dispersion of light

9. just before setting, the sun may appear to be elliptical. This happens due to 
(a) reflection              (b) refraction
(c) dispersion             (d) diffraction

10. The refractive index of glass with respect to air is 3/2. The refractive index of air with respect to glass is :
(a) 3/1                     (b) 3/2
(c) 2/3                     (d) 1/3

Answers :
1. d
2. c
3. b
4. d
5. c
6. d
7. b
8. c
9. b
10. c

Refraction At The Time Of Sun Rise And Sun Set

Refraction At The Time Of Sun Rise And Sun Set :

Sun becomes visible about two minutes before the actual sun rise and remains visible for two minutes even after  sun set. Thus duration of the day is approximately increased by 4 minutes.

          
          

As the sun rays enter the atmosphere of earth, they travel from rarer to denser medium i.e., from vacuum to air (µ ≈ 1.003) and density of air go on increasing towards the surface of earth, hence rays of light when enter earth's atmosphere go on bending (Refracting) towards the normal and finally meet at a point i.e., observer on earth.

                          When these refracted rays are produced backward from observer, then it appears that as if they are coming from sun at a position above the horizon, both at the time of sun rise as well as sun set.

                          That is why sun rises 2 minutes early and sets 2 minute late. Also the surface of sun appears flattened at the time of sun rise and sun set due to refraction of light through atmosphere.

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Laws Of Refraction and Refractive Index

For a homogeneous and an isotropic medium, the phenomena of refraction is governed by two laws called Laws of Refraction.

1. First Law :  The incident ray,  the normal to the surface of separation, at the point of incidence and the refracted ray, all lie in the same plane perpendicular to the surface of separation.

2. Second Law : The ratio of sine of the angle of incidence to the sine of angle of refraction is a constant quantity for any two media for the same color of light. The constant is called the refractive index of the 2nd medium with respect to first medium and is denoted by µ.

So                                    

                    Sin i /sin r  =  constant  =  ^aµ_b 

                                                                        

where  ^aµ_b   shows that light is traveling from the medium 'a' into the medium 'b'.

i.e. ^aµ_b  represents the refractive index of medium 'b' with respect to medium 'a'.

Refractive index :

The refractive index of a medium with respect to another medium is the ratio of the sine of the angle of incidence to the sine of angle of refraction when light travels from first medium into second medium.   

                                                                      

Absolute Refractive index: When the first medium (a) is vacuum, then

sin i / sin r = µ  , gives the absolute refractive index of medium b.

Refractive Index in terms of Velocity of Light :

When light passes from one medium to another medium, there is change in velocity of light, thus due to this change in velocity of light, there is refraction (bending of light), hence refractive index may also be defined in terms of velocity of light.

The ratio of velocity of light in vacuum to the velocity of light in a medium is known as the refractive index of the medium.

i.e.,                µ = Velocity of light in vacuum / Velocity of light in medium

                      µ = C / v

Velocity , of light in vacuum (C) = 3 x 10⁸ m/s (Approximately)

There are two types of medium depending upon the value of refractive index.

(i) Optically denser medium : The medium in which value of refractive index is more than 1 is called optically denser medium.

i.e.,           
           µ = C/v >> 1 

i.e., value of v (velocity of light in medium is much less than velocity of light in vacuum (C).

(ii) Optically rarer medium : The medium in which value of refractive index is small i.e., less than 1 is called optically rarer medium.

i.e.,         

            µ = C/v << 1                 

Refraction Of Light

When a ray of light travels from one medium to another medium, either it bends towards the normal or away from normal.

This beam is hitting the glass at a 45-degree angle, with respect to the normal.
                Some of the light bounces off, "reflected". Some of the light goes through, but the angle changes "refracted".

When the refracted beam passes through the other side of the glass, it is refracted again. 





The process of bending of light ray from its straight path as it enters from one medium into another medium is called Refraction of light.

OR

The change in direction of straight line path of ray of light when it travels from one medium to another medium is called refraction.

e.g., When a ray of light incident beam falls on a surface separating a rarer medium (air) and denser medium (glass), it is derived from its path and bends towards normal, but when it travels from denser medium (e.g., glass in this case) to rarer medium (air), it bends away from the normal as shown in fig above.

Angle of incidence : The angle made by incident ray with normal at the point of incidence is called angle of incidence.

Angle of Refraction : The angle made by the refracted ray with the normal is called angle of refraction.

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Optics - Test Questions (Objective type Questions)

1. A convex mirror forms image of a real object . Th image will be
(a) real, erect                  (b) real,inverted
(c) virtual, erect              (d) virtual, inverted

2. In case of spherical mirrors, the focal length and the radius of curvature are related as-
(a)  f = R/2                 (b) f = 2R
(c)  f = R                    (d) None of these

3. A dentist's concave mirror has a focal length of 3 cm. The mirror is placed 2 m from the tooth. The magnification of the image will be
(a)  +6                      (b) +1/6
(c)  -3                       (d) +3

4. In search lights and head lights of automobiles the mirror used is
(a) Concave mirror          (b) Plane mirror
(c) Convex mirror           (d) None of these

5. A ray of light is incident on a plane mirror at an angle of incidence 30°.The ray after reflection is deviated through
(a) 30°                            (b) 60°
(c) 90°                            (d) 120°

6. A virtual image larger than the object can be produced by
(a) A concave mirror      (b) A convex mirror
(c) A plane mirror           (d) Concave  lens

7. A diminished virtual image can be obtained only in the
(a) Concave spherical mirror      (b)  Plane mirror 
(c) Convex spherical mirror       (d)  Concave Parabolic mirror

8. Reflectors used in solar cookers are
(a)  Plane                 (b) Concave
(c) Convex              (d) Cylinderical

9. Which of the following mirrors is also known as shaving mirror
(a) Concave           (b) Convex 
(c)  Plane               (d) None of these

10. Light waves have 
(a) Particle nature    (b) Wave nature
(c) Dual nature        (d) none of these

11.which of the following mirror cannot form a virtual image ?
(a) Concave      (b) Convex
(c) Plane           (d) All of the above can produce a virtual  image

12.While using an electric bulb in a street lighting, the reflector should be a
(a) Concave mirror    (b) Convex  mirror
(c) Plane mirror         (d) Cylinderical  mirror

13. When a number of rays starting from a point after suffering reflection or refraction converge to meet or diverge from a second point, then this second point is called
(a) Pole                   (b) Focal point
(c) Object               (d) Image

14. Focal length of a plane mirror is
(a)  0                      (b)  1
(c)  ∞                     (d)  None of these 

15. A man is looking his magnified image in a mirror placed in front of him. The Kind of the mirror he is using is
(a) Plane mirror                       (b) Convex mirror
(c) Concave mirror                  (d) None of these

16. Which of the following is not the case with the image formed by a convex mirror ?
(a)   It is erect                     (b) It is virtual
(c)   It is diminished           (d) It lies beyond the focus

17. A ray of light is reflected at angle 30°. If the angle of incidence becomes double, then the angle of reflection will be

(a) 90°               (b) 60°

(c) 45°               (d) 30°

18.What is the radius of curvature of a plane mirror ?

(a) Zero                                            (b)Between Zero and One

(c) Varies from surface to surface       (d) infinite

Image Formation in Concave Mirror and Convex Mirror

Image Formation in Concave Mirror :

Let us discuss the various positions and sizes of images formed by a concave  mirror i.e., when objetct is moved from infinity to the pole of concave mirror.

Properties of images :
a) Real image : Image which can be obtained by actual meeting of reflected rays
b) Virtual image: Image resulting from the ( back intersection) of diverging reflected rays.
c) Erect: When the image is straight
d) Inverted : Image upside down
e) Magnified : Larger than object
f) Diminished : Smaller than object

(i) When object is at infinity i.e., at infinite distance from the pole of the concave mirror. or when u = .

Image formed will be real and point in size as shown in fig (a).

Case (a): Object placed at infinity
  

Nature of image: Real, inverted, diminished, at the focus

Case (b): Object placed beyond C, center of curvature i.e., (u>2F)

Nature of image: Real, inverted, diminished image, beyond F and C

Case (c): Object placed at center of curvature
Nature of image: Real, inverted, same size as of object, at centre of curvature

Case (d ): Object placed between Centre of curvature and focus

Nature of image: Real, inverted, magnified, beyond centre of curvature

Case (e): Object placed at Focus
Nature of image: Real, inverted, highly magnified, at infinity.

Case (f): Object placed between pole and focus

Nature of Image: This is the only case in which we get virtual and erect image. Image is magnified and behind the mirror.

Image Formation in Convex Mirror :

In case of convex mirror for all positions of object ; virtual, erect and diminished image will be formed between pole of the lens and principal focus of the mirror  on the other side of the mirror.

(i) When object is at infinity :

In fig (a) below, A virtual point image will be formed at the principal focus of the mirror on the other side of the mirror.

(ii) When Object is placed at 2F :

A virtual, erect and diminished  image is formed between P and F on the other side of the mirror in fig.(b)below:

Points to remember: 

A plane mirror and a convex mirror always form virtual image whatever be the position of the object.

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Image in Optics (Physics)

When a number of rays starting from a point after suffering reflection or refraction converge to meet or appear to meet(diverge from) a second point, then this second point is called the Image of the first point.

                   

                    
Image formed is of two types:
Real image
Virtual image

Real image :
If the rays of light starting from a point after suffering reflection or refraction actually meet at the second point, then the image formed  will be real as shown in fig.1. Real image can be taken on screen.
Virtual image :
If the rays starting from a point after suffering reflection or refraction do
not actually meet but appear to diverge from a second point, then this type of image formed is virtual in nature. e.g., when we look our face in a plane mirror, we see an virtual image of our face in the plane mirror. A virtual image is shown in fig.2.

Image formation in Spherical Mirrors:
Let us first discuss the rules for forming image of an extended object by a spherical mirror. while drawing a ray diagram for the formation of an image of an object, remember to keep this in mind:

(a) A ray parallel to principal axis after suffering reflection from-
(i) a concave mirror will pass through principal focus.
(ii) a convex mirror will appear to diverge (coming out) from the principal focus.

(b) A ray passing through the centre of curvature of mirror (concave as well as convex mirrors) will retrace its path after suffering reflection from mirror.

(c) A ray passing through the principal focus of-
(i) Concave mirror  the ray of light will be reflected, and will became parallel to principal axis.
(ii) Convex mirror the ray of light appearing to be diverging out of principal focus of the convex mirror will be reflected and will become parallel to principal axis of mirror.

(d) A ray of light falling on the pole of mirror (convex or concave) at some angle to the principal axis, is reflected by making same angle (as angle of incidence) with the principal axis.

Applications of Spherical Mirrors :
Spherical mirrors have the following important applications :
(a) Concave mirrors :
(1) They are used in search lights, head lights for vehicles, torches etc.
(2) These are used as paraboloids in dishes for receiving and sending radio signals.
(3) They are used as reflectors in cinema projectors, solar cookers and reflecting type telescopes.
(4) They are used by dentists.
(5) Concave reflectors are used in table lamps.
(6) They are preferred to a plane mirror for shaving because a concave mirror forms an erect, virtual and magnified image when placed close to the eye.

                         

 

(b) Convex mirrors : 
(1) It is used as rear view mirror in vehicles  i.e., for looking at the back traffic because a convex mirror always forms erect and diminished image of the traffic behind the automobile. It helps driver to get a wide view of the traffic coming from behind.
(2) Some camera phones use convex mirrors  to allow the user correctly aim the camera while taking self portrait.
(3) It can also be used as a reflector for street lighting purpose.
                
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Mirror in Optics (Physics)

Mirror:
Surface that reflects most of the light falling on it is called a mirror.

Mirrors are of two types:
1.  Plane
2.  Spherical

(i) Plane mirror: A plane mirror is a flat  surface that produces an erect and virtual image of a real object in which front and back are reversed.
                                                

Light rays from an object strike the surface of a plane mirror. The rays are reflected on the mirror and appear to come from behind the mirror. The reflected rays are drawn as if they are from the image. The image is virtual. Therefore the rays behind the mirror do not exist. They are virtual rays and are represented by dotted lines. The continuous lines from the mirror to the eye indicates the reflected rays. The distance between the object and the mirror is the same as the distance between the image and the mirror.



Characteristics of the image of a plane mirror:
1. VirtualSame 
2. Size as the object 
3. laterally inverted

(ii) Spherical mirror: A Spherical mirror is the portion of a sphere having a reflecting surface. Spherical mirrors form images of real objects in same way as lenses. These are of two types;
1. Convex mirror
2. Concave mirror

Convex mirror: The mirror in which the reflecting surface is formed from the exterior surface i.e., the outer surface away from centre, reflects light.
Concave mirror: A Concave mirror is one in which the reflecting surface is formed from the interior surface of the sphere i.e., the inner surface toward the centre, reflects light.

                    

Important Terms Related to Mirrors:
(i) Centre of curvature (C) : It is the centre of the sphere of which mirror forms a part. It is denoted by 'C'.
(ii) Radius of Curvature (R) : The radius of a sphere of which mirror forms a part  is called radius of curvature.
(iii) Pole : The middle point of the mirror is  called pole of the mirror.
(iv) Principal axis : The line joining centre of curvature to the pole of the mirror is called principal axis.
(v) Aperture : It is the diameter of the circular boundary of the mirror.
(vi) Principal focus : It is a point on the principal axis of the mirror to which incident rays parallel to the principal axis converge (Actually meet at a point) or from which they appear to diverge (Appears to be coming out of that point) after reflection from the mirror. It is clear from the following fig.
         


In case of concave mirror, rays of light after suffering reflection from mirror actually meet at principal focus while in case of convex mirror, they do not meet actually but meet virtually i.e., they appear to be meeting at 'F'.
(vii) Focal length : The distance between the principal focus of a mirror and its pole is called focal length of the mirror. PF is the focal lengths of concave mirror and convex mirror respectively. It is denoted by f.

(viii)Relation between focal length and Radius of curvature

                                                              f  = R/2
                              Radius of curvature = 2 X focal length
i.e., Radius of Curvature is twice the focal length of the mirror.

(ix) Linear magnification :
                                             m = h2 / h1
         where h2 is height of image
           and   h1 is height of object.
                                            m = I / O = -v / u
           I is size of image
          O is size of object
           v  :  distance of image from pole of mirror
           u  :  distance of object from pole of mirror

(x)  Mirror Formula : 

For Concave mirror :     1/u + 1/v = 1/f

For Convex mirror :       1/u + 1/v = 1/f                                 


Courtesy:oocities.org
Stressfreestudies
R.A.Banwat