CBSE NOTES CLASS 12 PHYSICS CHAPTER 9
RAY OPTICS AND OPTICAL INSTRUMENTS
Characteristics of light
Light waves are electromagnetic waves, whose nature is transverse. The speed of light is different in different media. In vacuum it is 3×10^{8} m.
The speed and wavelength of light change when it travels from one medium to another; but its frequency remains unchanged.
Important terms
Luminous objects: The objects which emits its own light, are called luminous objects, e.g., sun, other stars, an oil lamp etc.
Nonluminous objects: The objects which do not emit its own light but become visible due to the reflection of light falling on them, are called nonluminous objects, e.g., moon, table, chair, trees etc.
Ray of light: A straight line drawn in the direction of propagation of light is called a ray of light.
Beam of light: A bundle of the adjacent light rays is called a beam of light.
Image: If light ray coming from an object meet or appear to meet each other at a point after reflection or refraction, then this point is called image of the object.
Real image: If light rays coming from an object really meet each other at a point after reflection or refraction, then the image formed is called real image. Real image can be obtained on a screen. Real image is inverted.
Virtual image: If light rays coming from an object do not really meet each other at a point after reflection or refraction, but only appear to meet, then the image formed is called virtual image. Virtual image cannot be obtained on a screen and it is erect.
Reflection of light
The bouncing back of light rays into the same medium on striking a highly polished surface such as a mirror is called reflection of light.
Laws of reflection
 The incident ray, the reflected ray and the normal at the point of incidence all three lie in the same plane.
 The angle of incidence (i) is always equal to the angle of reflection (r).
Types of reflection
 Regular reflection: When a parallel beam of reflected light rays is obtained for a parallel beam of incident light rays after reflection from a plane reflecting reflection is called regular reflection.
 Irregular or diffused reflection: When a nonparallel beam of reflected light rays is obtained for a parallel beam of incident light rays after reflection from a surface, then such type of reflection is called irregular or diffused reflection
Mirror
A smooth and highly polished reflecting surface is called a mirror.
Plane mirror: A highly polished plane surface is called a plane mirror.
Properties of image formed by plane mirror
Size of image = Size of object
 Magnification = Unity
 Distance of image = Distance of object
 A plane mirror may form a virtual as well as real image.
 A man may see his full image in a mirror of half height of man.
Lateral inversion: In the image formed by a plane mirror the right side of the object appears as left side and vice versa. This phenomenon is called lateral inversion.
 When two plane mirror are held at an angle θ, the number of images of an object placed between them is given as below
 n = $\left(\frac{360\xb0}{\mathrm{\theta}}\right)$ – 1, where$\frac{360\xb0}{\mathrm{\theta}}$ is an integer.
 n = integral part of$\frac{360\xb0}{\mathrm{\theta}}$, when$\frac{360\xb0}{\mathrm{\theta}}$is not an integer.
 An image formed by a plane mirror is virtual, erect, laterally inverted, of same size as that of object and at the same distance as the object from the mirror.
[A plane mirror may form a real image, when the pencil of light incident on the mirror is convergent. Children, during their play form an image of sun on wall by a strip of plane mirror.]
 Kaleidoscope and periscope employ the principle of image formation by plane mirror.
 If keeping an object fixed a plane mirror is rotated in its plane by an angle θ, then the reflected ray rotates in the same direction by an angle 2θ.
 Focal length as well as radius of curvature of a plane mirror is infinity. Power of a plane mirror is zero.
Spherical mirror
A highly polished curved surface whose reflecting surface is a cut part of a hollow glass sphere is called a spherical mirror. Spherical mirrors are of two types
 Concave mirror: A spherical mirror whose bent in surface is reflecting surface, is called a concave mirror.
 Convex mirror: A spherical mirror whose bulging out surface is reflecting surface, is called a convex mirror.
Some terms related to spherical mirrors
 Centre of curvature of spherical mirror (C): It is the centre of the sphere of which the mirror is a part. The line joining any point on the mirror to C is normal to the mirror.
 Radius of curvature of spherical mirror (R): The radius of the hollow sphere of which the mirror is a part, is called radius of curvature.
 Pole of spherical mirror (P): The central point of the spherical mirror is called its pole (P).
 Principal axis of spherical mirror: The straight line passing through the pole and the centre of curvature of a spherical mirror is called the principal axis.
 Focus of spherical mirror (F): When a parallel beam of light rays is incident on a spherical mirror, then after reflection it meets or appears to meet at a point on principal axis, which is called focus of the spherical mirror.
 Paraxial rays are those which are incident at points close to the pole P of the mirror and make small angles with the principal axis.
 Focal plane of spherical mirror: If the parallel paraxial beam of light were incident, making some angle with the principal axis, the reflected rays would converge (or appear to diverge) from a point in a plane through F normal to the principal axis. This is called the focal plane of the mirror.
 Focal length of spherical mirror: The distance between the pole and focus is called focal length (f). Relation between focal length and radius of curvature is given by
$$\mathrm{f}\mathrm{}=\frac{\mathrm{R}}{2}$$
The power of a mirror is given by
$$\mathrm{P}\mathrm{}=\mathrm{}\frac{1}{\mathrm{f}}\mathrm{}\left({\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{e}}^{1}\mathrm{}\mathrm{o}\mathrm{r}\mathrm{}\mathrm{D}\mathrm{i}\mathrm{o}\mathrm{p}\mathrm{t}\mathrm{r}\mathrm{e}\right)$$
Proof
Let C be the centre of curvature of the mirror. Consider a ray parallel to the principal axis striking the mirror at M. Then CM will be perpendicular to the mirror at M. Let θ be the angle of incidence, and MD be the perpendicular from M on the principal axis. Then,
Since the incident ray is parallel to the principal axis and MF is transversal,
∠MCP = θ and ∠MFP = 2θ
Now,
$$\mathrm{tan}\mathrm{\theta}=\frac{\mathrm{M}\mathrm{D}}{\mathrm{C}\mathrm{D}}$$
$$\mathrm{A}\mathrm{n}\mathrm{d},\mathrm{}\mathrm{tan}2\mathrm{\theta}=\frac{\mathrm{M}\mathrm{D}}{\mathrm{F}\mathrm{D}}$$
For small θ, which is true for paraxial rays;
tan θ ≈ θ,
tan 2θ ≈ 2θ.
Therefore,
$$\frac{\mathrm{M}\mathrm{D}}{\mathrm{F}\mathrm{D}}=\mathrm{}2\frac{\mathrm{M}\mathrm{D}}{\mathrm{C}\mathrm{D}}$$
$$\Rightarrow \mathrm{F}\mathrm{D}\mathrm{}=\frac{\mathrm{C}\mathrm{D}}{2}$$
Now, for small θ, the point D is very close to the point P, therefore,
FD = f and CD = R
$$\Rightarrow \mathrm{}\mathrm{f}\mathrm{}=\mathrm{}\mathrm{}\frac{\mathrm{R}}{2}$$
Sign convention for spherical mirrors and lenses
 All distances are measured from the pole of the mirror.
 Distances measured in the direction of incident light rays are taken as positive.
 Distances measured in opposite direction to the incident light rays are taken as negative.
 Distances measured above the principal axis are positive.
 Distances measured below the principal axis are negative.
The focal length of concave mirror is taken negative and for a convex mirror taken as positive
Types of rays for mirrors
 A ray parallel to the principal axis, after reflection, will pass through the principal focus in case of a concave mirror or appear to diverge from the principal focus in case of a convex mirror.
 A ray passing through the principal focus of a concave mirror or a ray which is directed towards the principal focus of a convex mirror, after reflection, will emerge parallel to the principal axis.
 A ray passing through the centre of curvature of a concave mirror or directed in the direction of the centre of curvature of a convex mirror, after reflection, is reflected back along the same path.
 A ray incident obliquely to the principal axis, towards a point P (pole of the mirror), on the concave mirror or a convex mirror is reflected obliquely. The incident and reflected rays follow the laws of reflection at the point of incidence (point P), making equal angles with the principal axis.
Position, size and nature of image formed by a concave mirror
Position of the object 
Position of the image 
Size of the image 
Nature of the image 
Ray diagram 
At infinity 
At the focus F 
Highly diminished, pointsized 
Real and inverted  
Beyond C 
Between F and C 
Diminished 
Real and inverted  
At C 
At C 
Same size 
Real and inverted  
Between C and F 
Beyond C 
Enlarged 
Real and inverted  
At F 
At infinity 
Highly enlarged 
Real and inverted  
Between P and F 
Behind the mirror 
Enlarged 
Virtual and erect 
Position, size and nature of image formed by a convex mirror
Position of the object 
Position of the image 
Size of the image 
Nature of the image 
Ray diagram 
At infinity 
At the focus F, behind the mirror 
Highly diminished, pointsized 
Virtual and erect  
Between infinity and the pole P 
Between P and F, behind the mirror 
Diminished 
Virtual and erect 
Mirror formula
$$\frac{1}{\mathrm{f}}=\frac{1}{\mathrm{v}}+\frac{1}{\mathrm{u}}$$
Proof of mirror formula
The A′B′ is the image (in this case, real) of an object AB formed by a concave mirror as shown in the diagram above.
Even though we have considered only 3 rays, it does not mean that only three rays emanate from the point A. An infinite number of rays emanate from any source, in all directions.
Thus, point A′ is image point of A if every ray originating at point A and falling on the concave mirror after reflection passes through the point A′.
The two rightangled triangles A′B′F and MPF are similar.
(For paraxial rays, MP can be considered to be a straight line perpendicular to CP.)
Therefore,
$$\frac{\mathrm{B}\mathrm{\text{'}}\mathrm{A}\mathrm{\text{'}}}{\mathrm{P}\mathrm{M}}\mathrm{}=\mathrm{}\frac{\mathrm{B}\mathrm{\text{'}}\mathrm{F}}{\mathrm{F}\mathrm{P}}\mathrm{}\Rightarrow \mathrm{}\frac{\mathrm{B}\mathrm{\text{'}}\mathrm{A}\mathrm{\text{'}}}{\mathrm{B}\mathrm{A}}=\mathrm{}\frac{\mathrm{B}\mathrm{\text{'}}\mathrm{F}}{\mathrm{F}\mathrm{P}}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}[\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{}\mathrm{P}\mathrm{M}\mathrm{}=\mathrm{}\mathrm{A}\mathrm{B}]\mathrm{}$$
Since ∠APB = ∠A′PB′, the right angled triangles A′B′P and ABP are also similar.
Therefore,
$$\frac{\mathrm{B}\mathrm{\text{'}}\mathrm{A}\mathrm{\text{'}}}{\mathrm{B}\mathrm{A}}=\mathrm{}\frac{\mathrm{B}\mathrm{\text{'}}\mathrm{P}}{\mathrm{B}\mathrm{P}}$$
Comparing both the equations, we get
$$\frac{\mathrm{B}\mathrm{\text{'}}\mathrm{F}}{\mathrm{F}\mathrm{P}}=\mathrm{}\frac{\mathrm{B}\mathrm{\text{'}}\mathrm{P}\mathrm{}\u2013\mathrm{F}\mathrm{P}}{\mathrm{F}\mathrm{P}}\mathrm{}=\mathrm{}\frac{\mathrm{B}\mathrm{\text{'}}\mathrm{P}}{\mathrm{B}\mathrm{P}}$$
Applying the sign conventions,
B′P = –v,
FP = –f,
BP = –u
Putting these values, we get,
$$\frac{\mathrm{v}\mathrm{}\u2013\left(\mathrm{f}\right)}{\mathrm{f}}\mathrm{}=\mathrm{}\frac{\mathrm{v}}{\mathrm{u}}\mathrm{}\mathrm{}$$
$$\mathrm{}\Rightarrow \mathrm{}\frac{\mathrm{v}\mathrm{}\u2013\mathrm{f}}{\mathrm{f}}\mathrm{}=\mathrm{}\frac{\mathrm{v}}{\mathrm{u}}\mathrm{}\mathrm{}$$
$$\Rightarrow \mathrm{u}\mathrm{v}\mathrm{u}\mathrm{f}=\mathrm{v}\mathrm{f}\mathrm{}\mathrm{}$$
$$\Rightarrow \mathrm{}\frac{1}{\mathrm{f}}\frac{1}{\mathrm{v}}=\frac{1}{\mathrm{u}}\mathrm{}\mathrm{}\mathrm{}\left[\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{}\mathrm{L}\mathrm{H}\mathrm{S}\mathrm{}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{}\mathrm{R}\mathrm{H}\mathrm{S}\mathrm{}\mathrm{b}\mathrm{y}\mathrm{}\mathrm{u}\mathrm{v}\mathrm{f}\right]$$
$$\Rightarrow \mathrm{}\frac{1}{\mathrm{f}}=\frac{1}{\mathrm{u}}+\frac{1}{\mathrm{v}}$$
Linear magnification due to spherical mirror
The ratio of height of image (h’) formed by a mirror to the height of the object (h) is called linear magnification (m).
$$\mathrm{L}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{}\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{}\mathrm{m}\mathrm{}=\mathrm{}\frac{\mathrm{h}\mathrm{\u2019}}{\mathrm{h}}$$
In similar triangles A′B′P and ABP, we have,
$$\frac{\mathrm{B}\mathrm{\text{'}}\mathrm{A}\mathrm{\text{'}}}{\mathrm{B}\mathrm{A}}=\mathrm{}\frac{\mathrm{B}\mathrm{\text{'}}\mathrm{P}}{\mathrm{B}\mathrm{P}}$$
With the sign convention, this becomes
$$\mathrm{}\frac{{\mathrm{h}}^{\mathrm{\text{'}}}}{\mathrm{h}}\mathrm{}=\mathrm{}\mathrm{}\frac{\u2013\mathrm{v}}{\mathrm{u}}\mathrm{}\mathrm{}\mathrm{}\mathrm{}$$
$$\Rightarrow \mathrm{}\mathrm{m}=\frac{{\mathrm{h}}^{\mathrm{\text{'}}}}{\mathrm{h}}=\mathrm{}\frac{\mathrm{v}}{\mathrm{u}}$$
Areal magnification due to spherical mirror
The ratio of area of image to the area of object is called areal magnification.
$$\mathrm{A}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{}\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{}{\mathrm{m}}^{2}=\mathrm{}\frac{\mathrm{A}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{}\mathrm{o}\mathrm{f}\mathrm{}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{e}}{\mathrm{A}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{}\mathrm{o}\mathrm{f}\mathrm{}\mathrm{o}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}}=\frac{{\mathrm{v}}^{2}}{{\mathrm{u}}^{2}}\mathrm{}$$
Uses of concave mirrors
 Concave mirrors are commonly used in torches, searchlights and vehicles headlights to get powerful parallel beams of light.
 They are often used as shaving mirrors to see a larger image of the face.
 The dentists use concave mirrors to see large images of the teeth of patients.
 Large concave mirrors are used to concentrate sunlight to produce heat in solar furnaces.
Uses of convex mirrors
 Convex mirrors are commonly used as rearview (wing) mirrors in vehicles. These mirrors are fitted on the sides of the vehicle, enabling the driver to see traffic behind him/her to facilitate safe driving. Convex mirrors are preferred because they always give an erect, though diminished, image. Also, they have a wider field of view as they are curved outwards. Thus, convex mirrors enable the driver to view much larger area than would be possible with a plane mirror.
Refraction of light
The deviation of light ray from its path when it travels from one transparent medium to another transparent medium is called refraction of light.
Cause of refraction
The speed of light is different in different media, but the frequency remains same, hence the wavelength also changes. Lights of different wavelengths travel different distances during the same time through a medium. This causes the light to bend.
Laws of refraction
 The incident ray, the refracted ray and the normal at the point of incidence, all three lies in the same plane.
 The ratio of sine of angle of incidence to the sine of angle of refraction is constant for a pair of two media,
$$\mathrm{i}.\mathrm{e}.,\mathrm{}\mathrm{}{\mathrm{n}}_{21}=\frac{\mathrm{sin}\mathrm{i}}{\mathrm{sin}\mathrm{r}}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{}\mathrm{}\left[\mathrm{A}\mathrm{l}\mathrm{s}\mathrm{o}\mathrm{}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{d}\mathrm{}\mathrm{b}\mathrm{y}\mathrm{}{\mathrm{\mu}}_{21}\right]$$
where n_{21} is called refractive index of second medium with respect to first medium.
This law is also called Snell’s law.
Refractive index
The ratio of speed of light in vacuum (c) to the speed of light in any medium (v) is called refractive index of the medium.
$$\mathrm{R}\mathrm{e}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}\mathrm{}\mathrm{o}\mathrm{f}\mathrm{}\mathrm{a}\mathrm{}\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{u}\mathrm{m}\mathrm{}\mathrm{n}\mathrm{}=\frac{\mathrm{c}}{\mathrm{v}}$$
The refractive index is maximum for violet colour of light and minimum for red colour of light. i.e., n_{v} > n_{R}.
Refractive index of water =$\frac{4}{3}$ = 1.33;
Refractive index of glass =$\frac{3}{2}$ = 1.50
Refractive index of second medium with respect to first medium.
$${\mathrm{n}}_{21}\mathrm{}=\mathrm{}\frac{\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{e}\mathrm{d}\mathrm{}\mathrm{o}\mathrm{f}\mathrm{}\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}\mathrm{}\mathrm{i}\mathrm{n}\mathrm{}\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{u}\mathrm{m}\mathrm{}1}{\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{e}\mathrm{d}\mathrm{o}\mathrm{f}\mathrm{}\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}\mathrm{}\mathrm{i}\mathrm{n}\mathrm{}\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{u}\mathrm{m}\mathrm{}2}\mathrm{}=\frac{{\mathrm{v}}_{1}}{{\mathrm{v}}_{2}}\mathrm{}\mathrm{}$$
Refractive index of first medium with respect to second medium.
$${\mathrm{n}}_{12}\mathrm{}=\mathrm{}\frac{\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{e}\mathrm{d}\mathrm{}\mathrm{o}\mathrm{f}\mathrm{}\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}\mathrm{}\mathrm{i}\mathrm{n}\mathrm{}\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{u}\mathrm{m}\mathrm{}2}{\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{e}\mathrm{d}\mathrm{o}\mathrm{f}\mathrm{}\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}\mathrm{}\mathrm{i}\mathrm{n}\mathrm{}\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{u}\mathrm{m}\mathrm{}1}\mathrm{}=\frac{{\mathrm{v}}_{2}}{{\mathrm{v}}_{1}}$$
Obviously,
$${\mathrm{n}}_{21}=\frac{1}{{\mathrm{n}}_{12}}\mathrm{}$$
If there are three media involved,
n_{32} = n_{31} × n_{12}
where n_{31} is the refractive index of medium 3 with respect to medium 1 and n_{32} is the refractive index of medium 3 with respect to medium 2.
Optical density
A medium with larger refractive index is called optically denser and a medium with smaller refractive index is called optically rarer medium.
If n_{21} > 1, r < i , i.e., the refracted ray bends towards the normal. In such a case medium 2 is said to be optically denser than medium 1.
If n_{21} <1, r > i, the refracted ray bends away from the normal. This is the case when incident ray in a denser medium refracts into a rarer medium.
Optical density and mass density are different. It is possible that mass density of an optically denser medium may be less than that of an optically rarer medium.
For example, mass density of turpentine is less than that of water but its optical density is higher than water.
The drowning child problem
Consider a rectangular swimming pool PQSR. A lifeguard sitting at G outside the pool notices a child drowning at a point C. The guard wants to reach the child in the shortest possible time. Let SR be the side of the pool between G and C. Should he/she take a straight line path GAC between G and C or GBC in which the path BC in water would be the shortest, or some other path GXC? The guard knows that his/her running speed v_{1} on ground is higher than his/her swimming speed v_{2}.
Suppose the guard enters water at X. Let GX = l_{1} and XC = l_{2}. Then the time taken to reach from G to C would be
$$\mathrm{t}=\frac{{\mathrm{l}}_{1}}{{\mathrm{v}}_{1}}+\frac{{\mathrm{l}}_{2}}{{\mathrm{v}}_{2}}$$
For time to be minimum,$\frac{\mathrm{d}\mathrm{t}}{\mathrm{d}\mathrm{x}}$ = 0.
This will happen if the guard enters water at a point where Snell’s law is satisfied.
Draw a perpendicular LM to side SR at X.
Let ∠GXM = i and ∠CXL = r, then t is minimum when
$$\frac{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{}\mathrm{i}}{\mathrm{sin}\mathrm{r}}=\frac{{\mathrm{v}}_{1}}{{\mathrm{v}}_{2}}\mathrm{}$$
That is whenever two media and two velocities are involved, Snell’s law must be statisfied for the shortest time.
Lateral shift
For a rectangular slab, refraction takes place at two interfaces (airglass and glassair) and r_{2} = i_{1}, i.e., the emergent ray is parallel to the incident ray  there is no deviation, but it does suffer lateral displacement/shift with respect to the incident ray.
Real depth vs apparent depth and bending of pencil at the interface
$$\frac{\mathrm{R}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{}\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}\mathrm{}\left({\mathrm{h}}_{1}\right)}{\mathrm{A}\mathrm{p}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{}\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}\mathrm{}\left({\mathrm{h}}_{2}\right)}=\mathrm{}{\mathrm{n}}_{21}$$
Early sunrise and late sunset
The sun is visible a little before the actual sunrise and until a little after the actual sunset due to refraction of light through the atmosphere
By actual sunrise we mean the actual crossing of the horizon by the sun. Due to this, the apparent shift in the direction of the sun is by about half a degree and the corresponding time difference between actual sunset and apparent sunset is about 2 minutes.
The apparent flattening (oval shape) of the sun at sunset and sunrise is also due to the same phenomenon.
Atmospheric refraction & apperent height of stars
The starlight, on entering the earth’s atmosphere, undergoes refraction continuously before it reaches the earth. The atmospheric refraction occurs in a medium of gradually changing refractive index.
Since the atmosphere bends starlight towards the normal, the apparent position of the star is slightly different from its actual position. The star appears slightly higher (above) than its actual position when viewed near the horizon.
Twinkling of stars
The twinkling of a star is due to atmospheric refraction of starlight. The starlight, on entering the earth’s atmosphere, undergoes refraction continuously before it reaches the earth. The atmospheric refraction occurs in a medium of gradually changing refractive index. This apparent position of the star is not stationary, but keeps on changing slightly, since the physical conditions of the earth’s atmosphere are not stationary. Also the stars are very distant, they approximate pointsized sources of light. As the path of rays of light coming from the star goes on varying slightly, the apparent position of the star fluctuates and the amount of starlight entering the eye flickers – the star sometimes appears brighter and some times, fainter, which is the twinkling effect.
Why don’t the planets twinkle?
The planets are much closer to the earth, and are thus seen as extended sources. If we consider a planet as a collection of a large number of pointsized sources of light, the total variation in the amount of light entering our eye from all the individual pointsized sources will average out to zero, thereby nullifying the twinkling effect.
Critical angle
The angle of incidence in a denser medium for which the angle of refraction in rarer medium becomes 90°, is called critical angle (C or i_{C}).
Refractive index of denser medium
$${\mathrm{n}}_{21}\mathrm{}=\frac{1}{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{}\mathrm{C}}$$
Critical angle increases with temperature.
CRITICAL ANGLE OF SOME TRANSPARENT MEDIA
Substance medium 
Refractive index 
Critical angle 
Water 
1.33 
48.75° 
Crown glass 
1.52 
41.14° 
Dense flint glass 
1.62 
37.31° 
Diamond 
2.42 
24.41° 
Total internal reflection (TIR)
When a light ray travelling from a denser medium into a rarer medium is incident at the interface at an angle of incidence greater than critical angle, then light ray is reflected back in to the denser medium. This phenomenon is called total internal reflection.
The refractive index is maximum for violet colour and minimum for red colour of light, i.e., n_{V} > n_{R}
Ttherefore critical angle is maximum for red colour of light and minimum for violet colour of light, i.e., C_{V} < C_{R}
Total internal reflection occurs if angle of incidence in denser medium exceeds critical angle.
Examples of total internal reflection in nature
 Mirage is an optical illusion observed in deserts and roads on a hot day when the air near the ground is hotter and hence rarer than the air above. When the layers of air close to the ground have varying temperature with hottest layers near the ground, light from a distant tree may undergo total internal reflection, and the apparent image of the tree may create an illusion to the observer that the tree is near a pool of water.
 Brilliance of diamond: The brilliance of diamonds is due to the total internal reflection of light inside them. The critical angle for diamondair interface (≅ 24.4°) is very small, therefore once light enters a diamond, it is very likely to undergo total internal reflection inside it. Diamond cutting enhances sparkle. By cutting the diamond suitably, multiple total internal reflections can be made to occur.
 Special prisms: Prism designed to bend light by 90º or by 180º make use of total internal reflection. Such a prism is also used to invert images without changing their size. In the first two cases, the critical angle for the material of the prism must be less than 45º.
 Optical fibres: Optical fibres are fabricated with high quality composite glass/quartz fibres. Each fibre consists of a core and cladding. The refractive index of the material of the core is higher than that of the cladding.
When a signal in the form of light is directed at one end of the fibre at a suitable angle, it undergoes repeated total internal reflections along the length of the fibre and finally comes out at the other end.
Since light undergoes total internal reflection at each stage, there is no appreciable loss in the intensity of the light signal. Optical fibres are fabricated such that light reflected at one side of inner surface strikes the other at an angle larger than the critical angle. Even if the fibre is bent, light can easily travel along its length. Thus, an optical fibre can be used to act as an optical pipe.
Uses of optical fibre
Optical fibres are extensively used for transmitting and receiving electrical signals which are converted to light by suitable transducers.
They are used as a ‘light pipe’ to facilitate visual examination of internal organs like esophagus, stomach and intestines.
They are used in decorative lamps.
Photometry and related terms
The measurement of light as perceived by human eye is called photometry.
The main physical quantities in photometry are
 the luminous intensity of the source
 the luminous flux or flow of light from the source, and
 illuminance of the surface.
The SI unit of luminous intensity (I) is candela (cd).
The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540×10^{12} Hz and that has a radiant intensity in that direction of 1/683 watt per steradian.
The luminous flux is the total luminous intensity passing through the given area.
SI unit of luminous flux is lumen (lm).
If a light source emits one candela of luminous intensity into a solid angle of one steradian, the total luminous flux emitted into that solid angle is one lumen (lm).
Illuminance is defined as luminous flux incident per unit area on a surface (lm/m^{2} or lux). It can be measured directly.
The illuminance E, produced by a source of luminous intensity I, is given by
$$\mathrm{E}=\frac{\mathrm{I}}{{\mathrm{r}}^{2}}$$
where r is the normal distance of the surface from the source.
Luminance (L), is used to characterise the brightness of emitting or reflecting flat surfaces. Its unit is cd/m^{2} (called ‘nit’).
Refraction at a convex or concave spherical surface
Let us consider formation of image I of an object O on the principal axis of a spherical surface with centre of curvature C, and radius of curvature R. The rays are incident from a medium of refractive index n_{1}, to another of refractive index n_{2}. The aperture (or the lateral size) of the surface is taken to be small compared to other distances involved, so that small angle approximation can be made and NM can be taken to be nearly equal to the length of the perpendicular from the point N on the principal axis.
For small angles,
$$\mathrm{\angle}\mathrm{N}\mathrm{O}\mathrm{M}=\mathrm{}\mathrm{tan}\mathrm{\angle}\mathrm{N}\mathrm{O}\mathrm{M}\mathrm{}=\frac{\mathrm{M}\mathrm{N}}{\mathrm{O}\mathrm{M}}\mathrm{}$$
$$\mathrm{\angle}\mathrm{N}\mathrm{C}\mathrm{M}=\mathrm{}\mathrm{tan}\mathrm{\angle}\mathrm{N}\mathrm{C}\mathrm{M}\mathrm{}=\frac{\mathrm{M}\mathrm{N}}{\mathrm{M}\mathrm{C}}$$
$$\mathrm{\angle}\mathrm{N}\mathrm{I}\mathrm{M}=\mathrm{tan}\mathrm{}\mathrm{\angle}\mathrm{N}\mathrm{I}\mathrm{M}\mathrm{}=\frac{\mathrm{M}\mathrm{N}}{\mathrm{M}\mathrm{I}}$$
For ΔNOC, i is the exterior angle.
Therefore,
$$\mathrm{i}\mathrm{}=\mathrm{}\mathrm{\angle}\mathrm{N}\mathrm{O}\mathrm{M}\mathrm{}+\mathrm{}\mathrm{\angle}\mathrm{N}\mathrm{C}\mathrm{M}\mathrm{}\mathrm{}\mathrm{}\mathrm{}$$
$$\Rightarrow \mathrm{i}\mathrm{}=\frac{\mathrm{M}\mathrm{N}}{\mathrm{O}\mathrm{M}}\mathrm{}+\mathrm{}\frac{\mathrm{M}\mathrm{N}}{\mathrm{M}\mathrm{C}}$$
$$\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{}\mathrm{}\mathrm{r}\mathrm{}=\mathrm{}\mathrm{\angle}\mathrm{N}\mathrm{C}\mathrm{M}\mathrm{}\u2013\mathrm{}\mathrm{\angle}\mathrm{N}\mathrm{I}\mathrm{M}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}$$
$$\Rightarrow \mathrm{}\mathrm{r}\mathrm{}=\frac{\mathrm{M}\mathrm{N}}{\mathrm{M}\mathrm{C}}\mathrm{}\u2013\mathrm{}\frac{\mathrm{M}\mathrm{N}}{\mathrm{M}\mathrm{I}}$$
Now, by Snell’s law
n_{1} sin i = n_{2} sin r
or for small angles
n_{1 }i = n_{2 }r
Substituting value of i and r
$$\mathrm{}{\mathrm{n}}_{1}\left(\frac{\mathrm{M}\mathrm{N}}{\mathrm{O}\mathrm{M}}\mathrm{}+\mathrm{}\frac{\mathrm{M}\mathrm{N}}{\mathrm{M}\mathrm{C}}\right)=\mathrm{}{\mathrm{n}}_{2}\left(\frac{\mathrm{M}\mathrm{N}}{\mathrm{M}\mathrm{C}}\mathrm{}\u2013\mathrm{}\frac{\mathrm{M}\mathrm{N}}{\mathrm{M}\mathrm{I}}\right)\mathrm{}\mathrm{}$$
$$\Rightarrow \frac{{\mathrm{n}}_{1}}{\mathrm{O}\mathrm{M}}+\frac{{\mathrm{n}}_{2}}{\mathrm{M}\mathrm{I}}=\frac{{\mathrm{n}}_{2}{\mathrm{n}}_{1}}{\mathrm{M}\mathrm{C}}$$
Here, OM, MI and MC represent magnitudes of distances.
Applying the Cartesian sign convention,
OM = –u,
MI = +v,
MC = +R,
we get
$$\frac{{\mathrm{n}}_{2}}{\mathrm{v}}\frac{{\mathrm{n}}_{1}}{\mathrm{u}}=\frac{{\mathrm{n}}_{2}{\mathrm{n}}_{1}}{\mathrm{R}}$$
where,
f = focal length of the lens,
u = distance of object,
v = distance of image,
n_{1}= refrective index of medium 1
and n_{2}= refrective index of medium 2
Lens
A lens is a uniform transparent medium bounded between two spherical or one spherical and one plane surface.
Convex lens
A lens which is thinner at edges and thicker at middle is called a convex or converging lens.
Concave lens
A lens which is thicker at edges and thinner at middle, is called a concave or diverging lens.
Important terms related to lenses
The centre of the sphere is called centre of curvature of the lens.
Since there are two centres of curvature, we may represent them as C_{1} and C_{2}.
An imaginary straight line passing through the two centres of curvature of a lens is called its principal axis.
A parallel beam of light rays incident on a convex lens after refraction converges a point on the principal axis. This point is called principal focus of the convex lens.
A parallel beam of light rays incident on a concave lens after refraction appears to diverge from a point on the principal axis. This point is called principal focus of the concave lens.
The central point of a lens is called its optical centre. It is represented by the letter O.
The effective diameter of the circular outline of a spherical lens is called its aperture.
We shall assume in our discussion that the aperture of lenses is much less than its radius of curvature. Such lenses are called thin lenses with small apertures.
Important rays for spherical lenses
 A ray of light from the object, parallel to the principal axis, after refraction from a convex lens, passes through the principal focus on the other side of the lens. In case of a concave lens, the ray appears to diverge from the principal focus located on the same side of the lens.
 A ray of light, passing through the optical centre of the lens, emerges without any deviation after refraction.>
 A ray of light passing through the first principal focus (for a convex lens) or appearing to meet at it (for a concave lens) emerges parallel to the principal axis after refraction.
Nature, position and relative size of the image formed by a concave lens
Position of the object 
Position of the image 
Relative size of the image 
Nature of the image 
Ray diagram 
At infinity 
At focus F1 
Highly diminished, pointsized 
Virtual and erect  
Between infinity and O 
Between F1 and O 
Diminished 
Virtual and erect 
Nature, position and relative size of the image formed by a convex lens
Position of the object 
Position of the image 
Relative size of the image 
Nature of the image 
Ray diagram 
At infinity 
At focus F_{2} 
Highly diminished, point sized 
Real and inverted  
Beyond 2F_{1} 
Between F_{2} and 2F_{2} 
Diminished 
Real and inverted  
At 2F_{1} 
At 2F_{2} 
Same size 
Real and inverted  
Between F_{1} and 2F_{1} 
Beyond 2F_{2} 
Enlarged 
Real and inverted  
At focus F_{1} 
At infinity 
Infinitely large or highly enlarged 
Real and inverted  
Between F1 and O 
On the same side of the lens as the object 
Enlarged 
Virtual and erect 
Lens maker’s formula
$$\frac{1}{\mathrm{f}}=\left({\mathrm{n}}_{21}\mathrm{}\u2013\mathrm{}1\right)\left(\frac{1}{{\mathrm{R}}_{1}}\u2013\frac{1}{{\mathrm{R}}_{2}}\right)$$
where, n = refractive index of the material of the lens and R_{1} and R_{2} are radii of curvature of the lens.
Proof of lens maker’s formula
Consider image formation by a double convex lens. The image formation can be seen in terms of two steps,
The first refracting surface forms the image I_{1} of the object O.
The image I_{1} acts as a virtual object for the second surface that forms the image I.
For the first interface ABC, we can write, [as discussed above]
$$\frac{{\mathrm{n}}_{1}}{\mathrm{O}\mathrm{B}}+\frac{{\mathrm{n}}_{2}}{{\mathrm{B}\mathrm{I}}_{1}}=\frac{{\mathrm{n}}_{2}{\mathrm{n}}_{1}}{{\mathrm{B}\mathrm{C}}_{1}}$$
For the second interface ADC,
$$\frac{{\mathrm{n}}_{2}}{{\mathrm{D}\mathrm{I}}_{1}}+\frac{{\mathrm{n}}_{1}}{\mathrm{D}\mathrm{I}}=\frac{{\mathrm{n}}_{2}{\mathrm{n}}_{1}}{{\mathrm{D}\mathrm{C}}_{2}}\mathrm{}$$
[Since DI_{1} and DC_{2} are –ve and first medium has n_{2} as index and second medium has n_{1} as index in this case].
For a thin lens, BI_{1} ≈ DI_{1}. Adding Eqs., we get
$$\frac{{\mathrm{n}}_{1}}{\mathrm{O}\mathrm{B}}+\frac{{\mathrm{n}}_{1}}{\mathrm{D}\mathrm{I}}=\mathrm{}({\mathrm{n}}_{2}\mathrm{}\u2013\mathrm{}{\mathrm{n}}_{1})\mathrm{}\left(\frac{1}{{\mathrm{B}\mathrm{C}}_{1}}+\frac{1}{{\mathrm{D}\mathrm{C}}_{2}}\right)$$
If the object is at infinity and DI = f, then
$$\frac{{\mathrm{n}}_{1}}{\mathrm{f}}=\mathrm{}\left({\mathrm{n}}_{2}\mathrm{}\u2013\mathrm{}{\mathrm{n}}_{1}\right)\left(\frac{1}{{\mathrm{R}}_{1}}\frac{1}{{\mathrm{R}}_{2}}\right)\mathrm{}$$
$$\Rightarrow \mathrm{}\frac{1}{\mathrm{f}}=\mathrm{}\left(\frac{{\mathrm{n}}_{2}}{{\mathrm{n}}_{1}}\mathrm{}\u2013\mathrm{}1\right)\left(\frac{1}{{\mathrm{R}}_{1}}\frac{1}{{\mathrm{R}}_{2}}\right)$$
$$\Rightarrow \frac{1}{\mathrm{f}}=\mathrm{}\left({\mathrm{n}}_{21}\mathrm{}\u2013\mathrm{}1\right)\left(\frac{1}{{\mathrm{R}}_{1}}\frac{1}{{\mathrm{R}}_{2}}\right)\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\left[{\mathrm{n}}_{21}=\frac{{\mathrm{n}}_{2}}{{\mathrm{n}}_{1}}\right]\mathrm{}$$
This is called lens makers formula.
The formula is true for concave lens also.
In that case R_{1} is –ve and R_{2} is +ve, so f becomes negative.
$$\frac{1}{\mathrm{f}}=\mathrm{}\left({\mathrm{n}}_{21}\mathrm{}\u2013\mathrm{}1\right)\left(\frac{1}{{\mathrm{R}}_{2}}\frac{1}{{\mathrm{R}}_{1}}\right)$$
Thin lens formula
Now,
$$\frac{{\mathrm{n}}_{1}}{\mathrm{f}}=\mathrm{}\frac{{\mathrm{n}}_{1}}{\mathrm{O}\mathrm{B}}+\frac{{\mathrm{n}}_{1}}{\mathrm{D}\mathrm{I}}\mathrm{}$$
$$\mathrm{O}\mathrm{B}=\mathrm{}\mathrm{u}\mathrm{}$$
$$\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{}\mathrm{D}\mathrm{I}=\mathrm{v},\mathrm{}$$
So, we have,
$$\frac{1}{\mathrm{f}}=\frac{1}{\mathrm{v}}\frac{1}{\mathrm{u}}\mathrm{}$$
Power of a lens
The power P of a lens is defined as the tangent of the angle by which it converges or diverges a beam of light falling at unit distance from the optical centre.
From the geometry, we can write,
$$\mathrm{P}\mathrm{}=\mathrm{tan\; \theta}=\frac{\mathrm{h}}{\mathrm{f}}\mathrm{}$$
If we put h = 1, we get,
$$\mathrm{tan\; \theta}=\frac{1}{\mathrm{f}}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{o}\mathrm{r}\mathrm{}\mathrm{P}\mathrm{}=\frac{1}{\mathrm{f}}$$
That is the reciprocal of the focal length of a lens, when it is measured in metre, is called power of a lens.
The SI unit for power of a lens is dioptre (D): 1D = 1m^{–1}.
The power of a lens of focal length of 1 metre is one dioptre.
Power of a lens is positive for a converging lens and negative for a diverging lens.
Focal length of a lens combination
The first lens produces an image at I_{1}. Since image I_{1} is real, it serves as a virtual object for the second lens B, producing the final image at I. Since the lenses are thin, we assume the optical centres of the lenses to be coincident. Let this central point be denoted by P.
For the image formed by the two lenses, we get
$$\frac{1}{{\mathrm{f}}_{1}}=\frac{1}{{\mathrm{v}}_{1}}\frac{1}{\mathrm{u}}$$
$$\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{}\mathrm{}\mathrm{}\frac{1}{{\mathrm{f}}_{2}}=\frac{1}{\mathrm{v}}\frac{1}{{\mathrm{v}}_{1}}$$
Adding the two we get,
$$\frac{1}{\mathrm{f}}=\mathrm{}\frac{1}{{\mathrm{f}}_{1}}+\frac{1}{{\mathrm{f}}_{2}}=\frac{1}{\mathrm{v}}\frac{1}{\mathrm{u}}$$
Therefore, the power of the combination
P = P_{1} + P_{2}
When lenses are separated by a distance ‘d’
$$\frac{1}{\mathrm{f}}=\frac{1}{{\mathrm{f}}_{1}}+\frac{1}{{\mathrm{f}}_{2}}\u2013\frac{\mathrm{d}}{{\mathrm{f}}_{1}{\mathrm{f}}_{2}}$$
Power of the combination
P = P_{1} + P_{2} – dP_{1}P_{2}
Linear magnification by a lens
Linear magnification (m) produced by a lens is defined as the ratio of the size of the image to that of the object.
$$\mathrm{m}\mathrm{}=\mathrm{}\frac{\mathrm{h}\mathrm{\u2019}}{\mathrm{h}}=\frac{\mathrm{v}}{\mathrm{u}}$$
Total magnification ‘m’ of the combination is a product of magnification (m_{1}, m_{2}, m_{3}, ...) of individual lenses
m = m_{1} m_{2} m_{3} ...
For a small sized object placed linearly along the principal axis, its axial (longitudinal) magnification is given by
Focal length of a convex lens by displacement method
$$\mathrm{F}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}\mathrm{}\mathrm{o}\mathrm{f}\mathrm{}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{x}\mathrm{}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{}\mathrm{f}\mathrm{}=\frac{{\mathrm{a}}^{2}\mathrm{}\u2013\mathrm{}{\mathrm{d}}^{2}}{4\mathrm{a}}$$
where, a = distance between the image pin and object pin and d = distance between two positions of lens.
The distance between the two pins should be greater than four times the focal length of the convex lens, i.e.,
a > 4f.
Height of the object
$$\mathrm{O}\mathrm{}=\mathrm{}\sqrt{{\mathrm{I}}_{1}{\mathrm{I}}_{2}}$$
where I_{1} and I_{2} are image heights
Cutting of a lens
 If a symmetrical convex lens of focal length f is cut into two parts along its optic axis, then focal length of each part (a plano convex lens) is 2f.

If a symmetrical convex lens of focal length f is cut into two parts along the principal axis, then focal length of each part remains unchanged as f.
Aberration of lenses
The image formed by the lens suffer from following two drawbacks

Spherical aberration: Aberration of the lens due to which the rays passing through the lens are not focused at a single point and the image of a point object placed on the axis is blurred is called spherical aberration.
It can be reduced by using lens of large focal length, planoconvex lens, crossed lens or combination of convex and concave lenses.

Chromatic aberration: Image of a white object formed by lens is usually coloured and blurred due to dispersion. This defect of the image produced by lens is called chromatic aberration.
Prism
Prism is uniform transparent medium bounded between two refracting surfaces, inclined at an angle.
Angle of deviation
The angle sub tended between the direction of incident light ray and emergent light ray from a prism is called angle of deviation (δ).
Consider a prism as shown in the diagram below.
In the quadrilateral AQNR, two of the angles (at the vertices Q and R) are right angles. Therefore, the sum of the other angles of the quadrilateral is 180º
∠A + ∠QNR = 180º
From the triangle QNR,
r_{1} + r_{2} + ∠QNR = 180º
Comparing these two equations, we get
r_{1} + r_{2} = A
The total deviation δ is the sum of deviations at the two faces,
δ = (i – r_{1} ) + (e – r_{2} )
that is,
δ = i + e – A
Angle of deviation depends on the angle of incidence and the angle of prism.
Any given value of δ, except for i = e, corresponds to two values i and e. That is δ remains the same if i and e are interchanged.
This is related to the fact that the path of ray can be traced back, resulting in the same angle of deviation.
At the minimum deviation D_{m}, the refracted ray inside the prism becomes parallel to its base and we have
δ = D_{m} and i = e
That is 2r = A or r = A/2
Also,
$$\mathrm{i}\mathrm{}=\mathrm{}\mathrm{e},$$
$$\mathrm{s}\mathrm{o},\mathrm{}\mathrm{}{\mathrm{D}}_{\mathrm{m}}\mathrm{}=\mathrm{}2\mathrm{i}\mathrm{}\u2013\mathrm{}\mathrm{A},$$
$$\mathrm{o}\mathrm{r}\mathrm{}\mathrm{i}\mathrm{}=\frac{\mathrm{A}\mathrm{}+\mathrm{}{\mathrm{D}}_{\mathrm{m}}}{2}$$
The refractive index of the prism is
$${\mathrm{n}}_{21}=\frac{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{}\mathrm{i}}{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{}\mathrm{r}}=\mathrm{}\frac{\mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{\mathrm{A}\mathrm{}+\mathrm{}{\mathrm{D}}_{\mathrm{m}}}{2}\right)}{\mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{\mathrm{A}\mathrm{}}{2}\right)}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\left[\mathrm{P}\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{m}\mathrm{}\mathrm{F}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{a}\right]$$
The angles A and D_{m} can be measured experimentally.
This provides a method of determining refractive index of the material of the prism.
For a small angle prism, i.e., a thin prism, D_{m} is also very small, so we can write,
$${\mathrm{n}}_{21}=\frac{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{}\mathrm{i}}{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{}\mathrm{r}}=\mathrm{}\frac{\mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{\mathrm{A}\mathrm{}+\mathrm{}{\mathrm{D}}_{\mathrm{m}}}{2}\right)}{\mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{\mathrm{A}\mathrm{}}{2}\right)}\mathrm{}=\frac{\frac{\mathrm{A}\mathrm{}+\mathrm{}{\mathrm{D}}_{\mathrm{m}}}{2}}{\frac{\mathrm{A}\mathrm{}}{2}}\mathrm{}\mathrm{}$$
Or D_{m} = A(n_{21 } 1),
That is thin prisms (with small A) do not deviate the light too much.
Dispersion of light
The splitting of white light into its constituent colours in the sequence of VIBGYOR, on passing through a prism, is called dispersion of light.
The pattern of colour components of light is called the spectrum of light.
The red light bends the least, while the violet light bends the most (refractive index n_{V} > n_{R})
Q: Does the prism itself create colour in some way or does it only separate the colours already present in white light?
Newton’s experiment
When two similar prisms are put inverted with respect to each other, the emergent beam is white light.
That means, the colours were split by the first prism and the second prism combined them back.
The colours are actually light with different wavelengths.
Red light has the longest and violet the shortest
Thick lenses could be assumed as made of many prisms; therefore, thick lenses show chromatic aberration due to dispersion of light.
Angular dispersion
The angle subtended between the direction of emergent violet and red rays of light from a prism is called angular dispersion.
Angular dispersion (θ) = δ_{V} – δ_{R} = (n_{V} – n_{R})A
where δ_{V} and δ_{R} are angle of deviation.
Dispersive power
$$\mathrm{W}\mathrm{}=\mathrm{}\mathrm{}\frac{\mathrm{\theta}}{{\mathrm{\delta}}_{\mathrm{Y}}}=\frac{{\mathrm{n}}_{\mathrm{V}}\mathrm{}\u2013\mathrm{}{\mathrm{n}}_{\mathrm{R}}}{{\mathrm{n}}_{\mathrm{Y}}\mathrm{}\u2013\mathrm{}1}$$
where n_{Y} =$\frac{{\mathrm{n}}_{\mathrm{V}}\mathrm{}+\mathrm{}{\mathrm{n}}_{\mathrm{R}}\mathrm{}}{2}$, is mean refractive index.
SOME EFFECTS DUE TO SUNLIGHT
Rainbow formation
This is a phenomenon due to combined effect of dispersion, refraction and reflection of sunlight by spherical water droplets of rain.
The sun should be shining in one part of the sky (say west) while it is raining in the opposite part of the sky (say east).
Rainbow can only be seen when the back of the observer is towards the sun.
Mechanism of rainbow formation
Sunlight is first refracted as it enters a raindrop, which causes the different wavelengths (colours) of white light to separate.
Primary Rainbow
Longer wavelength of light (red) bends the least while the shorter wavelength (violet) bends the most.
These component rays strike the inner surface of the water drop and get internally reflected if the angle between the refracted ray and normal to the drop surface is greater than the critical angle (48º, in this case).
The reflected light is refracted again as it comes out of the drop as shown in the figure.
The violet light emerges at an angle of 40º related to the incoming sunlight and red light emerges at an angle of 42º. For other colours, angles lie in between these two values.
If the light undergoes only one internal reflection, the rainbow formed is called, primary rainbow. The red light from drop 1 and violet light from drop 2 reach the observers eye. The violet from drop 1 and red light from drop 2 are directed at level above or below the observer. Thus the observer sees a rainbow with red colour on the top and violet on the bottom.
On the other hand if there is double internal reflection and the rainbow is seen as violet on top of red, it is called secondary rainbow.
Secondary rainbow is a 4 step process.
Secondary Rainbow
Scattering of light
As sunlight travels through the earth’s atmosphere, it gets scattered (changes its direction) by the atmospheric particles.
Light of shorter wavelengths is scattered much more than light of longer wavelengths.
(The amount of scattering is inversely proportional to the fourth power of the wavelength. This is known as Rayleigh scattering and is inversely proportional to fourth power of the wavelength of light
$$\mathrm{R}\mathrm{a}\mathrm{y}\mathrm{l}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{}\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{}\propto \frac{1}{{\mathrm{\lambda}}^{4}}$$
Hence, the bluish colour predominates in a clear sky, since blue has a shorter wavelength than red and is scattered much more strongly.
In fact, violet gets scattered even more than blue, having a shorter wavelength, but since our eyes are more sensitive to blue than violet, we see the sky blue.
Large particles like dust and water droplets present in the atmosphere behave differently. The scattering depends on relative size of the wavelength of light λ, and the scatterer (of typical size, say, a). (Why rain clouds are darker?)
For a << λ, Rayleigh scattering will be there which is proportional to$\frac{1}{{\mathrm{\lambda}}^{4}}$.
For a >> λ, i.e., large scattering objects (for example, raindrops, large dust or ice particles) this is not true; all wavelengths are scattered nearly equally. That is why the clouds which have droplets of water with a >> λ are generally white.
Why does sun look red at sunset or sunrise?
At sunset or sunrise, the sun’s rays have to pass through a larger distance in the atmosphere and most of the blue and other shorter wavelengths are removed by scattering. The least scattered light reaching our eyes, therefore, the sun looks reddish.
Human eye
Human eye is an optical instrument which forms real image of the objects on retina. Retina colours contains millions of cone and rod cells which of are sensitive to colour and intensities of light respectively and transmit electrical signals via the optic nerve to the brain which finally processes this information.
Accommodation of eye
The shape (curvature) and the focal length of the lens can be modified somewhat by the ciliary muscles. This property of the eye is called accommodation.
For example, when the muscle is relaxed, the focal length is about 2.5 cm and objects at infinity are in sharp focus on the retina. When the object is brought closer to the eye, in order to maintain the same imagelens distance ≅ 2.5 cm), the focal length of the eye lens becomes shorter by the action of the ciliary muscles.
Near point
The closest distance for which the lens can focus light on the retina is called the least distance of distinct vision, or the near point. The standard value for normal vision is taken as 25 cm (D).
This distance increases with age, because of the decreasing effectiveness of the ciliary muscle and the loss of flexibility of the lens.
Far point
Farthest distance for which the lens can light on the retina is called far point or largest distance of distinct vision (Infinity).
Eye defects
Myopia or shortsightedness
It is a defect of eye due to which a person can see nearby objects clearly but cannot see far away objects clearly.
In this defect, the far point of eye shifts from infinity to a nearer distance. This defect can be removed by using a concave lens of appropriate power.
Hypermetropia or longsightedness
In this defect, a person can see far away objects clearly but cannot see nearby objects clearly.
In this defect the near point of eye shifts away from the eye. This defect can be removed by using a convex lens of appropriate power.
Astigmatism
In this defect, a person cannot focus on horizontal and vertical lines at the same distance at the same time. This defect can be removed by using suitable cylindrical lenses.
Colour blindness
In this defect, a person is unable to distinguish between few colours. The reason of this defect is the absence of cone cells sensitive for these colours. This defect cannot be removed.
Cataract
In this defect, an opaque white membrane is developed on cornea due to which person lost power of vision partially or completely. This defect can be removed by removing this membrane through surgery.
Presbyopia
If an elderly person tries to read a book at about 25 cm from the eye, the image appears blurred. This condition (defect of the eye) is called presbyopia. It is corrected by using a converging lens for reading.
Camera
A camera consists of a light proof box, at one end of which a converging lens system is fitted. A light sensitive film is fixed at the other end of the box, opposite to the lens system. A real inverted image of the object is formed on the film by the lens system.
fnumber of a camera
The fnumber represent the size of the aperture.
$$\mathrm{f}\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{}=\frac{\mathrm{F}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}\mathrm{}\mathrm{o}\mathrm{f}\mathrm{}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{}\left(\mathrm{F}\right)}{\mathrm{D}\mathrm{i}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{}\mathrm{o}\mathrm{f}\mathrm{}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{s}\left(\mathrm{d}\right)}$$
The amount of light (L) entering the camera is directly proportional to the area (A) of the aperture, or square of the diameter of aperture, i.e.,
$${\mathrm{L}\propto \mathrm{}\mathrm{A}\mathrm{}\propto \mathrm{}\mathrm{d}}^{2}$$
Therefore,
Brightness of image ∝ d^{2}
Exposure time is the time for which light is incident on the photographic film.
Simple microscope
A simple magnifier or microscope is a converging lens of small focal length, the lens is held near the object, one focal length away or less, and the eye is positioned close to the lens on the other side.
This gives an erect, magnified and virtual image of the object at a distance so that it can be viewed comfortably, i.e., at 25 cm or more.
 If the object is at a distance f, the image is at infinity.
 If the object is at a distance slightly less than the focal length of the lens, the image is virtual and closer than infinity.
 Although the closest comfortable distance for viewing the image is when it is at the near point (distance D ≅ 25 cm), it causes some strain on the eye. Therefore, the image formed at infinity is considered most suitable for viewing by the relaxed eye.
Magnifying power of simple microscope
When final image is formed at ‘least distance of distinct vision (D)
In this case, we have.
$$\mathrm{m}\mathrm{}=\frac{\mathrm{v}}{\mathrm{u}}=\mathrm{}\mathrm{v}\left(\frac{1}{\mathrm{v}}\frac{1}{\mathrm{f}}\right)$$
$$\Rightarrow \mathrm{m}=\mathrm{}1\frac{\mathrm{v}}{\mathrm{f}}$$
Since in this case v =  D, we have,
$$\mathrm{m}\mathrm{}=\mathrm{}1+\frac{\mathrm{D}}{\mathrm{f}}$$
When the final image is formed at infinity, the object is at f.
In this case, we have,
$$\mathrm{A}\mathrm{n}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{}\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{}\mathrm{m}\mathrm{}=\frac{\mathrm{}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}\mathrm{}\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{d}\mathrm{}\mathrm{b}\mathrm{y}\mathrm{}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{e}}{\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}\mathrm{}\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{d}\mathrm{}\mathrm{b}\mathrm{y}\mathrm{}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{}\mathrm{o}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}}$$
$$\mathrm{O}\mathrm{r}\mathrm{}\mathrm{m}=\frac{{\mathrm{\theta}}_{\mathrm{i}}}{{\mathrm{\theta}}_{\mathrm{o}}}\cong \frac{\mathrm{tan}{\mathrm{\theta}}_{\mathrm{i}}}{\mathrm{tan}{\mathrm{\theta}}_{\mathrm{o}}}$$
Now,
$$\frac{\mathrm{h}\mathrm{\u2019}}{\mathrm{h}}=\frac{\mathrm{v}}{\mathrm{u}}=\mathrm{}\mathrm{m}\mathrm{}$$
$$\Rightarrow \mathrm{}\frac{\mathrm{h}\mathrm{\u2019}}{\mathrm{v}}=\frac{\mathrm{h}}{\mathrm{u}}$$
Also for small angle, we can write,
$${\mathrm{\theta}}_{\mathrm{o}}\cong \mathrm{tan}{\mathrm{\theta}}_{\mathrm{o}}=\frac{\mathrm{h}}{\mathrm{D}}$$
And
$${\mathrm{\theta}}_{\mathrm{i}}\cong \mathrm{tan}{\mathrm{\theta}}_{\mathrm{i}}=\mathrm{}\frac{\mathrm{h}\mathrm{\u2019}}{\mathrm{v}}=\frac{\mathrm{h}}{\mathrm{u}}=\frac{\mathrm{h}}{\mathrm{f}}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\left[\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{}\mathrm{o}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{}\mathrm{i}\mathrm{s}\mathrm{}\mathrm{a}\mathrm{t}\mathrm{}\mathrm{f}\mathrm{o}\mathrm{c}\mathrm{u}\mathrm{s}\mathrm{}\mathrm{u}\mathrm{}=\mathrm{}\mathrm{f}\mathrm{}\right]$$
Therefore,
$$\mathrm{m}\mathrm{}=\frac{{\mathrm{\theta}}_{\mathrm{i}}}{{\mathrm{\theta}}_{\mathrm{o}}}=\frac{\mathrm{D}}{\mathrm{f}}$$
Compound microscope
It is a combination of two convex lenses called objective lens and eye piece separated by a distance. Both lenses are of small focal lengths, but f_{o }< f_{e}, where f_{o} and f_{e} are focal lengths of objective lens and eye piece respectively
Magnifying power of compound microscope
Due to objective
$${\mathrm{m}}_{\mathrm{o}}=\mathrm{}\frac{\mathrm{h}\mathrm{\u2019}}{\mathrm{h}}=\frac{\mathrm{L}}{{\mathrm{f}}_{\mathrm{o}}},\mathrm{}\mathrm{}\mathrm{}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{tan}\mathrm{\beta}=\frac{\mathrm{h}}{{\mathrm{f}}_{\mathrm{o}}}=\mathrm{}\frac{\mathrm{h}\mathrm{\u2019}}{\mathrm{L}}$$
Due to eyepiece, when the final image is formed at D,
$${\mathrm{m}}_{\mathrm{e}}=1+\mathrm{}\frac{\mathrm{D}}{{\mathrm{f}}_{\mathrm{e}}}\mathrm{}$$
Total magnification,
$$\mathrm{m}={\mathrm{m}}_{\mathrm{o}}{\mathrm{m}}_{\mathrm{e}}=\frac{\mathrm{L}}{{\mathrm{f}}_{\mathrm{o}}}(1+\mathrm{}\frac{\mathrm{D}}{{\mathrm{f}}_{\mathrm{e}}})$$
When the final image is formed at infinity
$${\mathrm{m}}_{\mathrm{e}}=\frac{\mathrm{D}}{{\mathrm{f}}_{\mathrm{e}}}\mathrm{}$$
Total magnification in this case,
$$\mathrm{m}={\mathrm{m}}_{\mathrm{o}}{\mathrm{m}}_{\mathrm{e}}=\frac{\mathrm{L}}{{\mathrm{f}}_{\mathrm{o}}}\times \frac{\mathrm{D}}{{\mathrm{f}}_{\mathrm{e}}}$$
To achieve a large magnification of a small object (hence the name microscope), the objective and eyepiece should have small focal lengths.
In practice, it is difficult to make the focal length much smaller than 1 cm. Also large lenses are required to make L large.
Other factors such as illumination of the object also contribute to the quality and visibility of the image.
Multicomponent lenses are used for both the objective and the eyepiece to improve image quality by minimising various optical aberrations (defects) in lenses.
Astronomical telescope
It is used for observing distinct images of heavenly bodies like stars, planets etc.
Refractive telescope
It has objective lens and eye piece, separated by a distance.
The objective has a large focal length and a much larger aperture than the eyepiece.
Real image is formed at the second focal point of objective. The eyepiece magnifies this image producing a final inverted image.
The magnifying power m is the ratio of the angle β subtended at the eye by the final image to the angle α which the object subtends at the lens or the eye.
Hence
$$\mathrm{m}=\frac{\mathrm{h}}{{\mathrm{f}}_{\mathrm{e}}}\times \frac{{\mathrm{f}}_{\mathrm{o}}}{\mathrm{h}}=\frac{{\mathrm{f}}_{\mathrm{o}}}{{\mathrm{f}}_{\mathrm{e}}}=\frac{\mathrm{tan}\mathrm{\beta}}{\mathrm{tan}\mathrm{\alpha}}\mathrm{}$$
Also,
$$\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}\mathrm{}\mathrm{o}\mathrm{f}\mathrm{}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{}\mathrm{t}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{}\mathrm{t}\mathrm{u}\mathrm{b}\mathrm{e}={\mathrm{f}}_{\mathrm{o}}+\mathrm{}{\mathrm{f}}_{\mathrm{e}}$$
For large magnifying power of a telescope fo should be large and fe should be small. For large magnifying power of a microscope; fo < fe should be small.
Terrestrial telescopes have, in addition, a pair of inverting lenses to make the final image erect.
Refractive telescopes can be used for both terrestrial as well as astronomical purposes.
The main considerations with an astronomical telescope are its light gathering power and its resolution or resolving power. Both of these can be achieved by using an objective of large diameter.
Resolving power
The ability of an optical instrument to produce separate and clear images of two nearby objects is called its resolving power (d).
Limit of resolution
The minimum distance between two nearby objects which can be just resolved by the instrument, is called its limit of resolution (dθ).
$$\mathrm{R}\mathrm{e}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{v}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}\mathrm{}\mathrm{o}\mathrm{f}\mathrm{}\mathrm{a}\mathrm{}\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{}=\frac{1}{\mathrm{d}}=\frac{2\mathrm{n}\mathrm{sin}\mathrm{\theta}}{\mathrm{\lambda}}\mathrm{}$$
where, d = limit of resolution,
λ = wavelength of light used.
n = refractive index of the medium between the objects and objective lens
and θ = half of the cone angle.
$$\mathrm{R}\mathrm{e}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{v}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}\mathrm{}\mathrm{o}\mathrm{f}\mathrm{}\mathrm{a}\mathrm{}\mathrm{t}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{}=\frac{1}{\mathrm{d}\mathrm{\theta}}=\frac{\mathrm{d}}{1.22\mathrm{\lambda}}$$
where, dθ = limit of resolution,
λ = wavelength of light used and
d = diameter of aperture of objective
Reflective telescope
Telescopes with mirror objectives are called reflecting telescopes.
Advantages of reflective telescope
 No chromatic aberration in a mirror.
If a parabolic reflecting surface is chosen, spherical aberration is also removed.
 Mechanical support is easier, since a mirror weighs much less than a lens of equivalent optical quality, and can be supported over its entire back surface, not just over its rim.
Disadvantage of reflective telescope
The objective mirror focuses light inside the telescope tube. The eyepiece and the observer should be right there, obstructing some light.