# Dispersion, Spectra and Optical instrument

### Refraction through a Prism

A prism is a transparent medium enclosed by two plane refracting surfaces. Let EF be the monochromatic ray incident on the face PQ of prism PQR of refracting angle A at angle of incidence $i_1$ .

The ray is refracted along FG, $r_1$; being angle of refraction. The ray FG is incident on the face PR at angle of incidence $r_2$ and is refracted in air along GH. Thus GH is the emergent ray and $i_2$ is the angle of emergence. The angle between incident ray EF and emergent ray GH is called angle of deviation $\delta$ .

Refraction through a Prism

Refraction through a Prism

For a prism if A is the refracting angle of prism then,

$r_1 + r_2 = A \cdots equation \, \, 1$ and

$i_1 + i_2 = A + \delta \cdots equation \, \, 2$

If $\mu$ is the refractive index of material of prism then from Snell’s law:

$\mu = \dfrac{sin i_1}{sin r_1} = \dfrac{sin i_2}{sin r_2} \cdots equation \, \, 3$

If angle of incident is changed, the angle of deviation $\delta$ changes as shown in figure. For a particular angle of incidence the deviation is minimum called angle of deviation $\delta _m$ .

### Minimum Deviation

At minimum deviation the refractive ray with in prism is parallel to the base of prism. So,

$i_1 = i_2 = I ( say ) \\[3mm] r_1 = r_2 = r ( say )$

Then from equation (1) and (2),

$r + r = A$

$r = A / 2 \cdots equation \, \, \, 4a$

$I + I = A + \delta _m \\[3mm] I = \dfrac{A + \delta_m}{2} \cdots equation \, \, \, 4b$

Therefore, the refractive index of material of prism:

$\mu = \dfrac{sin i}{sin r} = \dfrac{sin \dfrac{A + \delta _m}{2}}{sin A / 2} \cdots equation \, \, 7$

For a thin prism,

$\delta _m = ( \mu - 1) A$

Maximum deviation: For maximum deviation produced by a prism either $i_1 \, \, \, or \, \, \, i_2 = 9060$

### Dispersion

The splitting of white light into constituent colors is called the dispersion. When white light falls on a prism, it is broken into constituent colors within the prism. So the emergent light has a number of colored beams, the violet being deviated most and red the least in visible region.

Dispersion

Thus the prism causes deviation as well as dispersion. If $\delta _v \, \, \delta _r \, \, and \, \, delta_y$ are the deviation caused by prism is violet, red and mean yellow rays, then for small angled prism.

$Angular \, \, dispersion \, \, = \delta _v - \delta _r = ( \mu _v - \mu _r ) A$

Dispersive power,

$\omega = \dfrac{Angular \, \, dispersion}{Mean \, \, deviation } = \dfrac{\delta _v - \delta _r}{\delta _y}$

$= \dfrac{ ( \mu _v - \mu _r ) A}{ ( \mu _y - 1 ) A} = \dfrac{ ( \mu _v - \mu _r}{\mu _y - 1}$

### Combination of two prisms

Two small prism may be combined to produce dispersion without deviation or deviation without dispersion.

(i) Dispersion without Deviation: In this arrangement of prism, this mean deviation $( \delta _y )$ caused by one prism is cancelled by the mean deviation$( \delta' _y )$ caused by the other prism.

I.e.

$\delta _y + \delta' _y = 0 \, \, \, or \, \, \, ( \mu _y - 1 ) A + ( \mu'_y - 1 ) A' = 0$

$Or \, \, \, A' = - \dfrac{ \mu _y - 1}{\mu'_y - 1} A$

Thus the angle of prism A’ in second prism has opposite sign as compared to angle A of first prism.

The net dispersion produced,

$= ( \delta _v - \delta _r ) + ( \delta'_v - \delta'_r )$

$= ( \mu _v - \mu _r ) A + ( \mu'_v - \mu'_r ) A'$

(ii) Deviation without Dispersion: In this arrangement of prism, the dispersion ( $\delta _v - \delta _r$ ) caused by one prism is cancelled by dispersion ( $\delta'_v - \delta'_r$ ) produced by the other prism.

I.e.

$( \delta _v - \delta _r ) + |delta'_v - \delta'_r ) = 0$

This gives,

$A' = - \dfrac{\mu _v - \mu _r}{\mu'_v - \mu'_r}A$

The net mean deviation $= \delta _y + \delta'_y = ( \mu _y - 1 )A + ( \mu'_y - 1 ) A'$

### Spectra

The orderly array of colors (i.e. wavelengths) is called the spectrum. The spectra of bodies may be divided into two categories:

i. Emission spectrum: The spectrum of radiations emitted by a luminous body is called the spectrum. For example when light from a luminous electric tungsten bulb, live candle, luminous sodium vapor lamp is allowed to fall on a prism (or grating), the emission spectrum of that source is obtained. It is bright spectrum on dark back-ground.

ii. Absorption spectrum: When white light from a luminous source is first passed through an unexcited transparent substance (gas, liquid or solid) and then transmitted light is allowed to fall on the prism (or grating); the spectrum obtained is the absorption spectrum of that substance. The substance in unexcited state absorbs some radiations emitted from a luminous source. Hence in absorption spectrum certain wavelengths are missing; which appear as black in the spectrum. Hence the absorption spectrum contains dark part / lines.

The emission and absorption spectra may be divided into three subgroups:

#### Emission Spectrum:

(i) Continuous emission spectrum: It consists of continuous wavelengths (colors) in a definite wavelength range. It is obtained by incandescent solid or liquid in bulk state. It is independent of substance but depends on temperature only. The spectrum obtained from incandescent tungsten filament, live candle, burning coal, red hot metals is continuous.

(ii) Line emission spectrum: It consists of distinct bright lines and is produced by excited source in atomic state. For example the spectrum obtained from luminous helium, sodium, argon, mercury vapors is line emission spectrum.

(iii) Band emission spectrum: It consists of distinct bright band and is obtained by excited source in molecular state. For example the spectrum obtained by oxygen, nitrogen, carbon, cynogen etc is band emission spectrum.

#### Absorption Spectrum:

(i) Continuous absorption spectrum: It consists of absence of continuous wavelengths (or colors) in a definite wavelength range and is produced when the substance between the luminous body and the prism in unexcited bulk state. For example if we place red glass in between the luminous body and the prism, then all wavelengths are continuously absorbed except for red part of spectrum.

(ii) Line absorption spectrum : It consists of absence of distinct lines (i.e. dark lines) and is produced when the substance between the luminous body and the prism is in unexcited atomic state. For example if we place a sodium vapor lamp between tungsten filament and the prism, then two dark lines in yellow region of spectrum, appear. This is absorption line spectrum of sodium.

(iii) Band absorption spectrum: It consists of absence of distinct bands (i.e. dark bands) and is produced when the substance between the luminous body and the prism in unexcited molecular state. For example if we place dilute solution of potassium per magnate between luminous tungsten filament and the prism; two dark bands are observed in the spectrum.

### Spectrometer

It is used to observe spectrum and to measure the deviation caused by the prism. A spectrometer essentially consists of three parts:

(i) Collimator

(ii) Prism table and

(iii) Telescope.

Spectrometer

The collimator renders the rays parallel from an extended source which falls on prism and emerges as a parallel beam. The emergent parallel rays are received by the telescope.

### Rainbow

Rainbow is an example of dispersion of sunlight by water drops in the atmosphere. The light suffers refractions and total internal reflections within the drops.

In the formation of primary rainbow: The light rays suffer two refractions and one total internal reflection. The innermost arc is violet and outermost is red.

Primary Rainbow

Secondary rainbow

In the formation of secondary rainbow: The light rays suffer two refractions and two total internal reflections. The innermost arc is red and outermost arc is violet. The secondary rainbow is broader than primary because light is weakened due to total reflections and angular dispersion is greater due to longer path within the drop. The secondary rainbow is situated above the primary because acute angles of deviations are greater than those for primary bow.

### Scattering of Light

When light ray interacts with air molecules, its direction changes. This phenomenon is called scattering. If $\lambda$ is the wavelength of light, then according to Lord Ray Leigh, the intensity of scattered light

$I \propto \dfrac{1}{ \lambda ^4}$

Accordingly blue light is scattered most and the red light is scattered least. Blue color of sky is due to scattering of light by air molecules. The red color of rising and setting sun is also due to scattering of light. At the time of rising and setting of sun, the red light traverses directly, while blue is scattered upward, hence rising and setting sun appears red.

### Magnification

The size of an object depends on the angle subtended by object on eye. This angle is called visual angle. Greater is the visual angle; greater is the size of object. Stars are bigger than sun; but appear smaller because stars are much farther than sun and they subtend smaller angles on eye.

The angle subtended on eye may be increased by using telescopes and microscopes. The telescopes and microscopes form image of object. The image subtends larger angle on eye; hence the object appears big. The magnification produced by optical instrument (telescope/microscope) is defined as the ratio of angle ( $\beta$ ) subtended by image on eye and the angle ( $\alpha$ ) subtended by object on eye.

I.e.

$Angular \, \, \, Magnification \, \, M = \dfrac{ \beta}{ \alpha}$

### Optical Instruments (Telescopes and Microscopes)

(i) Astronomical Telescope: It is used to see magnified images of distant objects. An astronomical telescope essentially consists of two co-axial convex lenses. The lens facing the object has large focal length and large aperture and is called objective, while the lens towards eye has small focal length aperture and is called eye lens.

[Capital letters symbolize for objective and small letters for eye lens i.e. F = focal length of objective, f = focal length of eye lens]

The magnifying power of telescope is:

$= \dfrac{Angle \, \, subtended \, \, by \, \, final \, \, image \, \, at \, \, eye}{Angle \, \, subtended \, \, by \, \, object \, \, on \, \, eye} = \dfrac{ \beta}{ \alpha}$

$= ( m _0 \times m_e ) = - \dfrac{F}{f} ( 1 + \dfrac{f}{v} )$

And length of telescope L = F + u

Where v = distance of final image from eye lenses

U =  Distance of real image A’B’ from eye lenses

Special cases (i) When final image is formed at a distance of distinct vision, then v = D

$\therefore M = \dfrac{F}{f} ( 1 + \dfrac{f}{D} ) \, \, and \, \, L = F + u$

(ii) When final image is formed at infinity, then $v = \infty$

### Simple Microscope

It consists of a convex lens of small focal length f.

If $\beta$ = angle subtended by image on eye

$\alpha$ = angle subtended by object on eye, when object is at a distance of distinct vision (D)

Magnifying power,

$M = \dfrac{ \beta}{ \alpha} = \dfrac{D}{v} ( 1 + \dfrac{v}{f} )$

If final image is at $\infty$ , v = $\infty$ then $M = \dfrac{D}{f}$ .

If final image is at distance of distance vision v = D, $M = 1 + \dfrac{D}{f}$

### Compound Microscope

A compound microscope essentially consists of two co-axial convex lenses of small focal lengths. The lens facing the object is called objective while that toward eye is called eye lens.

Compound Microscope

Therefore, magnifying power of microscope,

$M = \dfrac{ \beta}{ \alpha} ( m_0 \times m_e ) = \dfrac{v_0}{u_o} \dfrac{D}{v_e} ( 1 - \dfrac{v_e}{f_e} )$

(0 symbolizes for objective and e for eye lens)

The length of microscope,

L = length of tube

= separation between lenses = $v_0 + u_o$

### Special Cases

(i) When final image is formed at distance of distinct vision, $v_e = D$

$\therefore M = - \dfrac{v_0}{u_o} ( 1 + \dfrac{D}{f_e} ) \, \, and \, \, L = v_0 + u_e$

$= \dfrac{L}{f_o} ( 1 + \dfrac{D}{f_e} )$

(ii) When final image is formed at infinity, $v_e = \infty$ , then

$M_0 = - \dfrac{v_o}{u_o} \times \dfrac{D}{f_e} \, \, and \, \, L = v_0 + f_e$

$= - \dfrac{L}{f_o} \times \dfrac{D}{f_e}$

### Resolving power

The resolving power of an optical instrument is its ability to form distinct images of two neighboring objects. It is measured by the smallest angular separation between two neighboring objects whose images are just seen distinctly formed by the optical instrument. This smallest distance is called the limit of resolution.

Smaller is the limit of resolution, greater is the resolving power.

The angular limit of resolution of eye is 1′ or $( \dfrac{1}{60} ) ^0$. It means that if two objects are so close that angle subtended by them on eye is less than 1’ or $( \dfrac{1}{60} ) ^0$ , they will not be seen as separate.

Just Resolved

Not Resolved

The best criterion of limit of resolution was given by Lord Rayleigh. He thought that each object forms its diffraction pattern. For just resolution, the central maximum of one falls on the first minimum of the other. When the central maxima of two objects are closer than this objects appear over lapped and are not resolved but if the separation between then is more than this, they are said to be well resolved.

well Resolved

### Telescope

If a is aperture of telescope and $\lambda$ the wavelength, then resolving limit of telescope:

$d \theta \propto \dfrac{ \lambda}{a}$

For spherical aperture, $d \theta = \dfrac{1.22 \lambda}{a}$

Telescope

Microscope: For microscope, $\theta$ is the well resolved semi angle of cone of light rays entering the telescope, then limit of resolution $\propto \dfrac{ \lambda}{ \theta}$ .

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