Dispersion XI

aDispersionWhen a beam of white light is made to fall on one refracting face of the prism, it splits into seven colours i.e. violet, indigo, blue, green, yellow, orange and red from base. The phenomenon of the splitting of white light into its constituent colours is called dispersion of light.

Cause of DispersionWhen a beam of white light is made to fall on one refracting face of the prism, it splits into seven colours i.e. violet, indigo, blue, green, yellow, orange and red from base. The phenomenon of the splitting of white light into its constituent colours is called dispersion of light. 𝜇=a+b

𝜆2+c

𝜆4+...............where a, b and c are constant for the materials of the prism. We know for small angle of deviation, 𝛿 = A(𝜇 – 1). Since wave length of violet light is smaller than that of red light, refractive index 𝜇v>𝜇r. So, on entering the prism, the violet light is refracted through larger angle than red light, so 𝛿v>𝛿r. Thus, when while light is incident on the first face of the prism, each colour is refracted through different angles, i.e angle of refraction is maximum for a red light and least for violet. As a result, dispersion takes place.The SpectrometerA spectrometer is an optical device used to produce the spectrum of light from a source and to measure the refractive index of the material in the form of prism for different colours. A spectrometer essentially consists of three components: collimator, prism table and telescope as shown in figure.

Pure and Impure SpectrumWhen a ray of white light is incident on a prism, the light is split into different constituent colours and deviated through different angles. Such dispersed light is received on a screen or a photographic plate. A collection of dispersed light which gives its wavelength composition is called the spectrum. If the split rays overlap to each other, all colours can’t be seen distinctly on the screen. Such spectrum is called impure spectrum which is shown in figure. However, if the spectrum on the screen in which the split colours don’t overlap to each other, all colours can be seen distinctly and it is called the pure spectrum. A pure spectrum can be obtained by various-ways. A convex lens is introduced into the path of emergent rays which focus the rays on the screen forming a pure spectrum. It may be pointed out that it would be necessary to arrange the prism in minimum deviation otherwise the spectrum will be impure.

Angular Dispersion

When a beam of white light passes through a prism, light of different colours are deviated by different amounts. The mean deviation of light is measured by the deviation of yellow light as this deviation is nearly the average of all deviation. As shown in the figure, the red light is least deviated and the violet light is the most deviated. The angular dispersion is defined as angle between the two extreme colours of light (i.e. violet and red colour)angular dispersion =𝛿v-𝛿r…(i)Since the deviation is small prisma

𝛿=A(𝜇-1), then deviation for violet light, 𝛿v=A(𝜇v-1)

and deviation for red light, 𝛿r=A(𝜇v-1)

where, A is angle of prism, and 𝜇v and 𝜇rare refractive indices of violet and red light respectively. Substituting the values ofa

𝛿v and 𝛿r and equation (i), we get
angular dispersion	=𝛿v-𝛿r
	=A(𝜇v-1)-A(𝜇r-1)
	=A(𝜇v-𝜇r)
or ,𝛿v-𝛿r	=A(𝜇v-𝜇r)

Dispersive power, wDispersive power of a prism is defined as the ratio of angular dispersion to the deviation of mean colour. Dispersive power, a

𝜔	=𝛿v-𝛿r 𝛿
	=A(𝜇v-1)-A(𝜇r-1) A(𝜇-1)
Here, 𝛿	=A(𝜇-1)

is angle of deviation for mean color.a

or, 𝜔=A(𝜇v-𝜇r) A(𝜇-1)

∴𝜔=𝜇v-𝜇r 𝜇-1

Chromatic Aberrations in Lenses The inability of a lens to focus all colours of light at a single point is called chromatic aberration or axial or longitudinal chromatic aberration. It is measured by the difference in focal lengths between red and violet colours.

Chromatic aberration =fr-fvUsing lens maker's formula, for mean colour of light, we havea

1 f	=(𝜇-1)(1 R1+1 R2)
or, 1 R1+1 R2	=1 f(𝜇-1)…(i)

Where f is focal length of mean colour, 𝜇 is refractive index of mean colour, R1 and R2 are radii of curvature of two lens surfaces.For violet colour, we havea

1 f	=(𝜇v-1)(1 R1+1 R2)
or, 1 fv	=(𝜇v-1)1 f(𝜇-1)
or, 1 fv	=uv-1 f(𝜇-1)…(ii)

is refractive index of violet colour. Similarly for red colour, we have1

fr=𝜇v-1

f(𝜇-1)…(iii)Here 𝜇r is refractive index of red colour. Subtracting equation (iii)from equation (ii) we geta

1 fv-1 fr	=𝜇v-1 f(𝜇-1)-𝜇r-1 f(𝜇-1)
or, fr-fv fv⋅fr	=uv-1-𝜇r+1 f(𝜇-1)
or, fr-fv	=(𝜇v-𝜇r)fvfr f(𝜇-1)…(iv)

since focal length of seven colors are in geometric progression, so we can write f=fv.fror, f2=fv.fr so, a

or, fr-fv

=(𝜇v-𝜇r) f2 f(𝜇-1)

fr - fv = (𝜇v-𝜇r)

(𝜇-1). f

fr -fv = 𝜔 f

Achromatic Combination of LensesThe combination of two thin lenses in which their combination is free from chromatic aberration is called the achromatic combination of lenses. Consider two thin lenses l and L′ of dispersive power 𝜔 and 𝜔' respectively placed in contact with each other as shown in the figure. Let 𝜇v,𝜇 and 𝜇r are the refractive indices of L for violet, mean and red colour respectively, and fv,f and fr are the focal lengths of respective colours. Similarly 𝜇'v,𝜇' and 𝜇'r; f′v,f′,f′r are corresponding quantities of L '.For lens L, focal length of mean colour isa

1 f	=(𝜇-1)(1 R1+1 R2)
or, 1 R1+1 R2	=1 f(𝜇-1)

where R1 and R2 are radii of curvature of two lens surfaces. Focal length of lens L for violet colour isa

1 fv	=(𝜇v-1)(1 R1+1 R2)
or 1 fv	=𝜇v-1 f(𝜇-1)…(i)

Similarly, focal length of lens L ' for violet coloura

1 f′v=𝜇′v-1 f′(𝜇′-1)…(ii)

If Fv is the combined focal length of two lenses for violet colour, thena

1 Fv	=1 fv+1 f′v…(iii)
1 Fv	=𝜇v-1 f(𝜇-1)+𝜇′v-1 f'(𝜇'-1)…(iv)

In the same way for red colour,1

Fr=𝜇r-1

f(𝜇-1)+𝜇′r-1

f'(𝜇'-1)…(v)For achromatic combination, we havea

Fr=Fv

or 1 Fv=1 Fr

a

or, 𝜇v-1 f(𝜇-1)+𝜇′v-1 f'(𝜇'-1)	=𝜇r-1 f(𝜇-1)+𝜇′r-1 f'(𝜇'-1)
or, 𝜇v-1 f(𝜇-1)-𝜇r-1 f(𝜇-1)	=𝜇′r-1 f'(𝜇′-1)-𝜇′v-1 f'(𝜇'-1)
or, 𝜇v-1-𝜇r+1 f(𝜇-1)	=𝜇′r-1-𝜇′v+1 f'(𝜇′-1)
or, 𝜇v-𝜇r f(𝜇-1)	=-𝜇′v-𝜇′r f'(𝜇'-1)
where 𝜇v-𝜇r f(𝜇-1)=𝜔 and 𝜇′v-𝜇′r f'(𝜇'-1)=𝜔'
so, 𝜔 f=-𝜔' f'
∴𝜔 f+𝜔' f	=0

This is the condition for achromatic combination of two lenses. Spherical Aberration in a Lens

If a point object is placed on the axis of the large lens, images Ip and Im will be formed by the paraxial and marginal rays respectively. The paraxial rays of light from the image at a longer distance from the lens than the marginal rays. The image is not sharp at any point on the axis. If the screen is placed perpendicular to the axis at AB, the image appears to be a circular patch of diameter AB. The patch of diameter AB is called the circle of least confusion and corresponds to the position of the best image ( with least spherical aberration). The distance ImIp measures the longitudinal spherical aberration and the radius of the circle of least confusion measures the lateral spherical aberration. Removal of Spherical Aberration in lensesFor a single lens, spherical aberration cannot be entirely eliminated. However, it can be reduced by following methods:1. By using stopSpherical aberration can be reduced either by cutting off the paraxial rays or by cutting off the marginal rays.

2. By using plano-convex lensesIf parallel rays of light incident on the plane surface of the plano-convex lens, the spherical aberration will be maximum because incident rays entire deviation at the convex surface. Similarly if parallel rays of light incident on the convex surface, spherical aberration will be minimum.A telescope objective receives parallel rays of light from distant object. To reduce spherical aberration, the convex surface of the plano-convex lens is always towards the distant object. In a microscope objective, the rays fall on it from a very near point object and hence the incident rays are bound to be much oblique than the emergent rays. If the convex surface is towards the object, spherical aberration will be maximum and will be minimum if the plane surface faces the object.

3. By using two lenses separated by a distanceWhen two convergent lenses, separated by a distance are used, the refraction takes place at four surfaces. The spherical aberration will be least when there is an equal deviation at all a surface. It is achieved if,f2-f1=dwhere f2 and f1 are the focal length of the two lenses and d is distance between them. This arrangement is used in eye-pieces.4. By combining suitable concave and convex lensesIt is known that convex lens has positive spherical aberration and concave lens has negative spherical aberration. So by selecting suitable pair of concave and convex lens, spherical aberration can be minimised.Scattering of LightWhen an electromagnetic radiation is an incident on an electric charge at rest, the charges particle accelerates along the direction of the electric field of the incident radiation. Since the electric charge is at rest, it does not experience any force due to magnetic field of the electromagnetic radiations in all the directions, and this process is called scattering. The scattering of light by the matter and he found that the intensity of a particular wavelength of the scattered light depends on its wavelength. Lord Rayleigh found that the intensity of the light corresponding to a wavelength of the scattered light varies inversely as the fourth power of the wavelength.amount of scattering ∝1

𝜆4It is also called Rayleigh law of scattering.The Blue Colour of SkyWhen light from the sun travels through the earth's atmosphere, the different wavelength of light gets scattered from their path through different amounts obeying Rayleigh's law of scattering. Since, the wavelength of the blue colour is approximately half the wavelength of red colour, the scattering of blue light is about 24 times i.e. 16 times more than that of red light. Due to this, blue colour predominates and the sky appears blue.