Ray Optics

Second Year Physics study material 2015 Ray Optics (14 Marks weightage for Ray optics and Physical optics chapters)

1. Define Optics? A) The branch of physics which describes about the phenomenon and laws associated with generation and propagation of light and its interaction with matter is called Optics. Optics are broadly classified in to two categories 1. Geometrical optics ( if wavelength of light is smaller than size of obstacles) 2. Physical optics ( if wavelength of light is of the order of size of obstacles) 2. Define Reflection? State Laws of Reflection? A) Reflection : The phenomenon of return of light in the same medium when the light falls on a reflecting surface (say mirror) is known as reflection of light. Reflection obeys certain laws known as Laws of Reflection 1. The angle of incidence is equal to angle of reflection. 2. The incident ray, reflected ray and normal to the reflecting surface at a point of incidence all lie in the same plane. Note :

3. Define Spherical mirror? How many types of spherical mirrors present?

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Concave mirror : It is a part of hollow sphere having outer part (bulging surface) silvered and the inner part (depressed part ) as reflecting surface . Concave mirror is also known as convergent mirror because it converges the ray of light.

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A) Spherical mirror is a part of spherical reflecting surface. They are two types of spherical mirrors.

Second Year Physics study material 2015 Convex mirror : It is a part of hollow sphere having inner part (depressed surface) silvered and the outer part ( bulging surface) as a reflecting surface. Convex mirror is known as divergent mirror because it diverges ray of light. 4. Define Terminology associated with a spherical mirrors? A) Center of curvature : The center of sphere of which the spherical mirror forms a part is called center of curvature. Radius of curvature : The radius of a sphere of which the spherical mirror form a part is called Radius of curvature. Pole : Mid point of spherical mirror is called its pole Principle axis : The line joining the center of curvature and the pole of spherical mirror is called principle axis. Aperture : It is the effective diameter of light reflecting area of mirror. Principle focus : The point of the principle axis of a spherical mirror where the rays of light parallel to the principle axis meet or appear to meet after reflection from the mirror is called principle focus. Focal length : The distance between the pole and the principle focus of spherical mirror is called Focal length of a mirror. 5. Write a short note on sign conventions? A) 1. All distances are measured from the pole of spherical mirror 2. Distances measured in the direction of light are taken as positive , while the distances measured in the opposite direction of light is taken as negative. 3. The upward distances perpendicular to the principle axis are taken as positive, while the distances measured downward to the principle axis are taken as negative.

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6. Derive the relation between Radius of curvature and focal length of mirror?

Second Year Physics study material 2015 A) The distance between Pole and focal length of mirror is called Focal length of mirror. The distance between center of curvature and pole of mirror is called Radius of curvature of mirror. To find the relation between Focal length and radius of curvature of mirror Consider a Ray OA travelling parallel to principle axis and incident on concave mirror at point ‘A’. The light under goes reflection and passes through the focal point of mirror. ‘CA’ is the normal to the mirror. According to laws of reflection ∟ i = ∟r = ∟θ Also ∟ACF = ∟OAC = θ ∟AFP is the external angle of ∆ ACF , so ∟AFP = ∟ACF + ∟CAF = θ + θ = 2 θ Now draw AN perpendicular to the principal axis From right angled ∆ ANC , tan θ =

஺ே ே஼

From right angled ∆ ANF , tan 2 θ = From 1 and 2 , we get

2

஺ே ே஼

=

஺ே ேி

஺ே ேி

that is θ = that is

஺ே ே஼

2θ=

(if θ is small tan θ = θ) -----(1) ஺ே ேி

(if tan2 θ = 2 θ) ---------(2)

or NC = 2NF

As aperture of mirror is small, so point N lies very close to P . NF = PF and NC = PC Thus PC = 2PF -----(3) , using sign convection PC = -R and PF = -f , Hence equation (3) becomes

R = 2f that is f =

ࡾ ૛

7. Derive an expression for Mirror formula?

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Distance between pole of mirror and position of object on principle axis is represented by ‘u’. The distance between pole of mirror and position of

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A) The relation between position of object (u) , position of mirror (v) and focal length (f) of the mirror is known as mirror formula.

Second Year Physics study material 2015 image on principle axis is represented by ‘v’ and focal length of mirror is given by ‘f’. Let AB be an object lying beyond the focus (F) of a concave mirror. A ray of light BL which is parallel to the principle axis through the principal axis at F after reflecting from the concave mirror and goes along LB’ . Another ray from B passes through the center of curvature (c ) and strikes the mirror normally at point M. After reflection , this ray retraces its path and meets LB’ at B’. so A’B’ is the real image of the object AB. Now triangle ABC and A’B’C’ are similar , therefore A’B’/AB = CA’/CA

--------------------(1)

Now triangle ABP and A’B’P are similar , therefore A’B’/AB = PA’/ PA ------------------------(2) From 1 and 2 , we get CA’/CA = PA’/PA

---------------------(3)

As per New Cartesian sign convection all the distances are measured from the pole of mirror , so CA’ = (PC – PA’) and CA = (PA – PC) ------------- (4) Substituting the values of equations (4) in equations (3) we get, (PC – PC’)/(PA – PC) = PA’/PA ---------------------- (5) Applying New Cartesian sign convection , PA’ = -v , PC = -R , PA = -u

Hence equation (5) becomes ,

( -R + v)/(-u+R) = -v/-u

uR –uv = vu –vR, we get , uR + vR = 2uV dividing by uvR , we get 1/v + 1/u = 2/R or 1/f = 1/u + 1/v 8) Define linear Magnification and express formula in convex and concave mirrors?

If

I =size of the image and

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Linear Magnification produced by a mirror is defined as the ratio of the size of the image to the size of the object. It is denoted by m.

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A) Linear Magnification:

Second Year Physics study material 2015 O= size of the object Then, magnification,

m=I/O ----------------- (6)

Magnification produced by a concave mirror: In the case of concave mirror, magnification may be positive or negative depending upon the nature of the image of an object. (a) When real image is formed m=I/O=-v/u (b) When Virtual image formed m =I/O= v/-u Magnification produced by a convex mirror: m=I/O=-v/u 9) Define Refraction of light? A) The phenomenon of change in direction of path of light when it goes from one medium to another medium is called refraction of light. This phenomenon occurs because of fact that speed of light changes when it goes from one medium to another. When light goes from rarer medium to denser medium it bends towards the normal. When light goes from denser medium to rarer medium it bends away from the normal. 10) Write a short note on Refractive index of a material? A) Absolute refractive index : It is defined as the ratio of speed of light in air/vacuum to the speed of light in a medium. n = c/v where c = speed of light in vacuum = 3 x 108 m/s Relative refractive index : when light passes from one medium to other , the refractive index of medium 2 relative to medium 1 is written 1µ2 and is given by 1µ2 = µ1/µ2 =v1/v2 A medium having higher value of refractive index is called optically denser medium while a medium having lower value of refractive index is called optically rarer medium.

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A) 1. Incident ray , Refracted ray, Normal to the interface at a point of incidence lie on the same plane , they are all coplanar.

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11) what are Laws of Refraction ?

Second Year Physics study material 2015 2. The ratio of sine of angle of incidence to the sine of angle of refraction is constant for any two given media. This is also known as snells law. Sin i / sir r = constant This constant is also known as relative refractive index of second medium with respect to first medium Sin i / sin r = n2/n1 n1sini = n2 sin r 11) Write a short note on Lateral shift of light ray when it passes through a glass slab? A) Consider a glass slab. Consider a ray AO incident on slab at an angle of incidence ‘’i’ and passing through a slab of thickness ‘t’. After two refractions at the boundary, the ray emerge parallel to the incident ray. The perpendicular distance between the incident ray direction and emergent ray direction is called lateral shift or lateral displacement (x). From figure , the distance ‘PQ’ is called lateral displacement or lateral shift. From triangle PQO, sin (i-r) = PQ/OP PQ = OP sin (i –r) x = OP sin (i –r) ---------------- (1) But cos r = OM/OP, OP = OM/cos r = t/ cos r ------------------- (2) from (1) and (2) x = t((sin (i-r))/cos r)

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A) Let us consider a beaker filled with water up to certain level. Let an object O lie at the bottom of beaker. The depth AO is known as real depth. The ray of light OA is incident normally on the surface of water, so it is undeviated and goes along AB in to air.

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12) Write a short note on Apparent depth?

Second Year Physics study material 2015 Another Ray of light OC , incident on the surface of water at an angle i and after refraction bends away from the normal and travels along CD in to air. The ray CD meets the refracted ray AB at point I when produced backwards. So I is the image of object O. Thus the object appears at position I instead of O. The depth AI is known as apparent depth of object ‘O’. According to snells law wna = sin i/sin r -----------------(1) From ∆AOC , sin i = AC/OC and from ∆AIC , sin r = AC/IC substituting the values of sin i and sin r in equation (1) we get wn a

= (AC/OC) x (IC/AC) = IC/OC

Since point C lies very close to A , so IC ≈ AI and OC≈AO wna

= AI/AO

since anw = 1/wna anw

in general

1n 2

= AO/AI = Real depth/Apparent depth

= Real depth/Apparent depth

Normal shift in the object position is given by x = AO – AI = AO (1 – AI/AO) = AO (1-1/1n2) In general x =t(1-1/n) 13. Define Total internal reflection? A) The phenomenon of reflection of light that takes place , when a ray of light travelling in a denser medium gets incident at the interface of the two media at an angle greater than the critical angle for that pair of media. Critical angle : The critical angle for a pair of media may be defined as the angle of incidence in denser medium , for which angle of refraction in the rarer medium is 90 degree angle.

a

= sin i / sin r

when the ray of light is incident at critical angle that is when i = C , r = 90 degrees

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bn

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According to snell law the refractive index of rarer medium a with respect to denser medium b is

Second Year Physics study material 2015 then

b na

= sin C / sin 90

bn a

but

therefore

= 1/anb anb

= 1/sin C

Practically there will be no loss of light in total internal reflection. 14 . Explain briefly some applications of Total internal reflection? A) Mirages : It is an optical illusion observed in deserts and coal – tarred roads on a hot day. The object such as a tree appears inverted and the observer gets impression as if the inverted image has been formed by pool of water . This phenomenon is known as mirage.

Due to intense heat, surface of earth becomes quite hot and temperature of air near the surface of earth is maximum. Temperature of layers goes on decreasing when one goes up. Therfore density as well as refractive index of air increases slightly for higher layers. Therefore a ray of light travelling from a point on a tree passes through air of gradually decreasing refractive index and gets refracted more and more away from the normal and accordingly angle of incidence goes on increasing. At a layer , when angle of incidence is becomes greater than critical angle total internal reflection takes place. Then ray of light starts traversing layers of increasing refractive indices and goes on bending more and more towards the normal. ultimately when the ray reaches the eye from the observer , it appears to be coming from the different point. Hence the inverted image of the tree produces the impression of reflection from a pool of water. Brilliance of diamonds: Refractive index of diamond is 2.47 and the critical angle for diamond air interface is 23 degrees. Due to low value of critical angle , a diamond can be cut so as to have a large number of faces. As such each ray of light on entering the diamond from a face undergoes a series of total internal reflections from other faces , till the angle of incidence inside the diamond greater than critical angle. As a result it shines very brilliantly

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A right angled isosceles prism is called Porro prism , can be used in periscope or binocular. Refractive index of glass is 1.5 . Therefore critical angle is given by 41 degrees. When a ray of light falls on the face of right angled prism at angle greater than 41 degrees, it will suffer total internal reflection. Right angled prism are used to bend light through 90 and 180 degree respectively.

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Prisms :

Second Year Physics study material 2015

Optical fibers : Optical fibers are used to transmit light from one place to another place in a curved path in a more effective manner. The optical fibers consists of thousands of strands of a very fine quality glass or quartz of refractive index 1.7 or so. The thickness of strand is 10-6 cm. The strands are coated with a layer of some material of refractive index (about 1.5) . When light is incident at a small angle at one end , it gets refracted into the strands and gets incident on the interface of the fibers and the coating. Angle of incidence being greater than the critical angle the ray of light undergoes total internal reflections. It suffers total internal reflection again and again till the angle of incidence greater than critical angle. Optical fibers are used in variety of applications 1. They are used in the field of communication. They are used for transmitting and receiving electrical signals which are converted into light. 2. The optical fibers can be used for medical investigations like endoscope etc., 15. Explain briefly refraction at a spherical surface? A) Spherical refracting surface is a refracting medium whose curved surface is a part of sphere.

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Let α, β and γ be the angles made by the incident ray, refracted ray and the normal respectively with the principal axis. Draw AN perpendicular on the principal axis.

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A point object O lies on the principal axis at a distance u from the pole of the convex refracting surface in rather medium. A ray of light from O incident the convex surface at A. AC is the normal to the convex surface. After refraction at A, the ray enters the denser medium and bends towards the normal. The refracted way meets the principal axis at I which is the real image of the object O. The distance of the image I from the pole of the convex surface is v.

Second Year Physics study material 2015 From ∆AOC, and from ∆AIC,

i=α+γ

……………………… (i)

or

γ=r-β …………………..(ii)

γ=r+β

α,γ and β are small angles (assumption), Using the relation θ=l/r, we get α=AN/NO, γ=AN/NC and β=AN/NI …………..(iii) Then

i= AN/NO + AN/NC ≈ AN/PO + AN/PC

……………………(iv)

and

r= AN/NC - AN/NI ≈ AN/PC - AN/PI

……………………(v)

Since aperture of the spherical surface is assumed to be small, so point N lies very close to point P. NO ≈ PO, NC ≈ PC and NI ≈ PI Now according to Snell’s law, sin i/sin r =n2/n1

or

n1sin i=n2sinr

Since angles i and r are also small, so n1i=n2r

(sin i=i and sin r=r) ………(vi)

Using eqns. (iv) and (v) in eqn. (vi), we get n1[AN/PO + AN/PC] = n2[AN/PC - AN/PI] or n1/PO + n1/PC = n2/PC-n2/PI or n1/PO + n2/PC = n2 –n1/PC

………………. (vii)

Applying new Cartesian sign conventions, PO=-u, PC=R, PI=v Hence eqn. (vii) can be written as

A lens is a piece of transparent material bounded by two refracting surfaces out of which at least one is curved.

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16. Write a short note on Lenses ?

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- n1/u + n2/v = n2-n1/R

Second Year Physics study material 2015 If thickness of a lens is small and its curved surface is spherical then the lens is known as thin spherical lens. If the central portion of a lens is thinner than its edges it behaves as a divergent lens known as concave lens. similarly, if central portion of a lens is thicker than its edges then it behaves as a convergent lens known as convex lens.

Uses : Lenses are commonly used to correct vision defects of human eye. They are also used in microscopes, telescopes and cameras. They are used in cinematography and in photography.

Choice of rays for ray diagram to locate the image of an object Any two of the following may be selected for drawing ray diagrams. (i) A ray of light starting from the top of the object and parallel to the principal axis of lens passes through the second principal focus in the case of a convex lens or appear to diverge from first principal focus in the case of concave lens. (ii) A ray of light passes through first principal focus of a convex lens come sout parallel to the principal axis on refraction or a ray of light appears to meet at principal focus of a concave lens parallel to principal axis on refraction. (iii) A ray of light passes undeviated on refraction through optical center of the lens.

Assumptions made to derive Lens Maker’s Formula:

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The formula giving relation between the focal length (f) of the lens, refractive index of the material of the lens (n) and the radii of curvature of its surfaces (R1 & R2) is known as Lens Maker’s Formula.

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17. Derive an expression for Lens Maker’s Formula ?

Second Year Physics study material 2015 (i) The lens is thin and all the distances are measured from the optical center of the lens. (ii) The aperture of the lens is small. (iii) The object is a point object and lies on the principal axis. (iv) The angle made by incident ray and refracted ray with the principal axis are small. New Cartesian Sign Conventions: (i) All the distances are measured from the optical center of the lens. (ii) Distances measured in the direction of the propagation of incident light are taken as positive while the distances measured in the direction opposite to the direction of propagation of incident light are taken as negative. Derivation for Convex lens.

Consider a lens made of a material of absolute refractive index n2. This lens is placed in a medium of absolute refractive index n1(n1