Physics optics

Refraction

• When a ray of light travels from one medium to another, it undergoes a change in its direction.

• When light travels from a rarer medium to a denser one, it bends towards the normal (i > r) and when travels from a denser medium to a rarer one, it bends away from the normal (i < r).

• The bending of a ray of light on passing from one medium to another is called `"refraction"`.

• Every transparent medium has a property known as optical density. The optical density of a transparent medium is closely related to the speed of light in the medium.

• A medium in which the speed of light is more, is known as optically rarer medium and the medium in which speed of light is lesser is known as optically denser medium.

• The rays of light while going from air (rarer) to glass (denser) will bend towards the normal. The rays of light going from denser to rarer, say from glass or water to air, will bend away from the normal.

• When a ray of light suffers refraction, then its velocity and wavelength changes but frequency remain same
i.e. `n_1 = n_2` or `v_1/lamda_1 = v_2/lamda_2`

`text(Laws of Refraction)`
There are two laws of refraction:
(i) Suppose light travels from medium 1 to medium 2 in which the speeds of light are `v_1` and `v_2` respectively. The ratio of the sine of the angle of incidence to the sine of the angle of refraction for the given pair of media is constant and equal to the ratio of the speeds of light in medium 1 and medium 2.
`(sin i)/(sin r) = (v_1)/(v_2)` `"This relation is called Snell's law"`.
where `i=` angle of incidence
and `r=` angle of refraction

The ratio `(v_1)/(v_2)` is called the relative refractive index of medium 2 with respect to medium 1 and is denoted by `text()_1mu_2`

(ii) The incident ray, the refracted ray and the normal at the point of incidence, all lie in the same plane.
The Laws of refraction are valid for both types of surfaces i.e. for plane as well as spherical refracting surfaces.

`Atext(bsolute Refractive Index of a Medium)`
Absolute Refractive index of a medium is the ratio of velocity of light in air or vacuum to its velocity in a given medium.
`mu = "(velocity of light in vaccum)"/"(velocity of light in given medium)" => mu = c/v`
where `mu=` absolute refractive index.

If `mu_1` and `mu_2` are the refractive indices of two media and `v_1` and `v_2` are the velocity of light in the medium one and medium two respectively.
then `mu = (c)/(v_1)`.....(1)
and `mu_2=(c)/(v_2)`.......(2)
Dividing (2) by (1), we get
`(mu_2)/(mu_1) = (c//v_2)/(c//v_1) => (mu_2)/(mu_1)= (v_1)/(v_2)= text()^1mu_2 = (sin i)/(sin r)`

`text()^1mu_2` is the relative refractive index of medium 2 with respect to medium 1.

`text(Note)`
The focal length of a mirror does not depend on the refractive index of the surrounding medium but that a lens does. If a mirror and a lens are immersed in some liquid, the focal length of the mirror will remain unchanged while that of the lens would change.

Compound Slab

A compound slab is made of two or more than two media (say air-water-glass-air) bounded by parallel faces and is placed in air. A compound slab can be made by placing a glass tray completely filled with water on a glass slab.

When an incident ray AB travelling in air (medium 1) strikes the water surface (medium 2) at B, it is refracted along BC. In figure `2.35 angleABN = i` "(incident angle)" and `angleN'BC = r_1` (angle of refraction).

Now the ray BC acts as an incident ray for the surface separating glass slab and water.
So the incident ray BC after striking this surface at C is refracted along CD in glass (medium 3).
`angleBCN_1 = r_1`, which is equal to angle of refraction, now acts as angle of incidence. `angleDCN'_1 = r_2 =` angle of refraction.

The ray CD acts as an incident ray for the surface separating glass slab and air. So the incident ray CD after striking this surface at D is refracted along DE in air. The rays DE and AB are parallel, so `angleN'_2DE = angleABN = i.` In this case, `angleCDN_2 = r_2` is incident angle and `angleN'_2DE = i,` angle of refraction.

`text(Note)`
In case of a slab placed in a homogeneous medium, the Incident Ray and the Emergent are parallel to each other.
`text(Lateral Shift)`
The perpendicular distance between incident and emergent ray is known as lateral Lateral shift d = BC and t = thickness of slab

In `Delta BOC Sin (i-r) = (BC//OB) = (D//OB)`
`d =OB sin (i-r)`.....(i)
In `Delta OBD cos r = (OD//OB)= (t//OB)`
`=> t/(cosr)`.....(ii)
From (i) and (ii) `d = (t)/(cosr) sin(i-r)`

`text( Critical Angle )`

When a light ray goes to rarer medium from denser medium, then as we increase the angle of incidence, angle of refraction also increases, so the angle of incidence for which the angle of refraction becomes 90° is called critical angle.

`sin C = mu_r/mu_d `
`=> C = sin^-1 (mu_d/mu_d) = sin^-1 (1/(text()_r mu_d ))`

Refraction through Glass Slab

Consider a glass slab ABCD placed in air. A ray PQ incident from air to glass at an angle `i_1` is refracted along QR at an angle `i_2`.
The ray QR is incident at glass- air interface, hence it undergoes refraction along RS at an angle `i_3` (RS is called as an emergent ray)

From Snell's law
`mu_2/mu_1 = (sini_1)/(sin i_2)`.....(i)
where `mu_2` is refractive index of glass and `mu_1` is refractive index of air.

From equation (i)
`mu_1 sin i_i = mu_2 sin i_2`.....(ii)

At glass air interface `(mu_1)/(mu_2) = (sin i_2)/(sin i _3)`
`mu_i sin i_3 = mu_2 sin i_2`

Comparing (ii) and (iii)
`mu_1 sin i_1 = mu_2 sin i_3`
Since `i_1` = 1 (refractive index of air)
`sin i_1 = sin i_3 => i_1 =i_3`
Thus the emergent ray is parallel to the incident ray.

`text(Note)`
For Total Internal Reflection to take place the light should travel from denser to rarer medium and the angle of incidence must be greater than the critical angle for the given pair of media. During total internal reflection of light, the whole (i.e. 100%) incident light energy is reflected back to the parent optically denser medium. Image formed due to Total Internal Reflection is much brighter then reflection from a mirror because total light is reflected without any loss in intensity.

Total Internal Reflection

When light travelling in a denser medium incident on the interface between denser medium ( medium with higher refractive index ) and rarer medium ( medium with lesser refractive index) , then refracted ray bends away from the normal forming a greater angle of refraction than angle of incidence . In this situation, when the angle of incidence exceeds a particular angle, the incident ray comes back into first medium (denser medium). This angle is called critical angle `(theta_C)`

At critical angle, there is no refracted ray found because `angle r = 90^o`

`(sin theta_C)/(sin 90^o) = (mu_text(rarer))/mu_text(denser) = mu_r/mu_d`

`=> sin theta_C = mu_r/mu_d => theta_C = sin ^-1 (mu_r/mu_d)`

where, `mu_r = ` refractive index of rarer medium

and `mu_d =` refractive index of denser medium

Some Phenomena with Total Internal Refraction

`text(Working of Porro Prism)`
A right angled isosceles prism called Porro-Prism can be used in periscope or binocular. The refractive index of glass is 1.5 and the critical angle is equal to `41.8°`. When the ray of light falls on the face of a right angled prism at angle greater than `41.8^o`, it will suffer total internal reflection.

Right angle prisms used to bend the light through `90^o` and `180^o` are shown in figure 2.41 respectively. A right angled prism used to invert the image of an object without changing its size as shown in figure 2.41.

Mirrors can also be used for bending the rays of light. But the intensity of the beam reflected by mirrors is low because even a highly polished mirror does not reflect whole light. On the other hand, in Porro-prism the whole light is reflected. Therefore, there is no loss in intensity of light and hence image is bright.

`text(Sparkling or brilliance of a diamond)`
The refractive index of diamond is `2.5` which gives, the critical angle as `24°`. The faces of the diamond are cur is such a way that whenever light falls on any of the faces, the angle of incidence is greater than the critical angle i.e. `24°`. So when light falls on the diamond, it suffers repeated total internal reflections. The light which finally emerges out from few places in certain directions makes the diamond sparkling.

`text(Shining of air bubble in water)`
The critical angle for water-air interface is `48° 45'`. When light propagating from water (denser medium) is incident on the surface of air bubble (rarer medium) at an angle greater than `48°45`', the total internal reflection takes place. Hence the air bubble in water shines brilliantly.

`text(Mirage)`
Mirage is an optical illusion of water observed generally in deserts when the inverted image of an object (e.g. a tree) is observed along with the object itself on a hot day.

Due to the heating of the surface of earth on a hot day, the density and hence the refractive index of the layers of air close to the surface of earth becomes less. The temperature of the atmosphere decreases with height from the surface of earth, so the value of density and hence the refractive index of the layers of air at higher altitude is more. The rays of light from distant objects (say a tree) reaches the surface of earth with an angle of incidence greater than the critical angle. Hence the incident light suffers total internal reflection as shown in the figure 2.43. When an observer sees the object as well as the image he gets the impression of water pool near the object.

(a) The mirage formed in hot regions is called inferior mirage.
(b) Superior mirage is formed in cold regions. This type of mirage is called looming.

`text(Note)`
Critical angle of a medium depends upon the wavelength of light. Greater the wavelength greater will be the critical angle thus angle of' a medium will be maximum for red color and minimum for violet color. Also, critical angle depends upon the nature. of the pair of media. Greater the relative refractive index, lesser will be the critical angle.

General Terms Used in lenses

`(i) "Optical Centre":` Optical center is a point for a given lens through which any ray passes un-deviated.

`(ii) "Principal Axis":` `C_1` and `C_2` are the Centre's of curvature of the two curved surfaces of the lens. The line passing through `C_1` and `C_2` is called the principal axis. It also passes through the optical centre.

`(iii) "Principal Focus":` A lens has two surfaces and hence two focal points. First focal point is an object point on the principal axis for which image is formed at infinity, while second focal point is an image point on the principal axis for which object lies at infinity.

`(iv) "Focal Length f"`: lt is defined as the distance between Optical Centre of a lens and the point where the parallel beam of light converges or appears to converge after passing through the lens.

The distance between first principal focus and the Optical Centre is called the first focal length.

The distance between the second principal focus and the Optical Centre is called the second focal length.

If the medium on both sides of a lens is same, then the numerical values of the first and second focal length are equal.

`(v) "Aperture:"` In reference to a lens, aperture means the effective diameter. Intensity of image formed by a lens which depends on the light passing through the lens will depend on the square of aperture, i.e.,
`I prop "Aperture"^2`

`text(Rules for Image formation)`
(i) A ray passing through Optical Centre proceeds un-deviated through the lens.
(ii) A ray passing through first focus or directed towards it, after refraction from the lens, becomes parallel to the principal axis.
(iii) A ray passing parallel to the principal axis after refraction through the lens passes or appears to pass through `F_2`

Image Formation for Convex Lens (Convergent Lens)

`"(i) Object is placed at infinity"`
Image: real (at F), inverted, very small in size. `(- m < < 1)`

`"(ii) Object is placed in between" `oo` "and" 2F`
Image: real `(F- 2F),` inverted, small in size (diminished). `(- m < 1)`

`"(iii) Object is placed at"` `2F`
Image: real (at `2F`), inverted, equal (of same size). `(m =- 1)`

`"(iv) Object is placed in between"` `2F` "and" `F`
Image: real `(2F- oo )`, inverted, enlarged. `(- m > 1)`

`"(v) Object is placed at"` `F`
Image: real (at `oo`), inverted, enlarged.` (- m > > 1)`

`"(vi) Object is placed in between"` `F` and `O`
Image: virtual (in front of lens), erected, enlarged. `(m > 1)`

• Where f is the average value of focal length for all the colors. `mu` is the refractive index of the material of the lens with respect to air.

• `R_1` and `R_2` are the radii of curvature of the curved surfaces respectively.

• `R_1` is positive and `R_2` is negative for convex lens.

• `R_1` is negative and `R_2` positive for concave lens.

`text(Lateral Magnification Formula for Lens)`
The lateral or transverse magnification is defined as the ratio of the height of the image and the height of the object. It is represented by m.

`m = (h_i)/(h_0) = v/u`

where `h_i =` height of image,
`h_0 =` height of the object,
`v =` distance of image from the lens, and
`u =` distance of the object from the lens
While using this formula put the correct signs of variables according to the sign conventions.

Power of Lens

• It may be defined as the reciprocal of its focal length in meters
Power of a lens, `P = 1/text{ Focal length of the lens (f) in meters}`
`P = 1/f = (100)/{ftext( in cm)}`

• The power of a lens is inversely proportional to its focal length, hence the lens of short focal length has more power and the lens of long focal length has less power.

• SI unit of power of lens is diopter. It is denoted by D.

`"One diopte":`
• It is the power of a lens whose focal length is 1 meter.

• A convex lens has a positive focal length, hence its power is also positive (+D)

• A concave lens has a negative focal length hence its power is also negative (-D).

`"Power of Combination of Lenses"`
Two thin lens are placed in contact to each other
Power of combination `=> P = P_1+P_2; 1/F =1/(f_1) + 1/(f_2)`
Two thin lens are placed in contact to each other
`1/F = 1/(f_1) + 1/(f_2) - d/(f_1 f_2)`
`P=P_1 + P_2 - d P_1P_2`

`text(Note)`
Color of light is determined· by its frequency and not the wavelength. During refraction of light frequency and color of light do nor change.

Prism

`"Refraction through Prism"`
A homogeneous transparent and refracting medium bounded by two plane surfaces inclined at an angle is called a prism. AB and AC are refracting surfaces. `angleBAC = angleA` is called refracting angle or the angle of prism. (Also called Apex angle).

`delta = "angle of deviation". [ delta = delta_1 +delta+2]`

For refraction of a monochromatic (single wave length) ray of light through a prism;
`delta = (i_2+i_2) - (r_1+r_@) and r_1+r_2=A`

For a Prism, there is one and only one angle of incidence for which the angle of deviation is minimum.
When `delta = delta_m, i_1= i+2 and r+1 = r_2 r` (say) and `r = A//2,` the ray passes symmetrically through the prism, and then

`mu=sin{(A+delta_m)/(2)}/sin(A/2)..........(i)`
`mu=` absolute Refractive Index of prism material.

`'Note"`
When the prism is dipped in a medium then `mu` is Relative Refractive Index of prism material with respect to medium.
If angle of prism A is small, `delta_m` is also small. Equation (i) then becomes

`mu approx [(A+delta_m)/(2)]/(A/2)` or `delta_m =(mu-1)A`

Dispersion of Light

The angular splitting of a ray of white light into a number of components when it is refracted in a medium other than air is called Dispersion of Light.

`"Angle of Dispersion:"`
Angle between the rays of the extreme colors in the refracted (dispersed) light is called angle of dispersion.` theta = delta_V - delta_R` ,. (Figure 2.59 a)

Successive Refraction, (Figure 2.59 b, c)

`"Dispersive Power"` `(omega)` of the medium of the material of prism:
It is defined as the ratio of angular dispersion to the average deviation (for yellow color) when a light beam is transmitted through a thin prism placed in a position so that the mean ray (yellow color ray having the mean wavelength) passes symmetrically through it.

`omega = text(Angular dispersion)/text[Deviation of mean ray (yellow)]`

For small angeld prism `(A leq 10^0)`

`omega = (delta_V- delta_R)/(delta_V) = (mu_V - mu_R)/(mu-1); mu = (mu_V+mu_R)/(2)`

`mu_V, mu_R` and `mu` are R.I of material for violet, red and yellow colors respectively.

`"Note"`
The speed of light in vacuum is the same for all wavelengths, but the speed in a material substance is different for different wavelengths. Therefore, the index of refraction of a material depends on wavelength. In most materials the value of refractive index `mu` decreases with increasing wavelength.

Combination of Prisms

As the dispersive powers of the different materials are different, two or more prisms of different materials can be combined such that the rays of composite light on passing through the combination may suffer either dispersion without deviation or deviation without dispersion.

Figure 2.60 shows two thin prisms place in contact in such a way that the two refracting angles are reversed with respect to each other.

Suppose, the refracting angles of the two prisms are A and A' and their dispersive powers are `omega` and `omega`' respectively. Consider a ray of light for which the refractive indices of the materials of the two prisms are `mu` and `mu`'.

`(i) "Achromatic combination (or deviation without dispersion)"`

If the combination is not to produce a net dispersion, `delta_V- delta_R =0`
`(mu_V-mu_R) A= (mu_V' - mu_R')A'`
or `(mu_Y -1) omegaA= (mu_Y'-1)omega'A'`

`"Net mean deviation"` `delta= [ (mu_V+mu_R)/(2) - 1] A- [ (mu_V'+mu_R')/(2) -1 ]A`

`(ii) "Dispersion without deviation (Direct vision combination")`

If the combination is not to produce a net average deviation in the beam, `delta_Y` should be 0.
`(mu_Y-1)A= (mu_Y -1)A'`

The net angular dispersion produced is `delta_Y - delta_R = ( mu_Y - 1)A (omega- omega') = (mu_V-mu_R) A+(mu_Y'-mu_R')A'`
This combination is used for dispersion without deviation.

Colors

`"Color of Objects in White and Colored Light"`
We know that white light is a mixture of several colors, Light can be of different colors. Let us understand that why different objects appear to have different colors. A rose appears red because when white light falls on rose, it reflects only the red component and absorbs the other components. We conclude that the color of an object depends upon the color of light it reflects.

`"Primary Colors of light"`
Red, green and blue are primary colors of light and they produce white light when added in equal proportions. All colors can be obtained by mixing these three colors in different proportions.

`S"econdary Colors or Composite Colors of light"`
The Colors of light produced by adding any of primary colors are called secondary colors. Cyan, magenta and yellow are secondary colors of light. The method of producing different colors of light adding the primary colors is called color addition.

`"Complementary Colors of light"`
The light of two colors which when added in equal proportions produce white light are called complementary colors of light and the two colors are called complements of each other. For example, yellow and blue light are complementary colors of light because when they are mixed in equal proportions, they produce white light. We can also find the pairs of complimentary colors of light as follows.

(Red + Green) + Blue = Yellow + Blue = White
Red + (Green + Blue) = Red + Cyan =White
(Red + Blue) + Green = Magenta + Green = White

The above result can be diagrammatically represented in the form of a triangle as shown in Figure 2.63. The outer limbs of the Figure show the results of the addition of primary colors red, green and blue. "The complementary color pairs such as red and cyan are opposite to each other.

`"Primary Colors of Pigment"`
Pigments are those substances that give color to an object. The color of a pigment as seen by us depends on what components of light it absorb or subtract from white before reflecting the rest to our eyes. A primary color (cyan, magenta, yellow) of a pigment is due to a primary color of light being subtracted from white light. Mixing CMY (cyan, magenta, yellow) pigment in the correct proportions can produce millions of color. If equal amount of pure CMY pigments are mixed, we should get black pigment. However, printers use black ink in addition to CMY inks to get good results.

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