RAY RA Y & WAVE OPTICS
These topic s are taken from our Book: ISBN : 9789386320117 Product Name : Ray & Wave Optics for JEE Main & Advanced (Study Package for Physics) Product Description : "Disha's Physics Physics series by North Nort h India's popular faculty for IIT-JEE, Er. D. C. Gupta, have achieved a lot of acclaim by the IIT-JEE teachers and students for its quality and indepth coverage. To make it more accessible for the students Disha now re-launches its complete series in 12 books based on chapters/ units/ themes. themes. These books would provide opport unity to students to pick a particular particular book in a particular particular t opic. Ray & Wave Wave Optics for JEE Main & Advanced Advanced (Study Package for Physics) Physics) is the 11th book of the 12 book set. • The chapters provide detailed theory which is followed by Important Formulae, Strategy to solve problems and Solved Examples. Examples. • Each chapter covers 5 categories of New Patt Pattern ern practice exercises exercises for JEE - MCQ 1 correct, correct , MCQ more than 1 correct, Assertion & Reason, Passage and Matching based Questions. •
The book provides Previous years' questions quest ions of JEE (Main and Advanced). Past years KVPY questions are ar e also incorporated at their appropriate places. • The present format of the book would be useful for the students preparing preparing for Boards Bo ards and various competitive competitive exams."
Contents 1. Reection of Light 1.1 What is light? 1.2 Sources of light 1.3 The electromagnetic spectrum 1.4 Reection of light 1.5 The image 1.6 Perverted image 1.7 Spherical mirrors 1.8 Mirror formula 1.9 Magnication 1.10 Uses of spherical mirrors 1.11 Spherical aberration in mirrors Review of formulae & important points Exercise 1.1 - Exercise 1.6
3.4 3.5. 3.6 3.7 3.8
1 - 44
2 2 3 4 5 11 12 14 15 19 19 25
Hints & solutions (Ex. 1.1 - Ex. 1.6)
2. Refraction and Dispersion 2.1 2.2 2.3 2.4
Introduction : refraction Optical path Image formation by refraction Practical phenomena based
45-106 46 49 49
on refraction 2.5 Total internal reection 2.6 Phenomena based on TIR 2.7 The prism 2.8 Deviation produced by prism 2.9 Dispersion of light 2.10 Dispersive power 2.11 Combination of prisms
61 61 64 66 66 72 73 74
2.12 Line, band and continuous Spectrum 2.13 Rainbow 2.14 Scattering and blue sky 2.15 Colour of an object Review of formulae & important points Exercise 2.1 - Exercise 2.6
77 77 78 79 81
Hints & solutions (Ex. 2.1 - Ex. 2.6)
3. Refraction at Spherical Surface Lenses and Photometry
107-182
3.1
Refraction at a spherical Surface
108
3.2
Principal foci
109
3.3
Magnication
109
Lenses The thin lens formulas Principal foci Magnication Least possible distance between an object and its real image for a convex lens 3.9 Deviation produced by a lens 3.10 Power of a lens 3.11 Combined focal length 3.12 Silvering of lenses 3.13 Defects of images : aberration 3.14 The human eye 3.15 Defects of vision 3.16 Simple microscope or magnier 3.17 Compound microscope 3.18 Telescope 3.19 Photometry : an introduction 3.20 Luminous intensity 3.21 Illuminance 3.22 Photometer Review of formulae & important points Exercise 3.1 - Exercise 3.6 Hints & solutions (Ex. 3.1 - Ex. 3.6)
4. Wave Optics 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13
Wave optics : an introduction Huygens’ principle Interference Displacement of fringes Fresnel’s biprism Lloyd’s mirror arrangement Interference in thin lms Diffraction Fraunhoffer diffraction at Single slit Diffraction grating Transverse nature of light Polarisation Polarisation by reection : Brewster’s law 4.14 Malus’ law Review of formulae & important points Exercise 4.1 - Exercise 4.6 Hints & solutions (Ex. 4.1 - Ex. 4.6) 98
114 116 118 120
121 124 125 125 129 138 141 142 143 145 148 152 153 153 154 155
183-244 184 184 185 193 194 195 199 205 205 208 210 211 211 212 213
Chapter 3 Refraction at Spherical Surface Lenses and Photometry 114
3.4 L ENSES Lenses play very important role in our life. They are used in microscopes, telescopes and movie cameras etc. We have natural lenses in our eyes. A lens consists of two refracting surfaces (at least on e spherical) inclin ed at some angle. In thin lens th e spacing between the refracting surfaces is negligibly small. In thick lens the spacing between the refracting surfaces at the centre of the lens is large enough. Basically a lens is the combination of many prisms. Thus lens can also produce deviation and dispersion (aberration). Lenses a re of two types. We shall study the special case of thin lens in which the thickest part is thin compared to the object distance or focal length of the lens.
Convex lens A lens which is thicker at the middle and th inner at the edges is known as convex or converging lens.
Concave lens A lens which is thin ner at the middle and thicker at the edges is known as concave or diverging lens.
Fig. 3.16
Note:
For convex lens R1= + ve, R2 = – ve. For concave lens R1 = – ve, R2 = + ve.
Fig. 3.17 The following are the terms used with the lenses:
(i)
Aperture : The effective width of a lens from which refraction takes place is called aperture. In figure LL is the aperture of the lens.
(ii)
Optical centre : The centre of a lens is called its optical c entre. It is denoted by letter P. A ray of light passing through optical centre does not suffer any deviation.
(iii)
Principal or optic axis : The line joining the centres of curvatures of the lens is known as principal axis ( PA).
(iv)
Principal focus and focal length : A point on the principal axis at which parallel rays of light after refraction from the lens converge or appear to diverge from it is known as focus. It is denoted by a letter F . The distance of focal point from optical centre is known as focal length of the lens. It is denoted by f .
Guidelines for image formation
Refraction in thin lenses. Fig. 3.18
On the basis of laws of refractions, the following rays coming from the object are usually used for constructing ray diagram for image: (i) A ray of light coming parallel to the principal axis; after refraction from the lens will pass or appears to pass through focus and vice-versa. (ii) A ray of light passing through the optical centre of the lens goes straight without deviation. This is however, is true for a thin lens because the two sides of a lens at its centre are almost parallel only when the lens is thin.
R EFRACTION AT S PHERICAL SURFACES AND P HOTOMETRY
Image formation by convex lens Object position
Ray diagram
Image at focus. Real, inverted and diminished image.
At ¥
Between 2F and
Position and nature of image
Between F and 2F. Real, inverted and diminished.
¥
At 2F. Real, inverted and same size of the object.
At 2F
Object position
Ray diagram
Position and nature of image
Between 2F and F
Beyond 2F. Real inverted and larger than object.
At F
At ¥. Real, inverted and very larger than object.
Between F and P
On the side of the object. Virtual, erect and larger than object.
115
116
O PTICS AND M ODERN P HYSICS Image formation by concave lens Object position
Position and nature of image
Ray diagram
At ¥
At focus. Virtual erect and diminished.
Anywhere between ¥ and P
Between P and F . Virtual, erect and smaller than object.
3.5. THE
THIN LENS FORMULAS
Consider a thin lens made of a material of refractive indexm2 and situated in a medium of refractive index m1 on its both sides. Let R1 and R2 be the radii of curvature of the two co-axial spherical surfaces. Suppose an object O is placed at a distance u from the optical centre of the lens. An image I ¢ is formed by refraction at the first surface of the lens, at a distancev¢ from the pole of the surface. Then by refraction formula, we have
m2 v'
-
m1 u
=
m 2 - m1 R1
.
...(i)
The image I ¢ becomes the virtual object for the second surface of the lens, an d which forms the image I at a distance v from this surface. Then
m1
Fig. 3.19
v
-
m2 v'
=
m1 - m2 R2
.
...(ii)
In this case rays are going from medium of refractive index m2 to the medium of refractive index m1. Moreover do not place the sign with R1 and R2, because they have already signed. Adding equations (i) and (ii), we have
m1 m1 v
u
=
æ1 1 ö ( m 2 - m1 ) ç - ÷ è R1 R2 ø
æ m2 ö æ 1 1 ö ç - 1÷ ç - ÷ v u è m1 ø è R1 R2 ø If the lens is placed in air, then m1 = 1, and putting m2 = m, we have or
1
-
1
1
-
1
v
u
=
=
æ1 1 ö ( m -1) ç - ÷ ...(1) è R1 R2 ø
R EFRACTION AT S PHERICAL SURFACES AND P HOTOMETRY
Note:
1.
The equation derived will hold only for paraxial rays and for a thin lens.
2.
While solving numerical problems, proper signs are to be placed for all the given values, and no sign for unknowns.
Equation (1) is known as the thin lens formula and i s usually written in the form 1 v
-
1 u
1. f
=
...(2)
where f is known as focal length of the lens, and is given by 1 f
æ1 1 ö ( m - 1) ç - ÷ ....(3) è R1 R2 ø
=
The above formula is known as lens maker's formula.
1 v
-
1 u
1
=
f
Graph of u vs. v for a lens : According to lens formula it is a hyperbola, as shown in figure. (a)
Convex lens
u = – ¥
-2 f
-f -
= +f
+2 f
+¥ - f
v (b)
f 2
-
Fig. 3.20
f 4 f 3
0
+f
0
+
f 2
+2 f +
2 f 3
+¥
+f
Concave lens
u = – ¥ v
= -f
-2 f -2 f 3
-f
-
f
-
-
2
f 2 f 3
f
0
+
0
+f
2
+ f + f + 2 f +¥
+¥ +¥ - 2 f - f
Lens with different mediums on its sides In case when there ar e different mediums on both sides of the lens saym1 and m3, then we can write for first surface;
and for second surface;
m2 v'
-
m1 u
m3 m 2 v
v'
=
=
Fig. 3.21
m2 - m1
...(iii)
R1
m3 - m2 R2
.
..(iv)
Adding equations (iii) and (iv), we have
m3 m1 v
u
=
m2 - m1 m3 - m2 ....(4) + R1
R2
Fig. 3.22
117
118
O PTICS AND M ODERN P HYSICS
3.6 PRINCIPAL FOCI There are two principal foci of any lens. These are : First principal focus : For the first focus F 1, v =
(i)
1
=
f 1
-
¥ , u = f 1. Thus by equation (4)
æ m 2 - m1 m3 - m2 ö + ÷ ...(5) m1 çè R1 R2 ø 1
Here f 1 is called first focal length. For the convex lens it will be on object side and for concave lens it will on ima ge side. (ii)
Second principal focus : For the second focus F 2, u = –
1
=
f 2
¥ , v = f 2. Thus
é m2 - m1 m3 - m 2 ù + ú ...(6) m3 êë R1 R2 û 1
Now from equations (5) and (6), we get f 1
=
f 2
-
m1 m3
...(7)
In case when m1 = m3, f 1 = – f 2. In this case we will simply use f as the focal length.
Fig. 3.23
Fig. 3.24
Note:
1. In case, if t is the thickness of the lens at the centre, then we can solve the
problem in two steps :
m1 -u
=
m2 - m1 + R1 ...(i)
m2 ( v '- t )
=
m1 - m 2 - R2 ...(ii)
m2 v' Fig. 3.25
and
m1 v
-
-
On solving (i) and (ii), we can get v. 2.
If distances of the object and the image are measured from first and second focus respectively, then x1 x2 = f 1 f 2. This known as Newton¢s formula.
Fig. 3.26
R EFRACTION AT S PHERICAL SURFACES AND P HOTOMETRY
More about focal length According to our sign conventions; For convex lens, R 1
\
1 f
=
+ R,
R2 = – R;
=
æ1 1 ö ( m -1) ç - ÷ è R - R ø R
or
f
=
2 ( m -1) .
=
– R,
=
1 ö æ 1 ( m -1) ç - ÷ è - R + R ø
For concave lens, R 1
\ or
1 f
f
=
-
R2 = + R.
R 2 ( m -1)
Fig. 3.27
.
Thus the focal length of convex lens is positive and that of concave is negative. Note:
Fig. 3.28
In case when parallel ray are not parallel to principal axis, they intersect at a point which is not on the axis. Plane thr ough this point is called focal plane.
Fig. 3.29
119
Chapter 4 Wave Optics 4.3 INTERFERENCE When two or more coherent waves superimpose, the r esultant intensity in the region of superposition is different from the int ensity of individual waves. This modification in the distribution of intensity in the region of superposition is called interference.
Young's double slit experiment (YDSE) Thomas Young in 1801 devised an ingenious method of producing coherent sources. In this method a single wavefront is divided into two; these two split wavefronts act as if they originated from two sources having a constant phase relationship and therefore, when they were allowed to interfere, a stationary int erference pattern was obtained. In the experiment light from a source S fell on a cardboard which contai ned two pinholes (or slits) S 1 and S 2 which were very close to one another. The spherical waves origina ting from S 1 and S 2 were coherent and so beautiful interference fringes or bands were obtained on the screen.
Fig. 4.4. Young's double slits arr angement.
Coherent sources Two sources of light are said to be coherent if th ey emit light waves of same frequency and having constant phase difference (may be zero). It means the two sources must emit waves of the same wavelength. In practice it is not possible to have two independent sources which are coherent an d so for practical purposes, two virtual sources formed from a single source can act as coherent sources. Young¢s double slits arrangement, Fresnel¢s biprism method, Llyod¢s mirror arrangement are the methods of producing two coherent sources from a sin gle source. Note:
1. 2.
Two independent laser sources of equal wavelengths can be coherent. Because they can maintained the constant phase difference for long time. Two ordinary sources can not maintain the constant phase difference so they can not be coherent an d hence will not interfere.
Analytical treatment of interference Consider a monochromatic source of light S emitting light waves of wavelength l and two narrow slits S 1 and S 2. S 1 and S 2 are separated a distance d and equidistance from S . S 1 and S 2 then becomes two virtual coherent sources of light waves. Let f is the phase difference between the two waves reaching at point P . The equation of wave for any fixed position (say screen at x = 0) can be written as : y = a sin (wt –k x), where x = 0 and so, we get y = a sin wt. Thus for two coherent waves, we can write
185
186
O PTICS AND M ODERN P HYSICS y 1 = a1 sin wt and y 2 = a2 sin (wt + f). By principle of superposition, we have y = y1 + y2
Substituting and
a1 + a2 cos f a2 sin f
=
a1 sin wt
+ a2 sin ( wt + f )
=
a1 sin wt
+ a2 [sin wt cos f + cos wt sin f]
=
( a1 + a2 cos f ) sin wt + a2 sin f cos wt
= R cos q
...(i)
= R sin q, we get
...(ii)
y = R cos q sin wt or
y = R sin ( wt
+ R sin q cos wt
+ q) .
...(1)
This shows that the resultant wave at any point P is simple harmonic of amplitude R. The amplitude R can be obtained as : Squaring equations (i ) and (ii), we have R2 =
a12 + a22 + 2 a1a2 cos f .
...(2)
As intensity I of wave is proportional to square of the amplitude, and so I = I1 + I 2 + 2 I1 I 2 cos f .
...(3)
Also dividing equation (ii) by (i), we get tan q =
a2 sin f a1 + a2 cos f
.
...(4)
Fig. 4.5 In Young's interference experiment, incident monochromatic light is diffracted by slit S o, which then acts as a point source of light that emits semicircular wavefronts. As that light reaches screen B, it is diffracted by slits S 1 and S 2, which then act as two point sources of light. The light waves traveling from slits S 1 and S 2 overlap and undergo interference, forming an interference pattern of maxima and minima on viewing screen C .
AVE O PTICS W
Depending on the phase difference f between the two waves, the intensity of resulting wave may be minimum or maximum. Accordingly there are two types of interference. These are : (i) Constructive interference (bright point) The intensity I will be maximum, when cosf = + 1, or n = 0, 1, 2,..... f = 2pn,
l f; 2p D x = nl
D x
=
2 I max = Rmax
=
a12 + a22 + 2a1a2
2 I max = Rmax
=
( a1 + a2 )
As path difference
\ Now or (ii)
2
.
...(5)
Destructive interference (dark point) The intensity I will be minimum, when
cosf = –1 =
( 2n - 1) p,
D x
=
( 2n - 1)
2 I min = Rmin
=
a12
2 I min = Rmin
=
( a1 - a2 )
=
( a1 + a2 ) . 2 ( a1 - a2 )
Also Now or
Thus
n = 1, 2, 3,...
f
or
I max I min
2
=
l 2
+ a22 - 2a1a2 2
...(6)
2
Rmax 2 Rmin
...(7)
Special cases : When two identical waves interfere, a1 = a2 = a
\ Also
or
I max
= 4a2 and I min = 0.
I =
a2 + a2
+ 2 aa cos f
=
2a 2 ( 1 + cos f )
=
2a
=
4a 2 cos 2
2
´ 2 cos 2
f 2
f 2
2 I = I max cos
f 2
.
...(8)
Intensity distribution It has been obtained that intensity at bright points is 4a2 and at dark points is zero. According to law of conservation of energy, the energy of the intefering waves as a whole remains constant. Thus th e energy from points of minimum in tensity transfers to
Fig. 4.6
187
188
O PTICS AND M ODERN P HYSICS the points of ma ximum intensity. The intensity variation with phase difference is shown in fig. 4.7.
Fig. 4.7
Fringe width Consider two sources S 1 and S 2 emitting monochromatic light of wavelength l. The separation between them is d . The interference fringes are obtained on a screen placed at a distance D from the sources. The fringes are of equal width and altern atively bright and dark. The centre to centre d istance between two consecutive bright or dark fringes is called fringe width . Consicer a point P on the screen at a distance yn from the centre of the screen O. The angular position of the point P is q from the centr e of the sources (see fig. 4.8).
The path difference between the waves on arriving at point P , is S 2 P – S 1 P , which is equal to D x. From the figure D x = d sinq. For small q, we can write sin q tan q. Thus ;
D x From the triangle SOP ,
\ (i)
;
tan q =
D x
=
d tanq. n
D
,
d yn D
… (i)
Bright fringes
There will be bright fringe at P , when D x = nl. Thus path difference dyn D
= nl
AVE O PTICS W
or
yn =
n Dl d
;
n = 0, 1, 2, ......
...(9)
Equation (9) represents the position ofnth bright fringe. The (n – 1)th fringe will be at a distance
\
yn -1
=
b
=
Fringe width
=
b
or (ii)
=
( n - 1) n
Dl d
- yn-1
n Dl d Dl d
- ( n - 1)
Dl d
.
...(10)
Dark fringes
There will be dark fringe at P , when D x = (2n –1) d yn D or
=
yn =
( 2n - 1)
l 2
. Thus
l 2
( 2n - 1)
Dl
2
d
; n = 1, 2,...
...(11)
Equation (11) represents the position of nth dark fringe. The (n –1)th fringe will be at a distance yn -1
\
Fringe width
b
=
= yn –y n–1 =
or
b
é 2 ( n - 1) - 1ù Dl ê ú d 2 ë û
=
é 2n - 1ù Dl - é 2 ( n - 1) - 1ù Dl ú êë 2 úû d ê 2 ë û d Dl d
.
It shows that the fringe width is equal for bright and dark frin ge. Note:
The maximum path difference D xmax = d, when sinq = 1. If n are the number of brights fringes on one side of the central bright, then d = nl or n =
d
l
. Thus
total number of fringes that can be on the screen are = 2n + 1, including central central fringe.
Angular fringe width Sometime it is required to represent fringe width in terms of angle subtended at the centre of the sources. If a is the angular fringe width, then
a
=
b D Fig. 4.9
189
190
O PTICS AND M ODERN P HYSICS
=
or
a
=
Dl / d D
l d
radian.
Special case : If YDSE is performed in water, and observer is in ai r, then fringe width
b water
=
D l ater . d
As
lwater
=
l air , mw
\
b water
=
1 é D l air ù
mw ëê
d
ú û
=
bair . mw
Important points : 1.
In YDSE, the central fringe is bright, and all the bright fringes are of same intensity. Colour of bright fringes are of the colour of incident light .
2.
If slits are of equal size, the intensity of all the dark frings are zero.
3.
If slits are of unequal size, then the intensity of dark fringe is not zero.
4.
All the fringes are of equal width.
5.
If sources have random phase difference, then there will be no interference. The intensity at any point will be I = a2 + a2 = 2a2.
6.
If white light is used in the experiment, then the central fringe will be white, and other fringes are overlapped colour fringes.
Condition of obserable interference 1.
The sources must be coherent.
2.
The separation between the slits should be small (order of mm), so that size of fringe is large enough to observe.
3.
The amplitudes of interfering waves are equal or nearly equal, otherwise the intensities of bright and dark frin ges are not differentiable.
