RAY OPTICS.pdf

Ray optics chapter

1. Chapter -9
Ray optics and
optical instruments

2. RAY OPTICS
1.Refraction ofLight
2.Laws ofRefraction
3.Principle of Reversibility ofLight
4.Refraction through a ParallelSlab
5.Refraction through a Compound Slab
6.Apparent Depth of aLiquid
7.Total Internal Reflection
8.Refraction at Spherical Surfaces -Introduction
9.Assumptions and SignConventions
10.Refraction at Convex and ConcaveSurfaces
11.Lens Maker’sFormula
12.First and Second PrincipalFocus
13.Thin Lens Equation (GaussianForm)
14.LinearMagnification

3. Reflection
Mostthingsweseearethankstoreflections,sincemostobjectsdon’t
producetheirownvisiblelight.Muchofthelightincidentonanobjectis
absorbedbutsomeisreflected.thewavelengthsofthereflectedlight
determinethecolorswesee.Whenwhitelighthitsanapple,forinstance,
primarilyredwavelengthsarereflected,whilemuchoftheothersare
absorbed.
Arayoflightheadingtowardsanobjectiscalledanincidentray.Ifit
reflectsofftheobject,itiscalledareflectedray.Aperpendicularline
drawnatanypointonasurfaceiscalledanormal(justlikewithnormal
force).Theanglebetweentheincidentrayandnormaliscalledtheangleof
incidence,i,andtheanglebetweenthereflectedrayandthenormalrayis
calledtheangleofreflection,r.Thelawofreflectionstatesthattheangle
ofincidenceisalwaysequaltotheangleofreflection.

4. Refraction ofLight:
Refraction is the phenomenon of change in the path of light as it travels
from one medium to another (when the ray of light is incidentobliquely).
It can also be defined as the phenomenon of change in speed oflight
Rarer
Rarer
N
N
Denser
r
i
r
i
from one medium toanother.
Laws of Refraction:
I.Law: The incident ray, the normal to
the refracting surface at the point of
incidence and the refracted ray all liein
the sameplane.
II.Law: For a given pair of media andfor
light of a given wavelength, the ratio of
the sine of the angle of incidence to the
sine of the angle of refraction is a
constant. (Snell’sLaw)
n=
sini
sinr
(The constant nis called refractive index of themedium,
i is the angle of incidence and r is the angle ofrefraction.)
µ

5. μ= 1 / sin i
c( when sin r = 1
for r = 90
o
)

6. Advancedsunriseanddelayedsunset
Thesunappearsabouttwominutesearlierthanactualsunriseandthesun
remainsvisibleforabouttwominutesafteractualsunsetbecauseof
atmosphericrefraction.Whenthesunisbelowhorizon,therayshaveto
passfromrarertodensermedium.Soraysbendtowardsthenormal.Asa
resultthesunappearshigherthanitsactualposition.Theapparent
flatteningoftheSun’sdiscatsunriseandsunsetisalsoduetothesame
phenomenon.

7. Total InternalReflection:
Total Internal Reflection (TIR) is the phenomenon of complete reflection of
light back into the same medium for angles of incidence greater than the
critical angle of that medium.
N N N N
O
r =90°
i
c i >i
c
i
Rarer
(air)
Denser
(glass)
µ
g
µ
a
Conditions forTIR:
1.The incident ray must be in optically densermedium.
2.The angle of incidence in the denser medium must be greater than the
critical angle for the pair of media incontact.

8. Mirage:Onhotsummerdays,theairnearthegroundbecomeshotterthantheairat
higherlevels.Therefractiveindexofairincreaseswithitsdensity.Hotterairisless
dense,andhassmallerrefractiveindexthanthecoolerair.Iftheaircurrentsaresmall,
thatis,theairisstill,theopticaldensityatdifferentlayersofairincreaseswithheight.
Asaresult,lightfromatallobjectsuchasatree,passesthroughamediumwhose
refractiveindexdecreasestowardstheground.Thus,arayoflightfromsuchan
objectsuccessivelybendsawayfromthenormalandundergoestotalinternal
reflection,iftheangleofincidencefortheairnearthegroundexceedsthecritical
angle.Toadistantobserver,thelightappearstobecomingfromsomewherebelow
theground.Theobservernaturallyassumesthatlightisbeingreflectedfromthe
ground

9. Diamond:Diamondsareknownfortheirspectacularbrilliance.Their
brillianceismainlyduetothetotalinternalreflectionoflightinsidethem.
Thecriticalanglefordiamond-airinterface(≅24.4°)isverysmall,therefore
oncelightentersadiamond,itisverylikelytoundergototalinternal
reflectioninsideit.Diamondsfoundinnaturerarelyexhibitthebrilliancefor
whichtheyareknown.Itisthetechnicalskillofadiamondcutterwhich
makesdiamondstosparklesobrilliantly.Bycuttingthediamondsuitably,
multipletotalinternalreflectionscanbemadetooccur.

10. iii)Prism:Prismsdesignedtobendlightby90ºorby180º
makeuseoftotalinternalreflectionSuchaprismisalsoused
toinvertimageswithoutchangingtheirsize.Inthefirsttwo
cases,thecriticalanglei
cforthematerialoftheprismmust
belessthan45º.

13. The mirror equation
(i) The ray from the point which is parallel
to the principal axis. The reflected ray
goes through the focus of the mirror.
(ii) The ray passing through the centreof
curvature of a concave mirror or
appearing to pass through it for a convex
mirror. The reflected ray simply retraces
the path.
(iii) The ray passing through (or directed
towards) the focus of the concave mirror
or appearing to pass through (or directed
towards) the focus of a convex mirror. The
reflected ray is parallel to the principal
axis.
(iv) The ray incident at any angle at the
pole. The reflected ray follows laws of
reflection

15. This relation is known as the mirror
equation.
Linear magnification (m) :
Is the ratio of the height of the image
(h) to the height of the object (h)

16. Spherical RefractingSurfaces:
A spherical refracting surface is a part of a sphere of refractingmaterial.
A refracting surface which is convex towards the rarer medium is called
convex refractingsurface.
A refracting surface which is concave towards the rarer medium is
called concave refractingsurface.
••
C C
R R
A
AB B
APCB –Principal Axis
C –Centre ofCurvature
P –Pole
R –Radius ofCurvature
•
PP
•
DenserMediumDenserMedium RarerMediumRarerMedium

17. Assumptions:
1.Object is the point object lying on the principalaxis.
2.The incident and the refracted rays make small angles with the principal
axis.
3.The aperture (diameter of the curved surface) issmall.
New Cartesian SignConventions:
1.The incident ray is taken from left toright.
2.All the distances are measured from the pole of the refractingsurface.
3.The distances measured along the direction of the incident ray are
taken positive and against the incident ray are takennegative.
4.The vertical distances measured from principal axis in the upward
direction are taken positive and in the downward direction are taken
negative.

18. Refraction at ConvexSurface:
•
C
R
O
DenserMediumRarerMedium
• •
I
P
•
M
2
n
1
α
βγ
i
r
i = α + γ
γ = r +β
A
tan α=
orr = γ -β
MA
tan β=
MO
MA
MI
MA
MC
orα=
MA
orβ=
MO
MA
tanγ= orγ=
MI
MA
MC
According to Snell’slaw,
n
2sini
sinr
= or
i
r
n
1 n
1
n
2
= or n
1i=n
2r
Substituting for i, r, α, β and γ, replacing M by P andrearranging,
n
1 n
2 n
2 -n
1
PO PI PC
+
=
Applying sign conventions with values,
PO = -u, PI =+vandPC = +R
v
n
u
n
2 -n
1
R
n
2 n
1
v
-
u
=
(From Rarer Medium to Denser Medium -RealImage)
N

19. Lens Maker’sFormula
The image formation can be seen in
terms of two steps:
(i) The first refracting surface forms
the image I
1of the object O
[Fig (b)].
The imageI
1acts as a virtual object
for the second surface that forms
the image at I [Fig.(c].
For the first interface ABC, we get

21. Itisusefultodesignlensesofdesiredfocal
lengthusingsurfacesofsuitableradiiof
curvature.
Notethattheformulaistrueforaconcave
lensalso.InthatcaseR1isnegative,R2
positiveandtherefore,fisnegative.

22. Thin Lens Formula (Gaussian Form of LensEquation):
f
u
C
•
For ConvexLens:
A
B
A’
B’
M
R
Triangles ABC and A’B’C aresimilar.
A’B’
=
CB’
AB CB
Triangles MCF
2 and A’B’F
2 aresimilar.
MC
A’B’
=
B’F
2
v
AB
A’B’
=
CF
2
B’F
2
CF
2
or
•
2F
2
•
F
2
•
F
1
•
2F
1
CB
CB’
=
B’F
2
CF
2
CB’
CB
=
CB’ -CF
2
CF
2
According to new Cartesian sign
conventions,
CB = -u, CB’ =+vandCF
2 = +f.
1
v
1
f
1
- =
u

23. LinearMagnification:
Linear magnification produced by a lens is defined as the ratio of the size of
the image to the size of theobject.
m=
I
A’B’
=
O
CB’
+I +v
-O -u
AB CB
According to new Cartesian sign
conventions,
A’B’ =+I,AB = -O, CB’ = + v and
CB = -u.
I
O
=
v
u
= m=or
Power of a Lens:
Power of a lens is its ability to bend a ray of light falling on it and is reciprocal
of itsfocal length.When f is in metre, power is measured in Dioptre(D).
P=
1
f

24. Thederivationisvalidforanynumberofthinlensesincontact.Ifseveral
thinlensesoffocallengthf
1,f
2,f
3,...areincontact,theeffectivefocal
lengthoftheircombinationisgivenby
Intermsofpower,
P=P
1+P
2+P
3+…
Total magnification m of the combination is a product of
magnification (m1, m2, m3,...) of individual lenses
m = m
1m
2m
3 ...

25. Refraction of Light through Prism:
A
Prism
i
A
B C
P
r
1Or
2
N
1 N
2
D
InquadrilateralAPOQ,
A + O=180°…….(1)
(since N
1 and N
2 arenormal)
In triangleOPQ,
…….(2)r
1 + r
2 + O =180°
In triangleDPQ,
δ = (i -r
1) + (e -r
2)
δ = (i + e) –(r
1 +r
2)…….(3)
RefractingSurfaces
From (1) and (2),
A = r
1 +r
2
From(3),
δ = (i + e) –(A)
ori + e = A +δ
δ
Q
e
µ
Sum of angle of incidence and angle
of emergence is equal to the sum of
angle of prism and angle ofdeviation.

26. Variation of angle of deviation with angle ofincidence:
δ
i
0
i =e
δ
m
When angle of incidence increases,
the angle of deviationdecreases.
At a particular value of angle of incidence
the angle of deviation becomes minimum
and is called ‘angle of minimumdeviation’.
Atδ
m,i=eandr
1 = r
2 = r(say)
After minimum deviation, angle of deviation
increases with angle ofincidence.
Refractive Index of Material ofPrism:
r = A /2
i + e = A +δ
2 i = A +δ
m
i = (A + δ
m) /2
A = r
1+r
2 According to Snell’s law,
A =2r sini
sinr
1
sini
sinr
n= =
n =
sin
sin
(A +δ
m)
2
A
2

27. δ
m = ( n –1 ) A
n =
(A +δ
m)
2
A
2

28. DispersionofWhite Light throughPrism:
Thephenomenonofsplitting a ray of white light into its constituent colours
(wavelengths) is called dispersion and the band of colours from violet tored
is called spectrum(VIBGYOR).
δ
r
A
C
D
White
light
δ
v
Screen
N

29. ScatteringofLight–Bluecolouroftheskyand
ReddishappearanceoftheSunatSun-riseand
Sun-set:
Themoleculesoftheatmosphereandotherparticlesthatare
smallerthanthelongestwavelengthofvisiblelightaremore
effectiveinscatteringlightofshorterwavelengthsthanlightof
longerwavelengths.Theamountofscatteringisinversely
proportionaltothefourthpowerofthewavelength.(RayleighEffect)
LightfromtheSunnearthehorizonpassesthroughagreaterdistance
intheEarth’satmospherethandoesthelightreceivedwhentheSun
isoverhead.The correspondinglygreaterscatteringofshort
wavelengthsaccountsforthereddishappearanceoftheSunatrising
andatsetting.
WhenlookingattheskyinadirectionawayfromtheSun,we
receivescatteredsunlightinwhichshortwavelengthspredominate
givingtheskyitscharacteristicbluishcolour.

30. Optical instruments
CompoundMicroscope:
•
o
••
F
F
e
2F
e
2Fo o
• •
f
o
f
o
f
e
Eye
AF
B
A’
B’
A’’
Objective
Eyepiece
2F
o
B’’
Objective:The converging lens nearer to theobject.
Eyepiece:The converging lens through which the final image isseen.
Both are of shortfocallength.Focal length of eyepiece is slightly greater
than that of the objective.
A’’’
α
•
β
D
L
v
ou
o
P
o P
e

31. Angular Magnification or Magnifying Power(M):
Angular magnification or magnifying power of a compound microscope is
defined as the ratio of the angle β subtended by the final image at the eye to
the angle α subtended by the object seen directly, when both are placed at
the least distance of distinctvision.
M =
β
α
Since angles aresmall,
α =tanαand β = tanβ
M=
tanβ
M=
D
tan α
A’’B’’
x
D
M=
A’’B’’
x
A’’A’’’
D
AB
M=
D
A’’B’’
M=
A’B’
AB
A’’B’’
x
A’B’
AB
M = M
e xM
o
M
e = 1+
D
f
e
and
o
M=
v
o
-u
o
M=
v
o
-u
o
( 1+
D
)
f
e
Since the object is placed very close to the
principal focus of the objective and the
image is formed very close to theeyepiece,
u
o≈f
oand v
o ≈L
M =
-L
f
o
( 1+
D
)
f
e
orM ≈
-L
x
D
f
o
f
e
(Normaladjustment
i.e. image atinfinity)
e
M=1-
v
e
f
e
or
(v= -D
e
= -25cm)

32. Imageat
infinity
α
α
Objective
I
Eyepiece
Astronomical Telescope: (Image formed at infinity –
NormalAdjustment)
f
o f
e
P
o P
e
Eye
F
o
F
e
•
β
f
o + f
e =L
Focal length of the objective is much greater than that of the eyepiece.
Aperture of the objective is also large to allow more light to pass throughit.

33. Angular magnification or Magnifying power of a telescope in normal
adjustment is the ratio of the angle subtended by the image at the eye as
seen through the telescope to the angle subtended by the object as seen
directly, when both the object and the image are atinfinity.
M =
β
α
Since angles are small, α =tanαand β = tanβ
M=
tanβ
tanα
(f
o+f
e = L is called the length of the
telescope in normaladjustment).
M= /
F
eIF
eI
M= /
P
oF
e
-I
f
o
P
eF
e
-I
-f
e
M=
-f
o
f
e

34. A
B
α
Objective
Astronomical Telescope: (Image formed atLDDV)
P
o
Eye
e
f
o
F
eF
o
••
βP
f
e
α
I
Eyepiece
u
e
D

35. Angular magnification or magnifying power of a telescope in this case is
defined as the ratio of the angle β subtended at the eye by the final image
formed at the least distance of distinct vision to the angle α subtended at
the eye by the object lying at infinity when seendirectly.
M =
β
α
Since angles aresmall,
α =tanαand β = tanβ
M=
tanβ
tanα
M=
F
oI
PF
eo
/
F
oI
PF
oo
M=
P
oF
o
P
eF
o
M=
+f
o
-u
e
Multiplying by f
o on both sides and
rearranging, weget
M=
-f
o
( 1 +
f
e
f
e
D
)
-
1
f
1 1
v u
=
e
1
f
e
1 1
-D-u
- =
or
LensEquation
becomes
or
+
1
u
e
1
f
e
1
D
=
Clearly focal length of objective must be
greater than that of the eyepiece for larger
magnifyingpower.
Also, it is to be noted that in this case M is
larger than that in normal adjustment
position.

36. CassegrainTelescope: (Reflecting
Type)
ConcaveMirror
PlaneMirror
Eyepiece
Eye
Light
fromstar
M=
f
o
f
e
MagnifyingPower:

37. Reflecting Telescope

38. TheCassegraintelescopeisanastronomicalreflecting
telescope,inwhichthelightisincidentonalargeconcave
parabolicmirror(primarymirror)andreflectedontoa
smallerconvexmirror(secondarymirror).Thisreflectedlight
isreflectedagainthroughaholeintheconcavemirrorto
finallyformtheimageattheeyepiece.

39. Advantages of Reflecting type telescope
•Thereisnochromaticaberrationastheobjectiveisamirror.
•Sphericalaberrationisreducedusingmirrorobjectiveintheformofa
parabolic.
•Theimageisbrightercomparedtothatinarefractingtypetelescope.
•Mirrorrequiresgrindingandpolishingofonlyoneside.
•Highresolutionisachievedbyusingamirroroflargeaperture.
•Amirrorweightsmuchlessthanalensofequivalentopticalquality.Therefore,
mechanicalsupportofmirrorismuchlessofaproblemcomparedtothe
supportrequiredforthelens.Furthermirrorcanbesupported
Limitations of refracting telescope over a reflectingtype telescope
•Refracting telescope suffers from chromatic aberration uses large sized
lenses.
•It is difficult and expensive to make such large sized lenses.