Optics

Introduction The Human eye is sensitive to visible light (electromagnetic radiation belonging to wavelength range 400 nm to 750 nm). It is through this sensation produced in the eye that we are able to interpret the world around us. Light as it is known, travels with a speed in a straight line. This happens when the wavelength of light is very small compared to the size of ordinary objects. The straight line path along which light travels is called a ray of light and a bundle of such rays constitute a beam of light. Let us familiarise ourselves with phenomena of reflection, refraction and dispersion using the ray picture of light. Reflection of Light by Spherical Mirrors It is the phenomenon of change in the path of light without any change in medium.

According to the laws of reflection,  

Angle 'i' =Angle 'r' i.e., angle of incidence = angle of reflection. The incident ray A0, reflected ray OB and normal ON to the mirror, all lie in the same plane. These laws are valid at each point on any reflecting surface, plane or curved.

Spherical Mirrors

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It is a part of a hollow sphere, whose one side is reflecting and other side is opaque. The types of mirrors are:  

Concave mirror Convex mirror

Concave Mirror Concave mirror whose reflecting surface is towards the centre of the sphere of which the mirror is a part.

Convex Mirror Convex mirror is one whose reflecting surface is away from the centre of the sphere of which the mirror is a part.

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In the above diagram,    

'C' is the centre of curvature of the spherical mirror, which is the centre of the whole sphere of which the mirror forms a part. 'P' is called the vertex or pole of the mirror, which is the mid point or centre of the spherical mirror. 'CP' is called the radius of curvature (R) M1M2 is called the aperture of the mirror. The straight line joining the pole and the centre C extended on both sides is called the principal axis of the mirror.

Definitions Principal Focus F is a point on the principal axis of the mirror at which, rays incident on the mirror in a direction parallel to the axis actually meet or appear to diverge 3

from, after reflection from the mirror. F is a real point in case of concave mirror and F is a virtual point in case of convex mirror. Focal Length The distance of principal focus from the pole of the spherical mirror is called focal length (f) of the mirror. i.e., PF = f Radius of Curvature The distance of C from P is called radius of curvature of the mirror. i.e., PC = R Sign Conventions and Rules for Drawing Ray Diagrams The Cartesian sign conventions adopted during measurements are as follows:

 



All distances are measured from the pole of spherical mirror. Distances measured in the direction of incidence of light are taken as positive, and when measured in a direction opposite to the direction of incidence of light is taken as negative. The heights measured upwards to the principal axis are taken as positive and negative if measured downwards.

Rules for Drawing Ray Diagrams 

Any ray of light traveling parallel to the principal axis, after reflection passes through the focus or appears to diverge from the focus.

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Concave Mirror

Convex Mirror (Curvature Reflects) 

Any ray of light passing through the centre of curvature reflects along the same path.

Concave Mirror

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Convex Mirror



A ray of light initially passing through the focus after reflection travels parallel to the principal axis.

Concave Mirror

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Convex Mirror

Relation between Focal Length and Radius of Curvature in Spherical Mirrors

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Consider a ray of light AB, parallel to the principal axis, incident on a spherical mirror at point B. The normal to the surface at point B is CB and CP = CB = R, is the radius of curvature. The ray AB, after reflection from mirror will pass through F (concave mirror) or will appear to diverge from F (convex mirror) and obeys law of reflection, i.e.,

. From the geometry of the figure,

If the aperture of the mirror is small, B lies close to P,  BF = PF or FC = FP = PF or PC = PF + FC = PF + PF or R = 2 PF = 2f

or Similar relation holds good for convex mirror also. In deriving this relation, we have assumed that the aperture of the mirror is small. Concave Mirror

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Object at Infinity

Object at 2F or C

Object between C and F

Object at F

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Object beyond C

Object between F and P

Table depicting the position and nature of the object and image Sl.No.Position Position of

Nature of the Image 10

of Object 1 2 3 4 5 6

Image

Image is Real, inverted and small in formed at size focus Image is At C Real, inverted and same size formed at 'C' Image is Between Real, inverted and large in formed beyond C and F size C Image is at Real, inverted and large in At F infinity size Image is Between Virtual, erect and large in behind the F and P size mirror Image between Real, inverted and Beyond C F and C diminished At infinity

Image for a convex mirror is virtual, erect and diminished. Real ../content/CB12P1/content/topic/ch599/images form on the same side of a mirror where the object is and the virtual ../content/CB12P1/content/topic/ch599/images form on the opposite side. The Mirror Equation A formula giving the relation between focal length of the mirror, object distance, image distance and radius of curvature, is called the mirror equation.

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Depending on the position of the object, the image formed may be real or virtual. When the object is placed beyond 'C', the image is formed between 'C' and 'F'. Since the reflected rays actually meet at A', the image is said to be real.

Again

From the above two equations we have

Since all distances are measured from P CB = PB - PC CBl = PC - PBl Therefore,

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Using the sign convention, PB = - u (object distance) PC = -R PBl = -v (Image distance) On substitution,

+uR - uv = uv - vR uR + vR = 2uv Dividing both sides by uvR, we have

where f is the focal length of the mirror. Note: The above mirror formula is same for convex mirror for which image is always virtual.

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Linear Magnification of Spherical Mirror Linear magnification is ratio of the size of the image to the size of the object.

We know,

Concave Mirror For Real Image using sign convention, AlBl = -h2 ; AB = +h1 PBl = -v, PB = - u For Virtual Image AlBl = +h2 ; AB = h1 PBl = +v ; PB = -u

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Convex Mirror Only Virtual Image AlBl= +h2; AB = +h1 PBl = +v ; PB = -u

Note: If m>1, image is enlarged. If m is +ve , image is erect and virtual. If m is -ve, image is inverted and real. Other formulae for magnification:

or

Applications of Spherical Mirrors

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

  



A convex mirror is used as a rear view mirror in vehicles as ../content/CB12P1/content/topic/ch599/images are small, erect. This gives us a wider view of the traffic behind. Convex mirrors are also used in reflecting telescopes. They are also used as reflectors in street lamps, as a result the light from the lamp diverge over a large area. A concave mirror is used as a reflector in search light, head light of motor vehicles, telescopes, solar cookers, shaving mirrors, microscopes. Concave mirror is used in ophthalmoscope, for reflecting light on to the retina of the eye.

Multiple Images When an object is kept between two plane mirrors inclined at an angle , more than one image forms. The total number of ../content/CB12P1/content/topic/ch599/images formed in such a case can be related to the angle as follows :

Question If  = 90o, find out the number of ../content/CB12P1/content/topic/ch599/images formed

=4-1=3 Three ../content/CB12P1/content/topic/ch599/images will be formed.

Refraction of Light Light not only bounces off surfaces but also goes through some of them often slowing down and changing direction in the process called refraction. It occurs at the point where light travels from one medium to another of different density. Refraction produces mirages and rainbows.

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When a ray of light (i.e., the incident ray) goes from rarer to denser medium, the ray (refracted ray) bends towards the normal in the denser medium.

If ray of light travels from denser to rarer medium the refracted ray bends away from the normal.

The Laws of Refraction

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

The incident ray, refracted ray and the normal, all lie in one plane.



Snell's Law: The ratio of the sine of angle of incidence to the sine of angle of refraction is constant for the pair of media in contact. This is denoted by 1 2 and is refractive index of medium 2 with respect to medium 1.

i.e., Where  2 and  1 are the absolute refractive indices of the media with respect to free space. This is Snell's law. The refractive index is characteristic of the pair of media and does not depend on the angle of incidence. If the refractive index is >1, then r<1, the medium 2 in which refracted rays travels is optically denser than 1. Consider the following diagram:

a

g

represents refractive index of glass with respect to air, then

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Similarly if refraction occurs from denser to rarer medium,

Then multiplying the two, or

Note:

The above expression is an important result of the principle of reversibility of light which states that when a ray of light after suffering a large number of reflections and refractions has its final path reversed, it travels back along the same path in the opposite direction as shown below.

Principle of reversibility

Relation between Relative Refractive Index and Absolute Refractive Index

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Let light travel from air to medium 1. If c and V1 are the velocities of light in these media, the refractive index of medium 1 with respect to air, or the absolute refractive index of medium 1 is given by

Similarly, when light travels from air to medium 2, we can write

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

Comparing the above equation with equation

, we get

Thus the relative refractive index between a pair of media is the ratio of their absolute refractive indices. While the absolute refractive index of any material medium is always greater than unity, its relative refractive index may be greater or lesser than unity. Practical Examples of Refraction Relation between Real and Apparent Depths The apparent depth of an object lying deeper in an optically denser medium appears to be lesser than its actual depth, due to refraction at a plane surface. This can be seen as follows.

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Apparent depth for nearly normal viewing

Consider a ray of light incident on XY, normally along OA, it passes straight along OAAl. Consider another ray from O (the object) incident at an angle on XY, along OB. This ray gets refracted and passes along BC. On producing this ray BC backwards, it appears to come from the point I and hence AI represents the apparent depth, which is less than the real depth AO.

In

In When angles are small, B lies close to A such that

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Now, the apparent shift in the position of the object IO = AO - AI

 

Its due to this reason that water tank appears shallower on account of refraction of light. Sun is visible to us before actual sunrise and after actual sunset due to atmospheric refraction of light.

Advanced sunrise and delayed sunset due to atmospheric refraction 

Rays from sun (S) entering the earth's atmosphere travel from rarer to denser medium. This results in the rays bending towards the normal and appears to come from S1, the apparent position. Therefore sun

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

appears above the horizon. It is for the same reason it continues to be seen a few minutes after it actually sets. Hence the day becomes longer by about 4 minutes due to refraction effect. The twinkling effect of a star is due to atmospheric refraction. As the refracting media are not steady, the rays bend through fluctuating masses of air in motion and this causes fluctuations in the apparent position of the star and hence gives the twinkling effect.

Total Internal Reflection

Total Internal Reflection

Consider an object at O in a denser medium. A ray of light incident normally to XY goes undeviated along AB. As the angle of incidence increases, the angle of refraction also increases. A1B1 and A2B2 bend towards the surface XY. On increasing the angle of incidence for a particular value say i = C, the angle of refraction is found to be 90o. The ray A3B3 travels along XY. When i > C, the ray goes along A4B4 i.e., the ray is reflected into the denser medium itself. This phenomenon is called total internal reflection. Conditions for Total Internal Reflection  

Light should travel from a denser to rarer medium. i > C in denser medium for a pair of media in contact. 23

Relation between Refractive Index and Critical Angle According to Snell's law,

When i = C, r = 90o

Or

Or Note:  depends on wavelength and therefore 'c' for a same pair of media in contact will be different for different colors. Applications of Total Internal Reflection 

Mirage is an optical illusion, which occurs usually in deserts on hot summer days. On such a day, temperature of air near the earth is maximum and hence is rarer or lighter. The upper layers of air, which are relatively cool, are denser. A ray of light from the top of a tree travels from denser to rarer and bends away from the normal. At a particular layer, if the angle of incidence is greater than 'C', total internal reflection occurs. To far away observer, this ray i.e., AE appears to be coming from I i.e., mirror image of O. Thus inverted

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image of tree creates an optical illusion of reflection from a pond of

water.



Right angled isosceles prism can turn light through 90o or 180o. This is based on total internal reflection. Since  for glass-air is 1.5, the value of 'C' is 42o. In such a prism, the angle of incidence in the denser medium is 45o(>C) and hence light suffers total internal reflection.

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

The brilliance of diamond is due to total internal reflection. Now  for diamond is 2.42 and C (the critical angle) is 24.4o for diamond-air interface. The faces of the diamond are so cut that a ray of light entering the diamond fall at angle greater than 24.4o. This results in multiple, total internal reflections at various angles and remains within the diamond. Hence

diamond sparkles.



Optical Fibres Optical fibres consist of a very fine quality of glass or quartz fibres. They are coated with thin layer of material of lower refractive index than that of the fibre. The thickness of the strand is 10-4 m. The optical fibre works on the principle of total internal reflection.

Light through optical fibre

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The word total means that reflection in the above case occurs with no loss of intensity. This phenomenon enables doctors to inspect many internal body sites. A Bundle of fibres transmit an image that can be inspected visually outside the body. Optic fibres are as thick as a human hair. If a beam of light is sent down a thin glass rod, total internal reflection traps the light inside the rod. This technique is called 'fibre optics'. Fibre optics finds its use in the medical field too. Endoscopes use fibre optics technique. A patient can swallow a tube containing a fine glass fibre through which a doctor can examine the internal stomach parts and hence unnecessary surgeries can be avoided. 'Fibre optics' is used to destroy tumors. If a fibre optic cable is passed into the organ, laser light can be directed along it. The laser is directed at the tumor cells and kills them. The red plastic reflector on the back of a bicycle uses total internal reflection. Spherical Refracting Surface A refracting surface, which forms a part of a sphere of transparent refracting material, is called spherical refracting surface. The two types are convex spherical refracting surfaces and concave spherical refracting surfaces. Convex Spherical Refracting Surface

Concave Spherical Refracting Surface

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XY is the refracting surface. P is the pole of spherical refracting surface. C is centre of curvature of spherical refracting surface. 1, 2 are the absolute refractive indices of the two media. Assumption: In dealing with refraction at spherical refracting surface, we assume,   

The object to be a point lying on the principal axis of the spherical refracting surface. The aperture of the spherical refracting surface is small. The incident and refracted rays make small angles with the principal axis of the surface so that sinii and sinr r

The sign convention used in mirror is applicable for spherical refracting surfaces. Refraction from Rarer to Denser Medium at a Convex Spherical Refracting Surface Real Image

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Consider a spherical surface XY convex to the incident ray OA. The point O is a point object and I is the image of the point object where the refracted rays actually meet.

From A draw a perpendicular on the axis so as to meet M. Let

In (Since exterior angle is equal to sum of the interior opposite angles)

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In triangle OAC, i =  +  According to Snell's law,

As the aperture is close, M is close to P.

Using the sign convention, we put PO = -u, PI = +v, PC = R

OR

Note: For the virtual image, the point lies close to the pole of refracting surface. In this case the refracted rays PC and AB do not meet actually at any point but appear to come from a point I as shown below. 30

Refraction from Denser to Rarer Medium at a Concave Spherical Refracting Surface

Let the point object lie on the principal axis. A ray of light meets the spherical surface concave to the incident ray at A. The refracted ray bends away from the normal C A N and moves along AI. Since the two refracted rays AI and PI actually meet, I represent a real image. Now, from Snell's law,

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(Since refraction occurs from denser to rarer) or  2 sin i =  1 sin r or  2 i =  1 r (as i and r are small angles) In  OAC i=- In  AIC r=+

From A, draw AM perpendicular to principal axis

Or For small aperture, M is close to P

Applying the sign convention Following the procedure as in previous case we have PO = -u, PI = +v, PC = -R 32

We have

or

Lenses Lens is a portion of transparent optical medium bound by two spherical surfaces or at least one spherical surface and the other plane surface. Lenses are divided into two classes: Convex or Convergent Lens Convex lenses are thin at the edges and thick at the centre.

Concave or Divergent Lens Concave lenses are thick at the edges and thin at the centre.

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Important Terms Principal Axis It is defined as a straight line passing through the centres of curvature of two surfaces of a lens. Optical Centre (C) It is a point lying on the principal axis of the lens so that a ray of light whose refracted path passes through this point will have its emergent path parallel to the direction of the incident ray. Principal Focus (F) When a beam of light is incident on a lens in a direction parallel to the principal axis of the lens, the rays after refraction through lens converge to or appear to diverge from a point on the principal axis. Focal Length The distance between the optical centre and the principal focus is known as focal length of a lens. Convex Lens In case of a convex lens, the rays after refraction actually come to a focus and hence a real focus is obtained at F.

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Concave Lens In case of a concave lens, the rays after refraction, appear to come from F and hence F is a virtual focus.

According to sign conventions, f is positive for convex lens and is negative for concave lens. Lens Formula This gives the relation between focal length, object distance and image distance from the optical centre of the lens. Convex Lens (Real Image)

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Consider an object AB held perpendicular to the principal axis at distance beyond the focal length of the lens. A real, inverted and enlarged image is formed as shown.

Also

But CD=AB

It follows

BC = -u BlC = +v using sign convention CF = +f FBl = CBl- CF = v - f

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vf = -uv + uf uv = uf - vf Dividing throughout by uvf,

Note: (1) The above formula is applicable for a convex lens even when a virtual image is formed. For this the following ray diagram is to be considered.

2) For concave lens, the ray diagram would be

Linear Magnification Produced by a Lens

Convex Lens Real Image

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Virtual Image

Concave Lens

Lens Maker's Formula It is a relation that connects focal length of a lens to radii of curvature of the two surfaces of the lens and refractive index of the material of the lens. The following assumptions are made for the derivation: 

  

The lens is thin, so that distances measured from the poles of its surfaces can be taken as equal to the distances from the optical centre of the lens. The aperture of the lens is small. Point object is considered which lies on the principal axis. Incident and refracted rays make small angles with the principal axis.

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Consider a convex lens (or concave lens) of absolute refractive index  2 to be placed in a rarer medium of absolute refractive index  1. Considering the refraction of a point object on the surface XP1Y, the image is formed at I1 which is at a distance of v1. CI1= P2I1 = v1 (as the lens is thin) CC1 = P1C1 = R1 CO = P1O = u It follows from the refraction due to convex spherical surface XP1Y The refracted ray from A suffers a second refraction on the surface XP2Y and emerges along BI. Therefore 'I' is the final real image of O. Here the object distance is (acts as virtual object) (Note P1P2 is very small) Let

(Final image distance)

Let R2 be radius of curvature of second surface of the lens.  It follows from refraction due to concave spherical surface from denser to rarer medium that

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Adding (1) & (2)

But Note: The lens maker's formula can be derived for a concave lens in the same way. The ray diagram is as follows:

Note: The lens maker's formula indicates that a convex lens can behave like a diverging one if  1 >  2 i.e., if the lens is placed in a medium whose  is greater than the  of lens. Similarly a concave lens can be made convergent.

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Magnification (m) produced by a lens is defined as the ratio of the size of the image to that of the object. That is

On applying sign convention, for erect and virtual image formed by lenses m is positive, while for an inverted and real image, m is negative. Ray diagrams for convex lens showing the formation and nature of image for different positions of the object Convex Lens Object at Infinity

Object Beyond 2F

Object at 2F

Object between F and 2F 41

Object at F

Object between F and O

Position of object

Position of Image

Image is formed at focus Image is formed 2) Beyond 2F between F and 2F 1) At infinity

Nature of the Image Real, inverted and small in size Real, inverted and smaller in size

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3) At 2F

Image is also at 2F

4) Between F Image beyond 2F and 2F 5) At F

Image is at infinity

6) Between F Image on the same and O side of the object

Real, inverted and same size Real, inverted and large in size Highly magnified (real, inverted) Virtual, erect and magnified

Power of a Lens Power of a lens is the extent to which lenses converge or diverge when light rays falls on it.

Power of lens is defined as the tangent of the angle by which it converges or diverges a beam of light falling at unit distance from optic centre. That is

Mathematically

the SI unit of power is dioptre (D).

The power of lens is positive for convex lens and negative for concave lens. Combination of Two Thin Lenses in Contact Let two thin lenses L1 and L2 of focal lengths f1 and f2 be placed in contact so as to have a common principal axis. It is required to find the effective focal length of this combination. Let O be a point object on the principal axis. The refractions through the two lenses are considered separately and the results are combined. While dealing with the individual lenses, the distances are to be measured from the respective optic centers; since the lenses are thin, these distances can also be measured from the center of the

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lens system (point of contact in the case of two lenses). Let u be the distance of O from the center of the lens system. Assuming that the lens L1 alone produces the refraction. Let the image be formed at I' at a distance v'. Writing the lens equation in this case, we get

The image I' due to the first lens acts as the virtual object for the second lens. Let the final image be formed at I, at a distance v from the center of the lens system. Writing the lens equation in this case, we get

Adding equations (1) and (2) we get

i.e.,

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Let the two lenses be replaced by a single lens which can produce the same effect as the two lenses put together i.e., for an object O placed at a distance u from it, the image I must be formed at a distance v. Such a lens is called an equivalent lens and its focal length is called the equivalent focal length. Writing the lens equation in this case, we get

Comparing equations (3) and (4) we get

Hence, when thin lenses are combined, the reciprocal of their effective focal length will be equal to the sum of the reciprocals of the individual focal lengths. Since the reciprocal of focal length represents the power, the above equation, in terms of power, may be written as P = P1 + P2 Therefore, the power of a combination of thin lenses is equal to the algebraic sum of the powers of the individual lenses.

Refraction and Dispersion of Light due to a Prism A prism is a portion of a transparent medium bounded by two triangular bases of ground glass and three rectangular surfaces. ADEB and CFEB are two refracting surfaces and EB is called the refracting edge.

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In above figure, ABC, DEF - Triangular bases ADEB, BCFC - Refracting rectangular faces EB - Refracting edge Angle A between the two refracting surfaces is called the angle of prism. A ray of light suffers two refractions on passing through a prism. If KL be a monochromatic light falling on the side AB, it gets refracted and travels along LM. It once again suffers a refraction at M and emerges out along MN. The angle through which the emergent ray deviates from the direction of incident ray is called angle of deviation ' ' i.e., Relation between Refractive index () Angle of Prism (A) and angle of deviation () Let

be the angle of incidence and

, the first angle of refraction at AB

is the angle of incidence at face AC and AC. Extend

to meet at O.

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then is the angle of refraction at

Draw LO and MO at L and M respectively. Extend KL and MN to meet at P.

In the

(exterior angle is sum of interior opposite angles) i.e.  =(i1- r1) + (i2- r2)

 = i1 + i2 - (r1+ r2) In

In the quadrilateral ALOM

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Now

(From Snell's law)

As the angle of incidence is increased, angle of deviation ' ' decreases and reaches minimum value. If the angle of incidence is further increased, the angle of deviation is increased. Let m be the angle of minimum deviation. The refracted ray in the prism in that case will be parallel to the base.

i-

curve

For minimum deviation position the incident ray and emergent rays are symmetrical with respect to the refracting surface and LM is parallel to BC.  i1 = i2 = i and r1 = r2 = r 2r1 = A , r1= A/2 m

= 2i1 - 2r1

m

= 2i1 - A

or 2i1 =  m + A

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From Snell's Law

This is the Prism formula when the prism is in the minimum deviation position. For a thin prism A is very small and if the light is incident at a small angle then i1, r1, r2, i2 are small.

From Snell's law

Similarly

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i.e., thin prism does not deviate light much.

Dispersion by a Prism It is the phenomenon of splitting of a beam of white light into its constituent colors on passing through prism. The order of colors from the lower end are violet, indigo, blue, green, yellow, orange and red. We find that the red light bends least while the violet light bends the most. The pattern of colour components of light is called the spectrum of light.

In a classic experiment, Newton put another similar prism in an inverted position as shown below to show that the emergent light was white light.

This confirmed that the first prism splits the white light into its components, while the inverted (second) prism recombines them to give back white light. Note:

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Thick lenses (having a shorter focal length) are assumed to be made of many prisms and so the formation of coloured image, a defect in thick lenses is due to dispersion of light. Cause of Dispersion The refractive index of prism depends on wavelength according to cauchys relation namely . Since different colors of light have different wavelengths, their refractive indices are also different. Now red light has a longer wavelength than violet and so  red< violet. This means the  v >  r; that is, violet deviates by a larger angle than red light. All colors having wavelengths in between red and violet deviate through intermediate angles leading to formation of a coloured spectrum. Since vacuum is a nondispersive medium, all the colours travel with the same speed. That is why sunlight reaches in the form of white light and not as its components.

Some Natural Phenomena Due to Sunlight The spectacle of colour we see around us is due sunlight. For example a rainbow is an example of the dispersion of sunlight by suspended water droplets of suitable size in the atmosphere. The Rainbow The combined effect of dispersion, refraction and reflection of sunlight by spherical water droplets results in the optical phenomenon called rainbow. The condition for observing a rainbow is that the sun should shine in the opposite part of the sky where it is raining. Thus, an observer can see a rainbow only when the sun is behind the observer.

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When the sunlight enters in to a water droplet, it undergoes refraction. The longer wavelength of light (red) is refracted least, while the shorter wavelength (violet) is refracted (bent) most. This refracted wavelength of different colour undergoes total internal reflection inside the droplets. The reflected light is refracted again as it comes out of the drop. Thus, the observer can see a rainbow with red colour on the top and violet on the bottom. This is called primary rainbow (figure b). The primary rainbow is a result of three step process: Refraction, Reflection and Refraction. When light rays under go two internal reflections inside a drop, then the rainbow formed is called secondary rainbow (figure c). The intensity of light is reduced in the second reflection, hence the secondary rainbow is fainter than the primary rainbow. Also, the order of the colour is reversed in the secondary rainbow as compared to the primary. Scattering of Light When light passes through a substance or gas, a part of it is absorbed and the rest scattered away by atoms or molecules of substance or gases. The basic process in scattering is absorption of light by the molecules followed by reemission in different directions. The strength of scattering can be measured by the loss of energy in the light beam as it passes through the medium. In absorption, the light energy is converted into the internal energy of the medium and in scattering the light energy is radiated in other directions. The strength of scattering depends on the size of the particle causing the scattering and the wavelength of light. The intensity of scattered light is proportional to . This is known as Raleigh's law of scattering. So the red light is scattered the least and the violet is scattered the most. This explains why red signals are used to indicate danger as they are least scattered due to their long wave length.

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Among the scattered wavelengths the colour with shorter wavelength, blue is present in the larger proportion in sunlight. This explains why the sky appears blue. When we look at the sky we see it blue because, blue is scattered the most. Another natural phenomenon related to the scattering of light is the appearance of the sun at the sunset and sunrise as a red ball of light. At these times sunlight has to travel a large distance through the atmosphere. The blue and the neighboring colors are scattered away and the red light reaches our eye. All these scattering is done by the atmospheric particles. Hence, if the earth had no atmosphere the sky would appear black. Not only the air molecules, but also the water particles and dust particles also scatter the sunlight. The change in the quality of color of sky is due to the various sizes of the scattering medium namely the water or the dust particles.

Optical Instruments A number of optical devices and instruments have been designed for specific purposes which utilise the property of reflection, refraction of mirrors, lenses and prism. This is done to magnify the object, to view distant object clearly, to analyse the spectrum of light from a distant source etc. The human eye is a marvellous organ which has the ability to interpret the incoming waves. It is through the eye that we perceive objects around us. Human Eye One of the most complicated optical devices is the human eye. Let us see the construction of the human eye and then the mechanism of image formation.

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Human Eye

Human eye is spherical in shape and diameter of about 2.5 cm. Sclerotic is a tough, opaque and white substance forming the outermost coating of the eye. The front portion is sharply curved and covered by a transparent protective membrane called the 'cornea'. Inner to the sclerotic there is a layer of black tissue called choroids consisting of a mass of blood vessels, which nourishes the eye. The black color does not reflect the light and hence rules out the blurring of image by reflection within the eyeball. Behind the cornea, the space is filled with a liquid called the aqueous humor and behind that a crystalline lens. 'Iris' is a muscular diaphragm lying between the aqueous humor and the crystalline lens. Iris has an adjustable opening in the middle called the pupil of the eye. The pupil appears black because all the light entering is absorbed by the 'retina', which covers the inside of the rear part of the ball. Iris controls the amount of light that enters because the retina absorbs nearly all the light, which falls upon it. This is done by varying the aperture of the pupil with the help of the iris. In dim light the iris dilates the pupil and so that more light can enter in. when the light is bright the pupil contracts. The crystalline lens divides the eyeball into two chambers. The chamber between the cornea and the lens is called the anterior chamber filled with a fluid called aqueous humour while the chamber between the lens and the retina is called the posterior chamber which is filled with a transparent gelatinous substance called vitreous humour. The refractive indices of the cornea, pupil lens and fluid portion of the eye are quite similar. So, when a ray of light enters the eye it is refracted at the cornea. This refraction produce a real inverted and diminished image of distant objects on the retina. When the object is kept at different distances then we may expect the image to be formed at different distances from the lens. It means it may not form on the retina always. But in reality it is not so. Image is always formed on the retina. This is possible because the curvature of the crystalline lens is altered by ciliary muscles. When the eye is focused on infinity the muscles are relaxed and the eye lens remains thin. If the object is brought nearby the curvature increases so that the image can be formed on the retina. This property of the eye lens is called accommodation. Then the question comes that how long does the image persist on the retina. It is as long as the eye focuses the object or does it linger on after that also? It is very surprising to find that the image persists on the retina even after the object is removed. This is called 'the persistence of vision'. The persistence of vision is approximately one eighth of a second. 54

Even though human eye has the property of accommodation, the muscles cannot be strained beyond a limit and hence if the object is very close to the eye, clear image is not formed on the retina. Thus, there is a minimum distance for the clear vision of an object. This distance is called 'least distance of distinct vision'. For a normal eye, this distance happens to be 25 cms. Image Formation and Accomodation The ciliary muscles contract and expand in order to change the focal length of eye-lens so that a sharp image of object always falls on the retina. This is accommodation of eye. Since the distance between the eye lens and retina is fixed, to see objects at different positions from eye lens, the focal length of the lens has to be changed.

For a normal eye, to view distant object, the focal length of lens should increase and so the ciliary muscles have to stretch. In this way the image is formed on the Retina.

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For a normal eye, if objects are close to the eye, the focal length should decrease and so the ciliary muscles have to contact. In this way the image is formed on the Retina. The following definitions would be useful to understand: Near Point: The nearest point from an eye at which an object can be placed so that its sharp image is formed on the Retina. For a normal eye, this is 25 cm. Far Point: The farthest point from an eye at which an object can be placed so that its sharp image is formed on the Retina. For a normal eye, this is infinity. Defects of Vision and their Correction: Due to various changes, like age or biological changes, the eye lens is not able to accommodate for clear vision of objects either near by or far off. Common defects of vision are    

Hypermetropia or Long-sightedness Myopia or Short-Sightedness Presbyopia Astigmatism

Hypermetropia or Long-sightedness An eye which can see far off objects but unable see the near by objects is said to suffer from Long-Sightedness or Hypermetropia. This could be due to increase in focal length of eye-lens or size of eyeball becoming too small for light rays from near by point. The image of an object at normal near point (i.e., 25 cm) is formed behind the Retina as shown below.

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Defect

The above defect can be corrected by using convex lens of suitable focal length. This convex lens converges rays entering the eye and focuses the rays on the Retina.

Correction

Myopia or Short-Sightedness An eye which can see the near by objects clearly but is unable to see far off objects or distant objects clearly. The image of a distant point is formed in front of the Retina of the eye as shown below. This defect arises due to elongation of the eyeball or excessive curvature of the eyeball.

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Defect

The above defect can be corrected by using concave lens of suitable focal length. The concave lens diverges the rays of light entering the eye from infinity as shown below.

Correction

Presbyopia An eye which can neither see near by objects nor far off objects clearly is said to suffer from presbyopia. This is due to ageing as the ciliary muscles are weakened. The defect be corrected by using bi-focal lenses which consist of concave lens which forms upper surface and convex lens which forms lower surface of the bi focal lens. Astigmatism An eye which cannot focus on both horizontal and vertical lines simultaneously is suffering from a defect called Astigmatism. For such an eye the horizontal part of the object will not be visible clearly as the vertical part

Defect

Correction

This arises when the Cornea of the eye has different curvature in different direction. This can be corrected by using glasses with cylindrical lens. Simple Microscope or Magnifying Lens

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Magnifying Lens

It consists of a converging lens of small focal length. By keeping the object close to the lens, a virtual, erect and magnified image is obtained as shown such that is formed at a distance 25 cm to be viewed comfortably. (This distance D=25 cm is called the distance of distinct vision)

O = optic centre FO = focal length of convex lens AB = object A1B1 = image Magnifying power or magnification is the ratio of the angle subtended by the image and the object on the eye, when both are at the least distance of distinct vision from the eye Let

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For small angles

and image distance v = -D (distance of distant vision) = 25 cm

Note: As f decreases, m increases. Let us see what is the magnification when the image is formed at infinity. Let be the height of object AB. When kept at a distance D (i.e., u=D), for an unaided eye as shown below

the maximum angle

also When a lens is placed between the eye and object as shown

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Then (this is because angle subtended by object and image is same at the lens)

This is one less than that obtained when image is formed at least distance of distinct vision. In practice the magnifying lens can have a maximum magnification as the decreasing focal length would only increase the defects in lens. Compound Microscope For increasing magnification, one uses two lenses, objective and eye piece. The objective is convex lens placed near the object, which forms real inverted magnified image of the object. The eye piece, that is, the second lens acts a simple microscope and enlarges this first image.

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A Laboratory Model of Compound Microscope

The ray diagram shows how the magnification can be calculated.

O1 - Objective O2 - Eyepiece D - Distance of distinct vision AB - Object A'B' - First image

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A"B" - Final image O1FO - Focal length of objective = fO O2 Fe - Focal length of eyepiece = fe Magnifying power of compound microscope is defined as the ratio of the angle subtended at the eye by the final image to the angle subtended at the eye by the object, when both the final image and the object are situated at the least distance of distinct vision from the eye.

Now

(-ve sign shows that final image is inverted with respect to the object) Note:

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

If the object is very close to Fo, the focus of lens O then and as AlBl is very close to the eye lens Vo = OlBl ; O1O2 = L = length of tube; If fo and fe both are small then magnifying power



will be large. If the image is formed at infinity then the total magnification

Telescope 'It is an instrument used for observing distinct ../content/CB12P1/content/topic/ch599/images of heavenly bodies and also to provide angular magnification of distant objects. This has an objective and eyepiece. Here the objective has a large focal length and a much larger aperture than the eye piece (More light gathering capacity). Telescope can be either refracting or reflecting.

Astronomical Telescope

Refracting Telescope It employs two convex lenses which act as objective and eye piece. Magnifying power is the ratio of the angle subtended at the eye by the final image to the angle subtended at the eye by the object directly, when both the final image and the object lie at infinite distance from the eye.

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When the final image is formed at 'D'

In this case the final image is at the least distance of distinct vision.

Now Now ve = -D; u = -ue; f = +fe

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Negative sign implies the final image inverted. When the image is formed at infinity as shown.

The total magnification m is

While constructing such huge telescopes two aspects have to be kept in mind, its light gathering power and its resolving power. With large diameters, the light gathering power is more and so fainter objects can be observed. Similarly the resolving power also depends on the diameter of the objective. But big lenses are very heavy and difficult to make and support. Further the ../content/CB12P1/content/topic/ch599/images formed are not free from any aberration. For these reasons modern telescopes use a concave mirror as objective. Such telescopes are called reflecting telescopes. Reflecting Telescope

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Refraction Telescope

The objective lens is replaced by a concave parabolic mirror of large aperture. The ../content/CB12P1/content/topic/ch599/images in such telescopes are brighter and have a higher resolving power compared to astronomical telescope.

Cassegrainan Reflecting Telescope

Reflecting telescope has several advantages. First, there is no chromatic aberration in a mirror, second, spherical aberration in mirror can be minimised by employing parabolic mirrors. Third, mirror weighs much less than the lens and can be supported over its back surface. In the cassegrain telescope, the viewer sits near the focal point of the mirror inside the tube or at the focus of the secondary mirror as shown. One disadvantage being that a 67

portion of incident light is obstructed which reduces the brightness of the image.

The magnification of the reflecting telescope the radius of the objective)

(where

is

Summary 

Ray optics is also called geometrical optics as it uses the geometry of straight-line paths (rays) to explain the optical phenomena. Ray optics, infact, is the limiting case of wave optics. This means for most practical purposes, we can ignore the deviation from straightline path as postulated by wave theory.



Reflection of light is the phenomenon of change in the path of light without any change in medium. A spherical mirror is a part of a hollow sphere whose one side is reflecting and other side is opaque. Two types of spherical mirrors are concave mirror and convex mirror. The centre of curvature (C) of spherical mirror is the centre of the sphere of which the mirror forms a part. Principal focus (F) of a spherical mirror is a point on the principal axis of the mirror at which rays incident on the mirror in a direction parallel to the principal axis actually meet or appear to diverge after reflection from the mirror.



While dealing with reflection of spherical mirrors, we use the following Cartesian sign conventions: - All distances are measured from pole of spherical mirror. - The distances measured in the direction of incidence of light are taken as positive and vice-versa. - The heights measured upwards and perpendicular to the principal axis of the mirror are taken as positive and vice-versa.



In case of both the spherical mirrors, convex and concave, f = R/2. Also the mirror formula for both the mirrors is

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Where, u is the distance of the object and v is the distance of the image from the pole of the mirror. Linear magnification in case of a spherical mirror is defined as the ratio of size of the image (h2) to the size of the object (h1). In a convex mirror, linear magnification is positive, because image is always virtual.

In a concave mirror, magnification can be both positive or negative, depending on the type of image formed (virtual or real).

Other formulae for magnification are

Spherical mirrors have several applications. A convex mirror is used as reflector in street lamps. It is also used as a rear view mirror. A concave mirror is used as a reflector in search light, telescopes, solar cookers, and ophthalmoscope. They can also be used as trick mirrors. 

Refraction of light is the phenomenon of change in the path of light, when it travels from one medium to another. When it travels from a rarer to a denser medium, a ray of light bends towards normal and while traveling from a denser to a rarer medium, a ray of light bends away from normal. This is because light travels slower in a denser medium than in a rarer medium. On account of refraction of light, a tank of water appears to be shallow, that is less deep than what it actually is. It is found that

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If i is the angle of incidence, r is the angle of refraction and  is the refractive index of denser medium with respect to rarer medium, then according to Snell's law,

(when light goes from rarer to denser medium) and

(when light goes from denser to rarer medium). Further, refractive index of medium a with respect to medium b is represented by



Total internal reflection is a phenomenon of reflection of light into the denser medium, on traveling from a denser medium to a rarer medium. Two essential conditions for the phenomenon of total internal reflection are: - Light should travel from a denser to a rarer medium. - Angle of incidence in the denser medium should be greater than the critical angle for the pair of media in contact. The critical angle for a pair of media in contact is defined as the angle of incidence in the denser medium corresponding to which angle of refraction in the rarer medium is 900. It is represented by C. If  is refractive index of denser medium with respect to rarer medium, then

Obviously, C would depend on colour of light.

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Some of the important applications of total internal reflection are brilliance of diamond, totally reflecting glass prisms, optical fibres, mirage (false appearance of water in deserts in hot summer season or optical illusion) etc. 

A surface, which forms a part of a sphere of transparent refracting material, is called a spherical refracting surface. It may be convex or concave. In dealing with refraction at such surfaces, we use the same new Cartesian sign conventions as in the case of spherical mirrors. The formula governing refraction at a spherical surface when light travels from a rarer to a denser medium.

(where, u and v are distances of object and image respectively from the pole of the spherical surface and R is the radius of curvature of the surface). When light travels from a denser to a rarer medium, we have to interchange  1 and  2 in the above formula. The relation becomes:

Or



A lens is bound by two spherical surfaces. Therefore, a ray of light suffers two refractions on passing through the lens. The lens maker's formula for both convex and concave lenses is

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(where, R1 and R2 are radii of curvature of the two surfaces of the lens and  is refractive index of material of lens with respect to medium in which lens is placed). The relation governing u, v and f in case of both the lenses is

Linear magnification (m) produced by a lens is the ratio of the size of image (h2) to the size of the object (h1). For concave lens, m is positive (when image is virtual) and m is negative (when image is real).

For convex lens, m is positive (when image is virtual) and m is negative (when image is real).

Power of a lens is defined as the ability of the lens to converge or diverge a beam of light falling on the lens. Power of lens is given by reciprocal of focal length of the lens i.e.,

When f = 1m, p = 1 dioptre For a converging lens or convex lens, P is positive (+) and for a diverging lens or concave lens, P is negative (-). 

When two lenses of focal length f1, f2 and linear magnification m1 and m2 are placed in contact with each other, then for the combination, focal length f, power, P and magnification, m are given by

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P = P1 + P2 (Sum has to be taken with proper sign.) and m= m1 x m2 

On passing through a prism, a ray of light suffers two refractions. The net deviation ( ) suffered by a ray in passing through a prism of small angle A is ,  = ( - 1) A The deviation through the prism is minimum ( m), when i1 = i2 and r1= r2 From i1 + i2 = A +  m, i+i=A+m 2i = A +  m

Or

From If  is refractive index of material of prism, then from Snell's law,

This formula is called prism formula. 

Dispersion of light is the phenomenon of splitting of white light into its constituent colors on passing through a prism. The band of seven colors so obtained when white light falls through the prism is called visible spectrum. The cause of dispersion is:  v<  r; therefore,  v >  r, As  = ( 1),  v >  r 73

That is why violet colour is at the lower end of the spectrum. 

Angular dispersion produced by a prism

Dispersive power of prism

Deviation ( ) for yellow colour is mean of red and violet. i.e.,

Similarly, mean refractive index of material of prism for yellow colour is



A simple microscope is used for observing magnified ../content/CB12P1/content/topic/ch599/images of tiny objects. It consists of a converging lens of small focal length. Object is held between principal focus and optical centre of the lens. The image formed is virtual, erect and magnified. Magnifying power of a simple microscope is defined as the ratio of the angles subtended by the image and the object on the eye, when both are situated at the least distance of distinct vision (D) from the eye.

Magnifying power is given by, In a compound microscope, the ../content/CB12P1/content/topic/ch599/images are highly magnified. The objective lens forms a real, inverted and magnified image of the object. This acts as an object for eye lens, which forms a virtual, erect

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and magnified image seen by the eye held close to the eye lens. Magnifying power of a compound microscope is given by

Where u and v are distances of object and image from optical centre of objective lens, fe is focal length of eye lens. 

An astronomical telescope is used for observing heavenly bodies like stars and planets etc. The objective lens forms a real, inverted and smaller image of distant object in its focal plane. This image serves as the object for eye lens, which forms a virtual, erect and magnified image seen by the eye held close to the eye lens. In normal adjustment, final image as seen by the eye is at infinity. In normal adjustment, magnifying power of astronomical telescope is given by

When final image is at the least distance of distinct vision from the eye, the magnifying power is given by



In a terrestrial telescope, final image has to be made erect with respect to the object. For this, we have to use an erecting lens between objective lens and eye lens. Expressions for magnifying power remains the same. But the length of telescope tube increases by 4f, where, f is the focal length of erecting lens. To overcome this difficulty, Gatile, a telescope is used, in which eye lens is concave. In normal adjustment, length of the tube becomes (fo - fe) but field of view is much smaller because of concave lens. In a reflecting telescope, the objective lens is replaced by a concave parabolic mirror. Magnifying power of a reflecting type telescope is

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Numericals Numerical 01 A 4.5 cm needle is placed 12 cm away from a convex mirror of focal length 15 cm. Give the location of the image and the magnification. Describe what happens as the needle is moved farther from the mirror. Suggested Solution: U = -12 cm, f = +15 cm, v = ?

On simplification, v = +6.7 cm The +ve sign indicates that the image is formed behind the mirror.

Again,

= 2.5 cm The positive sign indicates that the image is erect and virtual. As the needle is moved farther from the mirror, the image moves towards the focus (but never beyond). Moreover, it gets progressively diminished in size.

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Numerical 02 A square wire of side 3.0 cm is placed 25 cm away from a concave mirror of focal length 10 cm. What is the area enclosed by the image of the wire? Given: The centre of the wire is on the axis of the mirror, with its two sides normal to the axis. Suggested Solution: u = -25 cm, f = -10 cm v = ?

On simplification, v = -16.67 cm

Magnification,

Area enclosed by the image = (-2 cm)2 = 4cm2. Numerical 03 A small pin fixed on a table top is viewed from above from a distance of 50 cm. By what distance would the pin appear to be raised if it is viewed from the same point through a 15 cm thick glass slab held parallel to the table? Refractive index of glass = 1.5. Does the answer depend on the location of the slab? Suggested solution: If y is the distance through which the pin would appear to be raised, then y = Real thickness (of slab) - Apparent thickness (of slab) or

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Where,  is the refraction index of the material of the slab.

The location of the slab will not affect the answer in any way.

Numerical 04 A needle placed 45 cm from a lens forms an image on a screen placed 90 cm on the other side of the lens. Identify the type of lens and determine its focal length. What is the size of the image if the size of the needle is 5.0 cm? Suggested Solution: u = -45 cm v = +90 cm, f = ?

or

The lens is clearly a converging lens.

Again,

The negative sign indicates that the image is inverted.

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Numerical 05 An object of size 3.0 cm is placed 14 cm in front of a concave lens of focal length 21 cm. Describe the image produced by the lens. What happens if the object is moved farther from the lens? Suggested Solution: O = 3.0 cm, u = -14 cm, f = -21 cm

On simplification, v = -8.4 cm The image is erect, virtual and located 8.4 cm from the lens on the same side as the object.

Now,

As the object is moved away from the lens, the virtual image moves towards the focus of the lens but never beyond the focus. The image progressively diminishes in size.

Numerical 06 Double convex lenses are to be manufactured from a glass of refractive index 1.55, with both faces having the same radius of curvature. What is the radius of curvature required if the focal length of the lens is to be 20 cm? Suggested solution:  = 1.55, R1 = R and R2 = -R, f = 20 cm

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We know that

or

 R = 40 x 0.55 cm = 22 cm

Numerical 07 A glass lens has a focal length of 5 cm in air. What will be its focal length in water? Refractive index of glass is 1.51 and that of water is 1.33. Suggested Solution: If fa be the focal length of glass lens in air, then

If fw be the focal length of a glass lens in water, then

Dividing (1) by (2), we get or

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Numerical 08 (a) A screen is placed 90 cm from an object. The image of the object on the screen is formed by a convex lens at two different locations separated by 20 cm. Determine the focal length of the lens. (b) Suppose the object in (a) above is an illuminated slit in a collimator tube so that it is hard to measure slit size and its distance from the screen. Using a convex lens, one obtains a sharp image of the slit on a screen. The image size is measured to be 4.6 cm. The lens is displaced away from the slit and at a certain location, another sharp image of size 1.7 cm is obtained. Determine the size of the slit. Suggested Solution: (a) Distance between object and image, D = 90 cm Distance through which lens is displaced, d = 20 cm Now, D = u + v = 90 cm; d = u v = 20 cm  u = 55 cm or 35 cm and v = 35 cm or 55 cm

Using lens formula, (b) For first location of lens, u1 = x and v1 = y For second location of lens, u2 = y and v2 = x

In first case,

In second case,

Multiplying or

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Numerical 09 For a given source of light, the angle of minimum deviation of a 600 prism is 560. What is its refractive index? Suggested Solution:

Numerical 10 Deduce

for water (

) when the prism of

Suggested Solution:

or

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is used.

Numerical 11 Calculate the angle of dispersion between red and violet colours produced by a filter glass prism of refracting angle of 600. Given:  v = 1.663 and  r = 1.622 Suggested Solution: For minimum deviation position,

or

Similarly,

or

or

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Numerical 12 Calculate the dispersive power for crown and flint glass from the following data:

Suggested solution:

Numerical 13 Monochromatic light of wavelength 600 nm is incident from air to water. What are the wavelength, frequency and speed of (i) reflected, and (ii) refracted light. Refractive index of water is 1.33. Suggested Solution: (i) Reflected light travels in the same medium (air) as the incident light. Hence, speed of reflected light c = 3 x 108 ms-1. The wavelength ( ) of reflected light is 600 nm. The frequency f is

(ii) The refracted rays are in water. Therefore, speed of refracted rays

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The frequency of refracted light remains constant when it goes from one medium to other. Hence, f of refracted light is 5 x 1014 Hz. Let  1 be wave length of refracted light. Then,

= 0.45 x 10-6 m = 450 nm

Numerical 14 The critical angle of incidence of water for total internal reflection is 480 for a certain wavelength. What is the polarizing angle and the angle of refraction for light on water at this angle? Suggested Solution: Let  be the refractive index of water with respect to air and ic the critical angle. Then,

Let ip be the polarizing angle of incidence. From Brewster's law, tan ip = = 1.345 or ip = tan-1 (1.345) = 53022' At polarizing angle of incidence, if r is the angle of refraction,

Numerical 15

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A glass plate ( = 1.5) is used as a polarizer. Obtain the polarizing angle of incidence. What is the angle of refraction when the reflected light is plane polarised? Suggested Solution: According to Brewster's law, the polarizing angle of incidence ip is  = tan ip or 1.5 = tan ip ip = tan-1(1.5) = 56.30 When light is incident at the polarizing angle of incidence, the reflected beam is perpendicular to the transmitted beam. If r is angle of refraction, ip + r = 900

Numerical 16 (i) What type of lens is used to correct the defect of the eye of a person if the far point of this eye is 2 m? (ii) If the near point of the eye is 25 cm, then what will be the position of the near point of the eye with the given lens? Suggested Solution: (i)

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Therefore a concave lens is used to correct this defect. (ii) v=-25 cm; f = -2m = -200 cm; u = ?

= -28.6 cm

Numerical 17 A man with normal near point 25 cm reads a book with small print using a magnifying glass, a thin convex lens of focal length 5 cm. What is the closest and the farthest distance at which he can read the book while viewing through the magnifying glass? Suggested Solution: For closest distance V = -25 cm; f = 5 cm; u = ?

For farthest distance ; f= 5 cm; u = ?

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U = -5 cm

Numerical 18 A converging lens of focal length 6.25 cm is used as a magnifying glass. If the near point of the observer is 25 cm from the eye and the lens is held close to the eye, calculate (i) the distance of the object from the lens (ii) the angular magnification Suggested Solution: (i) f = 6.25 cm; v = -25 cm; u = ?

u = -5 cm (ii) Angular magnification

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